Topic outline

  • General

  • Introduction

    Changes in schools
    This text book is part of the reform of the school curriculum in Rwanda: that is 
    changes in what is taught in schools and how it is taught. It is hoped this will 
    make what you learn in school useful to you when you leave school, whatever 

    you do then.

    In the past, the main thing in schooling has been to learn knowledge – that is 
    facts and ideas about each subject. Now the main idea is that you should be 
    able to use the knowledge you learn by developing skills or competencies. 
    These skills or competencies include the ability to think for yourself, to be 
    able to communicate with others and explain what you have learnt, and to be 
    creative, that is developing your own ideas, not just following those of the 
    teacher and the text book. You should also be able to find out information and 
    ideas for yourself, rather than just relying on what the teacher or text book 

    tells you.

    Activity-based learning 
    This means that this book has a variety of activities for you to do, as well 
    as information for you to read. These activities present you with material or 
    things to do which will help you to learn things and find out things for yourself. 
    You already have a lot of knowledge and ideas based on the experiences you 
    have had and your life within your own community. Some of the activities, 

    therefore, ask you to think about the knowledge and ideas you already have.

    In using this book, therefore, it is essential that you do all the activities. 
    You will not learn properly unless you do these activities. They are the most 

    important part of the book.

    In some ways this makes learning more of a challenge. It is more difficult to 
    think for yourself than to copy what the teacher tells you. But if you take up 
    this challenge you will become a better person and become more successful 

    in your life.

    Group work
    You can also learn a lot from other people in your class. If you have a problem 
    it can often be solved by discussing it with others. Many of the activities in the 
    book, therefore, involve discussion or other activities in groups or pairs. Your 
    teacher will help to organise these groups and may arrange the classroom so 
    you are always sitting in groups facing each other. You cannot discuss properly 

    unless you are facing each other.

    Research
    One of the objectives of the new curriculum is to help you find things out 
    for yourself. Some activities, therefore, ask you to do research using books 
    in the library, the internet if your school has this, or other sources such as 
    newspapers and magazines. This means you will develop the skills of learning 
    for yourself when you leave school. Your teacher will help you if your school 

    does not have a good library or internet.

    Icons
    To guide you, each activity in the book is marked by a symbol or icon to show 

    you what kind of activity it is. The icons are as follows:

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    PHY

  • Unit 1 :Thin lenses

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    Key unit Competence 

    Explain the properties of lenses and image formation by lenses. 

    My goals

    By the end of this unit, I will be able to:

    * explain physical features of thin lenses
    * state the types of lenses and explain their properties

    * differentiate between lenses and curved mirrors
    * explain the phenomenon of refraction of light by lenses

    * construct the ray diagrams for formation of images by lenses
    * explain the defects of lenses and how they can be corrected

    * describe the daily applications of lenses

    INTRODUCTORY ACTIVITY

    Using a hand lens, candle (object) and a plain paper (screen).
    • Light the candle

    • Place the candle hand lens and plain paper on the same line respectively
     
    • Variate the position of the hand lens and sees the variation of the image 

    on the screen.

    Questions

    1. Discuss on the image formed on the screen (nature). 
    2. Try to draw a ray diagram of your observation and then discuss the 
    properties of the hand lens.
    3. Discuss on the changes of your observation on the screen that are taking 
    place as you variate the positions of the hand lens.

    4. Discuss other types of lenses and brainstorm their different uses.

    Introduction

    The scientific study of light and optical material is involved in the making of 
    spectacles, cameras, projectors and optical instrument.
    The most important optical materials are the various kinds of glass, but 
    many others such as plastics, polaroid, synthetics and natural crystals have 
    increasingly useful application. 
    In this unit we shall consider the behavior of certain component of lenses and 

    its images formation.

    Observe and think

    Look at yourself in a flat mirror and choose one of the following that identifies 
    your observation;

    a) my image is clearly seen without changes.

    b) my image shows some changes.

    What do you think

    a) What do you think about formation of your image by the mirror?

    b) What are the characteristics of this image formed?

    Key concept

    Image formation through a mirror.

    Discovery activity

    a) Look through a plain glass window and observe what happens. Discuss 
    with your neighbor on what is observed.
    b) Look through an open window and discuss with your neighbor about the 
    observations.
    c) Compare the observations in part (a) and (b) above.

    d) Look through the lenses and describe the nature of image formed.

    What I discover
    Just curved mirrors change images, certain transparent medium called lens 

    alter what you see through them.

    A lens is a transparent medium (usually glass) bound by one or two curved 
    surfaces. Different lenses give various natures of images depending on their 

    characteristics.

    Types of lenses and their characteristics

    A lens is a piece of glass with one or two curved surfaces. The lens which is 
    thicker at the centre than at the edges is called a convex lens while the one 
    which is thinner at its centre is known as a concave lens. The curved surface of 
    the lens is called a meniscus. The lens in the human eye is thicker in the centre, 

    and therefore it is a convex lens

    Activity 1

    Required Materials

    • Notebook 

    • 2 convex lenses

    • 2 concave lenses

    • Flashlight or a torch bulb

    • White paper

    Procedure

    1. Look closely at the lenses and answer these questions in your notebook: 

                 a. How are the lenses shaped?

                 b. How are the lenses alike?

                 c. How are the lenses different?

    2. Look through the lenses at the pages of a book, your hands, a hair, 

    and other things. Draw what you see in your notebook and label each 

    picture with the type of lens with which you observed the object. Be 

    sure to answer the following questions: 

               a. How does a concave lens make things look like?

               b. How does a convex lens make things look like?

    3. Lenses bend light in different directions. Shine a flashlight through 

    the lenses onto a piece of white paper and then answer the following 

    questions in your notebook: 

                   a. In what direction do convex lenses bend light?

                   b. In what direction do concave lenses bend light?

    4. Shine the flashlight through different combinations of lenses: two 

    convex lenses, two concave lenses, one concave and one convex lens. 

    Draw pictures of what you see and answer these questions: 

    a. What happens when you use multiple lenses at the same time?

    b. Can you use two different lenses to make things far away appear 

    closer?

    5. If you can, darken the room and place a convex lens between a sunlit 

    window and a white piece of paper. Place the lens close to the paper 

    and then slowly move the lens towards the window. Draw a picture of 

    what you see in your science notebook.

    Do you see that rays change the direction after the lens? How do the emergent 

    rays from each of the lenses behave?

    The light rays from the ray box change the direction after passing through the 
    lens. They are therefore refracted by the lens. Hence, lenses form images of 

    objects by refracting light.

    You can see that the rays from the convex lens are getting closer and closer 
    to a point. The rays are thus converging, and hence a convex lens is called a 
    converging lens. You can also see that the refracted rays from the concave lens 

    are spreading out. This kind of lens is called diverging lens.

    Summary: 

    1. A lens is a transparent medium (usually glass) bounded by one or two 
    curved surfaces. There are two types of lenses; a convex lens also called a 

    converging lens and a concave lens also known as a diverging lens.

    2. A convex lens is the one which is thicker at the centre than at the edges. A 
    concave lens is the one which is thinner at the centre than at the edges.

    The figure below shows three classifications of convex lenses and three 

    classifications of concave lenses.

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    Terms used in lenses

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    Lenses have two lines of symmetry, a vertical line and a horizontal line. The 
    vertical line is called the axis of the lens (already seen in activity 2). The 

    horizontal line is known as the principal axis of the lens.

    Notice that these lines meet at a point. This point is the centre of the lens, 

    called the optical centre of the lens denoted by O.

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    The centre of each sphere is called the centre of curvature of the surface of a 
    lens and the distance from the centre of curvature to the optical centre is the 
    radius of curvature of the surface. Since the convex lens forms part of the 

    spheres, its centre of curvature is real and hence its radius of curvature.

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    This point to which all parallel rays converge after refraction by a convex lens 

    is called the principal focus of the convex lens.

    The rays emerge from the lens when they are spreading out. They are diverged 
    and appear to come from a point. This point from which the rays appear to 

    diverge after refraction by the concave lens is the principal focus of the lens.

    Since rays converge to this point for the case of a convex lens, the principal 
    focus of a convex lens is real. The principal focus of a concave lens is virtual 

    as the rays appear to come from it.

    Repeat the above experiments by changing the lenses so that their right sides 

    become the left.

    Do you see that the same thing happens for each?

    Light can travel into the lens from the left or from the right. It therefore has 

    two principal foci on both sides of the lens.

    The principal focus of a lens is also called the focal point of the lens, and it is 

    denoted by F.

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    Since the image forms where the refracted rays meet and because the rays from 
    the distant tree are parallel, the piece of paper must then be at the principal 
    focus of the lens. This distance from the lens to the image is the focal length 
    of the lens. The focal length of the lens is thus the distance from the centre of 

    the lens to the principal focus. It is always denoted by f.

    The fatter lens has a shorter focal length, implying that the thicker the lens, the 

    shorter the focal length and vice versa. 

    We have already seen that the lens has two principal foci. It means that these 

    principal foci are at equal distances on the opposite sides of the lens.

    Repeat the experiment with the concave lens. 

    What do you notice?

    The image cannot be seen. This is because the concave lens has a virtual 

    principal focus.

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    Refraction of light through lenses

    Lenses can be thought of as a series of tiny refracting prisms, each of which 
    refracts light to produce an image. These prisms are near each other (truncated) 

    and when they act together, they produce a bright image focused at a point.

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    Each section of a lens acts as a tiny glass prism. The refracting angles of these 
    prisms decrease from the edges to its centre. As a result, light is deviated more 

    at the edges than at the centre of the lens.

    The refracting angles of the truncated prisms in a converging lens point to the 

    edges and so bring the parallel rays to a focus.

    The truncated prisms of the diverging lens point the opposite way to those of 
    the converging lens, and so a divergent beam is obtained when parallel rays 
    are refracted by this lens because the deviation of the light is in the opposite 

    direction.

    The middle part of the lens acts like a rectangular piece of glass and a ray 

    incident to it strikes it normally, and thus passes undeviated.

    Properties of images formed by lenses

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    Rays come from all points on the objects. Where these rays meet or appear to 

    meet after refraction by the lens is the position of the image.

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    Notice that an image cannot be seen on the screen irrespective of the position 
    of the object. The nature of the image formed by a convex lens depends on the 

    position of the object along the principal axis of the lens.

    The principal focus of a lens plays an important part in the formation of an 
    image by a lens since parallel rays from the object converge to it, and thus, 
    we consider points F and 2F when describing the nature of the images formed 
    by the lens. These images can be larger or smaller than the object or same 
    size as the object. When an image is larger than the object, we say that it is 
    magnified and when it is smaller, we say that it is diminished. Images which 
    can be formed on the screen are Real images. Because light rays pass through 
    these images, real images can be formed on the screen. All real images formed 

    by the convex lens are inverted.

    When an object is between F and the lens, there is no image formed on the 
    screen. The image formed is not real and is only seen by removing the screen 
    and placing an eye in its position. We say that it is a virtual image. For a
    virtual image, rays appear to come from its position. Unlike for a convex lens 
    where the nature of the image depends on the position of the object, a concave 
    lens gives only an upright, small, virtual image, and is situated between the 

    principal focus and the lens for all positions of the object. 

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    Critical thinking:

    1. Design an experiment to study images formed by convex lenses of 
    various focal lengths. How does the focal length affect the position and 
    size of the image produced?

    2. Suppose you wanted to closely examine the leaf of a plant, which type 

    of a lens would you use? Explain your decision.

    Ray diagrams and properties of images formed by lenses

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    We have already seen that an image is formed where rays from the object 
    meet. Rays come from all points on the objects. However, for simplicity, only 
    a few rays from one point are considered when drawing ray diagrams. Where 
    these rays meet or appear to meet after refraction by the lens is the position of 

    the image.

    To locate the position of the image, two of the following three rays are 
    considered.
    1. A ray parallel to the principal axis which after refraction passes through 

    the principle focus or appears to come from it.

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    The central part of a lens acts as a small parallel –sided block which slightly 
    displaces but does not deviate a ray passing through it and for a thin lens, the 

    displacement can be ignored.

    In ray diagrams, a thin lens is represented by a straight line at which all the 
    refraction is considered to occur. In reality, bending takes place at each surface 

    of the lens.

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    Ray diagrams for a convex lens

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    Nature of image 

    The image is virtual, erect, larger than the object and behind the object.

    Application activity 1.1 

    How is this lens useful when the object is in this position?

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    Nature of image 

    The image is formed at infinity. 

    Application activity 1.2 
    Can you think of how useful is the lens when an object is at its focal point? 

    What is it?

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    Nature of image 

    The image is real, inverted, larger than object (magnified) and beyond 2F.

    Application activity 1.3 

     How is the lens useful when the object is in the above position?

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    Nature of image 

    The image is real, inverted and same size as object.

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    The image is real, inverted, smaller than object (dimensional) and is formed 

    between F and 2F.

    Application activity 1.4 

    What can be a daily application of the lens when an object is in this 

    position?

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    Nature of image 

    The image is real, inverted, smaller than object and is formed at F.

    When an object is between the lens and the principal focus, the rays from the 
    object never converge, instead they appear to come from a position behind 
    the lens. In this case, the lens is used as a simple magnifying glass because it 

    forms an upright and magnified image (Figure 1.14).

    When an object is at the principal focus of the lens, refracted rays emerge from 
    the lens parallel to each other, and the lens is used as a search light torch, and 

    theatre spotlights (Figure 1.15).

    Figure 1.16 shows that when an object is between F and 2F, the lens forms a 

    magnified real image. In this case, a lens is used as a film projector.

    When an object is beyond 2F (Figure 1.18), a lens forms real and small image. 
    The lens is used as a camera because this small, real image can be formed on 

    a piece of film.

    Ray diagrams for a concave lens

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    Accurate construction of ray diagrams 
    Problems for locating the position of the image can be solved by constructing 

    a ray diagram as an accurate scale drawing on a graph paper.

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    Example 
    1. An object is placed 40 cm away from a diverging lens of focal length 
    20cm. If it is 2 cm high, determine graphically the position, size and 
    nature of the image.

    2. Let 1cm on the paper represent 10 cm on the horizontal axis and 1cm 

    on the vertical axis of the actual distance.

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    The image is virtual, erect, 0.7cm tall and is formed at 13cm from the lens on 

    the same side as the object.

    The thin lens formula

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    The sign convention
    From activity 13, we notice that all the distances are measured from the 
    optical centre and in activity 14, we substituted for u, v and f using positive 
    numerical values. It therefore follows that distances of real images and real 

    principal focus are positive.

    In activity 14, then you will notice that the image distance from the lens is 
    negative but equal to the distance determined graphically. This distance is 
    obtained by using a negative numerical value of the focal length. Since a 
    concave lens has a virtual principal focus, and forms virtual images, distances 
    of virtual images and virtual principal foci are negative. Sign convention 
    states that real is positive while virtual is negative. This should be put under 

    consideration when one is using the lens formula to solve problems.

    Derivation of the lens formula 

    Convex lens 
    Consider a point object O on the principal axis, at a distance, u greater than the 

    focal length from the lens. 

    Suppose that a ray from O is incident on the lens at a small height h above the 

    axis and is refracted to form an image I at a distance v from the lens.

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    Concave lens 
    Consider a point object O on the principal axis of the diverging lens at a 

    distance, u, so that its image is formed at a distance, v.

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    Application activity 1.5 

    1. An object is placed 12cm from a converging lens of focal length 18 
    cm. Find the nature and the position of the image.
    2. Find the nature and position of the image of an object placed 15cm 

    from a diverging lens of focal length 15cm.

    Critical thinking exercise

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    Least possible distance between object and real image with converging lens

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    Experiments show that it is not always possible to obtain a real image on a 
    screen although the object and the screen may both be at a greater distance 
    from a converging lens than its focal length. Theory shows that the minimum 
    distance between the object and the screen for an image to be formed is four 
    times the focal length, f. Therefore, the distance between an object and a 

    screen must be equal to or greater than four times the focal length.

    Consider a point object O on the principal axis of a converging lens forming an image I.

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    Since the image formed by the thicker lens is nearer, the thicker lens is more 
    powerful than the thinner lens of the same material. We have already seen that 
    an image of a distant object forms at the focus of the lens and the thicker the 
    lens the shorter the focal length. So the power of the lens depends on its focal 
    length, that is, as the focal length becomes shorter, the power increases. The 

    power of the lens is defined as the reciprocal of its focal length in metres.

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    Application activity 1.6 

    1. Calculate the power of the lens of focal length of 15 cm.

    2. A converging lens has a power of 0.02D, what is its focal length?

    Determination of the focal length of the lens

    Converging lens 

    Rough method

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    The distance from the lens to the screen is the focal length of the lens since 

    rays from a distant object strike the lens when they are parallel.

    Graphical determination of focal length of a convex lens

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    Diverging lens

    Determination of focal length of a diverging lens by Concave 

    mirror method

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    We have already seen that a concave lens forms virtual images of real images 
    which cannot be seen on the screen. So, to determine the focal length of a 
    diverging lens, we need to form a virtual object for the diverging lens so that 
    a real image is produced. This is achieved in the experiment by putting a 
    concave mirror behind the lens so as to reflect back the diverging rays from 

    the lens.

    As you saw in your lower secondary classes, when an object is placed at the 
    principal focus of a concave mirror, the image is formed at the same position 
    with it. Now, since the object and its image are coinciding, it means that they 
    are at the centre of curvature of the mirror; v is negative as I is a virtual image 
    for the lens, and as the object and image are coincident, the rays must be 
    incident normally on the mirror M. Thus, reflected rays from the mirror pass 

    through its centre of curvature which is the position of the virtual image.

    Combination of lenses

    In our next unit, we shall talk about instruments which use lenses to focus 
    objects. Among others, a microscope uses a combination of two lenses to 

    focus objects.

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    Application activity 1.7

    1. An object O is placed 12cm from a thin converging lens P of focal 
    length 10cm and an image is formed on a screen S on the other side of 
    the lens. A thin diverging lens, Q is now placed between the converting 
    lens and S, 50cm from the converging lens. Find the position and 
    nature of the final image if the focal length of the diverging lens is 

    15cm.

    2. An object is placed 6.0cm from a thin converging lens A of focal 
    length 5.0cm. Another thin converging lens B of focal length 15cm is 
    placed co-axially with A and 20cm from it on the side way from the 

    object. Find the position, nature and magnification of the final image.

    Defects of lenses and their corrections

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    Notice that the image has coloured patches. This defect where by an image 

    formed has coloured patches is called chromatic aberration.

    There are two kinds of defects; spherical aberration and chromatic aberration.

    Spherical aberration 

    This arises in lenses of larger aperture when a wide beam of light incident on 
    the lens, not all rays are brought to one focus. As a result, the image of the 
    object becomes distorted. The defect is due to the fact that the focal length 
    of the lens for rays far from the principal axis are less than for rays closer to a 
    property of a spherical surface and as a result, they converge to a point closer 

    to the lens.

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    This defect can be minimised (reduced) by surrounding the lens with an 
    aperture disc having a hole in the middle so that rays fall on the lens at a point 
    closer to its principal axis. However, this reduces the brightness of the image 

    since it reduces the amount of light energy passing through the lens.

    Chromatic aberration 

    This occurs when white light from an object falls on a lens and splits it into its 
    component colours. These colours separate and converge to different foci, and 

    this results into an image with coloured edges.

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    The separation takes place because the material of a glass of a lens has different 
    refractive indices for each colour. The colours travel at different speeds in
    glass: red colour with the greatest and the violet with the least. As a result, 

    violet is deviated most and red is the least deviated

    Thus, a converging lens produces a series of coloured images of an extended 

    white object as shown in the figure above (exaggerated for clarity).

    Chromatic aberration can be minimised by using an achromatic lens called an 
    achromatic doublet. This consists of a converging lens of crown glass combined 

    with a diverging lens of flint glass cemented together with Canada balsam.

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    The flint glass of the diverging lens produces the same dispersion as the 
    crown glass of the converging lens but in the opposite direction and the 
    overall combination is converging. As a result, the achromatic combination 

    converges the white light to one focus.

    Coma aberration

    Another type of aberration is coma, which derives its name from the comet-like 
    appearance of the aberrated image. In general, a bundle of parallel rays 
    passing through the lens at a fixed distance from the centre of the lens are 

    focused to a ring-shaped image in the focal plane, known as a comatic circle. 

    The sum of all these circles results in a V-shaped or comet-like flare. As with 
    spherical aberration, coma can be minimized by choosing the curvature of the 
    two lens surfaces to match the application. Lenses in which both spherical 

    aberration and coma are minimized are called best form lenses.

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    Refraction through prisms

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    In optics, a prism is transparent material like glass or plastic that refracts light. 
    Atleast two of the flat surfaces must have an angle less than 90o between them. 

    The exact angle between the surfaces depends on the application. 

    Terms associated with refraction through prism

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    The position and shape of the third side of the prism does not affect the 

    refraction under consideration and so is shown as an irregular in Fig.

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    Deviation of light by a prism

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    Light can be deviated by reflection and refraction. Since a prism refracts light, 

    it therefore changes its direction. 

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    A prism deviates light on both faces. These deviations do not cancel out as in a 
    parallel sided block where the emergent ray, although displaced, is parallel to 
    the incident ray surface. The total deviation of a ray due to refraction at both 
    faces of the prism is the sum of the deviation of the ray due to refraction at the 

    first surface and its deviation at the second face.

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    Angle of minimum deviation and determination 

    of refractive index n of a material of the prism

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    Application Activity 1.8

    A glass prism of refracting angle 72o
     and index of refraction 1.66 is immersed 
    in a liquid of refractive index 1.33. What is the angle of minimum deviation 

    for a parallel beam of light passing through the prism?

    Deviation of light by a small angle prism 
    Consider a ray incident almost normally in air in a prism of small refracting 
    angle A (less than about 60 or 0.1 radian) so that the angle of incidence i is 

    small.

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    The expression D = A (n – 1) shows that for a given angle A, all rays entering 

    a small angle prism at small angles of incidence suffer the same deviation.

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    Determination of refractive index of a material of a prism

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    The graph is a straight line graph and the gradient represents the mean value 

    which is the refractive index of the material. 

    Dispersion of light by a prism

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    A band of seven colours is formed on the screen. The colours are in order of 
    Red, orange, yellow, green, blue, indigo and violet (ROYGBIV) which are 
    colours of rainbow. This band of colours is called a spectrum. Thus, when 
    a narrow beam of white light falls on a glass prism, it splits into a range 
    of colours and these colours separate to form a spectrum, a process called 
    dispersion. This occurs because white is not a single colour but mixture of all 
    colours of the rainbow. The prism refracts each colour by a different amount 
    because the colours travel at different speeds in the glass and thus the glass 
    has different refractive indices for each colour. The speed of a red colour is 
    greatest and that of a violet colour is the least, and so the refractive index of a 
    material of the prism for red colour is the least and that of the violet colour is 
    the greatest. Now it follows that since the angle of incidence in air is the same 
    for all the colours, red in deviated least by the prism and the violet rays are the 
    most deviated as shown in the figure above (exaggerated for clarity because 
    the colours overlap).

    Applications of total internal reflection of light by a prism
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    Notice that light goes straight through the first surface and when it meets the 
    second surface, it is internally reflected. So, the long side of the prism acts as 
    a mirror and turns light through an angle of 90o
    . Two prisms of the same type 
    as above can be arranged in away and used in a periscope; an instrument used 
    to see the top of an obstruction.
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    Light is tuned through 90oat each prism and it emerges parallel to the incident 
    light. In prism periscopes, light from an object is turned through 90o
     at each prism ands reaches the observer at a different altitude to that of an object. So 

    the image of the object is formed at another altitude but is same size as object.

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    An arrangement of two prisms each turning light through an angle of 180o
     is used in prism binoculars; instruments used to view hidden objects. This will 

    be discussed in the next unit.

    Critical Thinking Exercise

    a) Give reasons why prism rather than plane mirrors are used in 
    periscopes and prism binoculars. 

    b) Explain why diamonds are cut with their sides flat and others slanting. 

    In periscopes and prism binoculars, plane mirrors can be used but prisms are 

    preferred because of the following reasons.

    In the first place, a prism allows light to undergo total internal reflection and 
    thus the images are formed by total internal reflection where as a mirror allows 
    light to both reflect and refract at its surface. So for a prism, all the light 
    (100%) from the object is reflected but for a mirror some light is absorbed 
    (about 95% is reflected) and thus a prism produces a brighter image than a 

    mirror

    The silvering on the mirrors wears off with time but with prism no silvering 

    is needed.

    Some mirrors, for example, thick plate mirrors produce multiple images of 
    one object because of reflections and refractions at the surfaces and inside the 

    glass but a prism produces anyone image.

    Diamonds are cut that way so as to make use of total internal reflection. The 

    multiple reflections inside diamond make it bright.

    END UNIT ASSESSMENT

    1. An object of height h = 7 cm is placed a distance p = 25 cm in front of a 

    thin converging lens of focal length f = 35 cm. 

    a) What is the height, location, and nature of the image? 
    b) Suppose that the object is moved to a new location a distance p 
    = 90 cm in front of the lens. What now is the height, location, 
    and nature of the image?

    2. How far must an object be placed in front of a diverging lens of focal 
    length 45 cm in order to ensure that the size of the image is fifteen 
    times less than the size of the object? How far in front of the lens is the 
    image located?

    3. An object is placed (a) 20 cm, (b) 5 cm from a converging lens of focal 
    length 15 cm. Find the nature, position and magnification of the image 
    in each case.

    4. Find the nature and position of the image of an object placed 10 cm from 
    a diverging lens of focal length 15 cm.

    5. A coin 3 cm in diameter is placed 24 cm from a converging lens whose 
    focal length is 16 cm. Find the location, size, and nature of the image.

    6. An object is placed 30.0 cm in front of a converging lens and then 12.5 
    cm in front of a diverging lens. Both lenses have a focal length of 10.0 
    cm. for both cases, find the image distance and describe the image.

    7. A 4.00 cm tall light bulb is placed a distance of 45.7 cm from a double 
    convex lens having a focal length of 15.2 cm. Determine the image 
    distance and the image size.

    8. A 4.00 cm tall light bulb is placed a distance of 8.30 cm from a double 
    convex lens having a focal length of 15.2 cm. Determine the image 
    distance and the image size.

    9. A 4.00 cm tall light bulb is placed a distance of 35.5 cm from a diverging 
    lens having a focal length of 12.2 cm. Determine the image distance and 
    the image size.

    10.A beam of parallel rays spreads out after passing through a thin diverging 
    lens, as if the rays all came from a point 20.0 cm from the center of the 
    lens. You want to use this lens to form an erect, virtual image that is the 
    height of the object. 
                 a) Where should the object be placed? Where will the image be? 
                 b) Draw a principal ray diagram.

    11.A ray of light incident at an angle i on a prism of angle, A, passes 
    through it symmetrically. Write an expression for the deviation, d, of 
    the ray in terms of i and A. Hence find the value of d, if the angle of the 
    prism is 60o and the refractive index of the glass is 1.48.

    12.A beam of monochromatic light in incident normally on the refracting 
    surface of a 60o glass prism of refractive index 1.62. Calculate the 
    deviation caused by the prism.

    13. a) Define the critical angle of a medium.
           b) One side of a triangular glass prism put in a pool of water of 
    refractive index 4/3 and the other side was left open to air. A ray
    of light from water was incident on the prism at an angle i = 21.7o. The 
    light just grazes as it emerges out of the prism. Given that the 
    refractive index of glass 1.52, determine the refracting angle A of 

    the prism.

    14.A monochromatic light is incident at an angle of 45o
     on a glass prism of refracting angle 70o
     in air. The emergent ray grazes the boundary of 
    the other refracting surface of the prism. Find the refractive index of 

    the material of glass.

    15.A prism of diamond has a refracting angle of 60o
    . A ray of yellow light is incident at an angle of 60o
     on one face. Find the angle of emergence 

    if the refractive index of diamond for yellow light is 2.42.

    16.A ray of light just undergoes total internal reflection at the second face 
    of a prism of refracting angle 60o
     and refractive index 1.5. What is its 

    angle of incidence on the first face?

    17.A sharp image is located 78.0mm behind a 65.0mm-focal-length 
    converging lens. Find the object distance (a) using a ray diagram, (b) 

    by calculation.

    18.What is (a) the position, and (b) the size of the image of a 7.6cm high 
    flower placed 1.00m from a 50.0mm focal length camera lens?

    19.An object is placed 10cm from a lens of 15m of focal length. 

    Determine the image position.

    20.Two converging lenses A and B, with focal lengths fA=20cm and fB = 
    -25cm, are placed 80cm apart, as shown in the figure (1). An object is 
    placed 60cm in front of the first lens as shown in figure (2). Determine 
    (a) the position, and (b) the magnification, of the final image formed by 

    the combination of the two lenses.

    C

    21.Where must a small insect be placed if a 25cm focal length diverging 
    lens is to form a virtual image 20cm in front of the lens?

    22.Where must a luminous object be placed so that a converging lens of 
    focal length 20cm produces an image of size four times bigger than the 

    object (Consider the case of a real image and the case of a virtual)

    23.From a real object AB we want to obtain an inverted image four 
    times bigger than the object. We place a screen 5m away the object. 
    Specify the kind, the position and the focus of the lens to use. Give the 

    graphical and the algebraic.

    24.In cinematography the film is located at 30m from the screen and the 
    image has a magnification of 100. Determine the focal length of the 

    lens used in projection

    25.An object AB of 1cm is placed at 8cm from a converging lens of focal 

    length 12cm. Find its image (Position, nature and the size).

    26.An object of 2cm is placed at 50cm from a diverging lens of focal 

    length 10cm. Determine its image.

    27.An object located 32.0 cm infront of a lens forms an image on a screen 
    8.00 cm behind the lens. (a) Find the focal length of the lens. (b) 

    Determine the magnification. (c) Is the lens converging or diverging?

    28.A movie star catches the reporter shooting pictures of her at home. She 
    claims the reporter was trespassing. To prove her point, she gives as 
    evidence the film she seized. Her 1.72m height is 8.25mm high on the 
    film and the focal length of the camera lens was 210mm. How far away 

    from the subject was the reporter standing?

    29.A lighted candle is placed 33cm in front of a converging lens of focal 
    length f1=15cm, which in turn is 55cm in front of another converging 
    lens of focal length f2=12cm. (a) Draw a ray diagram and estimate 
    the location and the relative size of the final image. (b) Calculate the 

    position and relative size of the final image.

    C
    30.When an object is placed 60cm from a certain converging lens, it forms 
    a real image. When the objet is moved to 40cm from the lens, the 
    image moves 10cm farther from the lens. Find the focal length of this 

    lens.

    

  • Unit 2: Optical instruments

    Key unit Competence 

    Describe and use optical instruments.

    My goals

    By the end of this unit, I will be able to:

    * explain an optical instrument.

    * explain the physical features of a human eye.

    * describe the image formation by the eye.

    * identify the physical features of a simple and compound microscope.

    * explain the applications of simple and compound microscopes.

    * differentiate between simple and compound microscopes.

    * explain the operation of a lens camera and its application.

    * explain the operation of a slide projector and its applications.

    * describe the physical features of a telescope.

    * list different types of telescopes.

    * demonstrate the operation of telescopes.

    * differentiate between telescopes and microscopes.

    * identify the physical features of prism binoculars.

    INTRODUCTORY ACTIVITY

    When a patient goes to hospital having a headache and fever, a doctor may 
    require a blood test for malaria. When a sample of blood is taken, it is not 
    possible to check whether a patient has malaria or not. But a laboratory 
    technician may need to test the blood using some instrument and decide 

    whether the patient has malaria or not.

    Questions:

    (i) Which instrument do you think may be used to test malaria from 
    blood sample?
    (ii) In summary, discuss how that instrument function.

    (iii) What other instrument do you think can be used for such purpose?

    Introduction

    Once the rules for predicting how rays travel through lenses have been 
    discussed; a fantastic range of practical devices began to appear which aided 
    the development of the modern world. The simple magnifying glass became 
    the basis for telescopes, microscopes and spectacles. These devices were 
    modified to improve the projection of images and with the discovery and 
    development of light-sensitive chemicals, gave birth to modern photography 

    and cinematography.

    Definition of an optical instrument

    Activity 2.1 

    In our daily activities and development, we observe different things in 
    environment or in universe. Sometimes, some objects cannot be easily 
    observed using our naked eyes. We need to see these very small things at 
    big distance.
    (i) What do you think we use to observe those distant or very small 
    bodies?
    (ii) Discuss the properties used by those instruments?
    (iii) Name at least four instruments that people use to observe distant or 

    very small objects.

    We use our eyes to see and view different objects. The eye cannot be used 
    to view clearly these objects at night, and some distant objects or hidden 
    objects. Objects which cannot be viewed by the eye can be focused using 
    other instruments. All the instruments used to aid vision are called Optical 

    instruments. 

    Man has always shown interest in observing things in a more detailed manner. 
    In your early secondary, you looked at the uses of mirrors. We have also learnt 
    in unit 1 of this book that lenses are used to focus objects. When the lenses 
    or mirrors or both are arranged in a way, the arrangement can be used to 
    observe objects in a more detailed manner. The arrangement makes what we 
    call a compound optical instrument. The compound instruments include a 

    compound microscope, telescopes, prism binoculars etc.

    THE HUMAN EYE

    The eye is a biological instrument used to see objects at different distances. It 
    uses a convex lens system to form a small, inverted, real image of an object 

    infront of it.

    Structure of the eye

    Activity 2

    (i) In groups of two, look at one another’s eye.
    (ii) Observe critically its external shape.
    (iii) Observe it carefully and note its behaviour as one tries to see some 
    objects in class. 

    Notice that the eye ball is round and fleshy.

    C

    Functions of the parts of the eye

    The cornea: It is made out of a fairly dense, jelly like material which provides 
                              protection for the eye, and seals in the aqueous humour. It also 
                             provides most of the power of the eye (59 Dioptres), having about 

                            46 Dioptres. So it provides most of the bending of light rays.

    The aqueous humour: This is a waterly liquid that helps to keep the cornea in 

                                                   a rounded shape, similar to that of a lens.

    The iris: This controls the amount of light entering the eye. The amount of 
                      light that enters the eye is one of the factors determining how 
                      focused an image is on the retina. The brighter the light the eye is 
                      exposed to, the smaller the iris’ opening will be. The brighter the 
                      light the eye is exposed to, the smaller the iris’ opening will be. 
                      The iris is the coloured part of the eye as seen from the outside. 
                       The iris opening or a gap through which light passes is called a 

                        pupil.

    The lens: This is used to focus an image on the retina. It controls the 
                        bending of light rays by change of its shape, a process called 

                        accommodation, which is done by the ciliary muscles.

    The ciliary muscles: These control the thickness of the lens during focusing. 

                                   By contracting or squeezing the lens, they make it thicker and 

                                   vice versa. Because the power of the lens is directly related to 

                                  its thickness, the ciliary muscles change the power of the lens by 

                                  their movement.

    The retina: This is the light sensitive part of the eye and it is where images 
                     are formed. It contains millions of tiny cells which are sensitive 
                      to light. The cells send signals along the optic nerve to the brain. 
                       So the retina is the screen of the eye and the image is formed by 
                      successive refraction at the surfaces between air, the cornea, the 
                     aqueous humour, the lens and vitreous humour. The retina is 
                       black, which prevents any light rays that hit it from reflections 
                        and thereby changing the image.

    The vitreous humour: This is a jerry like substance that helps the eye to 
                           keep its round shape. It is very close in optical density to the lens 

                             material.

    The yellow spot: This is a small area on the retina where the sharpest image, 

                                that is, the finest detail can be seen.

    The optic nerve: This is the nerve that transmits images received by the retina 
                                 to the brain for interpretation. The part of the eye where the optic 
                                nerve joins the retina is called the blind spot because no images 

                                can be observed at at this point.

    Angular magnification or magnifying power of an optical 

    instrument

    Accommodation of the eye 

    Accommodation of the eye is the ability of the eye to see near and distant 
    objects. The eye is capable of focusing objects at different distances by 
    automatic adjustment of the thickness of the eye lens which is done by the 
    ciliary muscles. To focus a distant object, the eye lens is made thinner, so less 
    powerful, and the rays from the object are brought to focus on the retina by 
    the eye lens. In this case, the ciliary muscles are relaxed and pull the lens. For 
    nearer objects, the eye lens must be made thicker and hence more powerful so 
    that the rays from the near object can be brought to a focus on the retina. In 

    this case, the ciliary muscles tighten and squeeze the lens.

    Near point and far point of the eye

    Activity 5

    (i) Hold a book at an arm’s length and move it closer to find the nearest 
    distance that you can focus the words clearly without straining your 
    eyes.
    (ii) Approximate the distance between your eyes and the book.

    (iii) What does this distance represent?

    The near point of the eye is the nearest point that can be focused by the un 
    aided eye. It is a closest distance that the ‘normal’ human eye can observe 
    clearly; without any strain to the eye. It is called the least distance of distinct 

    vision. The near point of a normal eye is 25 cm.

    Activity 6

    (i) Look at the trees around your school.
    (ii) Now, try to look at objects far from the school.
    (iii) Are you able to focus the distant objects?
    (iv) Measure this distance from the object to your eye.

    (v) Write down your observation in the notebook.

    Notice that you can not be able to measure this distance. The distance from a 
    distant object to the eye is the far point of the eye. The far point of the eye is 

    infinity. The far point is the farthest point that can be focused by the eye.

    The distance of 25 cm from the eye is called distance of most distinct vision or 
    least distance for distinct vision. The range of accommodation of the normal 
    eye is thus from 25 cm to infinity. This range is based upon the average human 
    eye which has an age of 40 years. Young persons have a much wider range but 

    the average 70 year – old has a reduced range.

    People with normal vision can focus both near and distant objects.

    c

    Defects of vision and their correction 

    Activity 7

    (i) Have you seen before some people putting on eye glasses?

    (ii) What do you think these glasses(spectacles) are used for.

    People put on eye glasses for different reasons. Some people wear them in 
    order to read a text, some put them on to see near objects if their eyes cannot 
    be able to do so while others put them on so as to focus distant objects; others 

    wear them for fan like sun goggles

    Short-sightedness (myopia)

    Activity 8

    (i) Hold a book at an arm’s length and move the lens so that the prints 
    are read without the eye getting strained.
    (ii) Now, try to read the words on a chalkboard a distance from the book.

    (iii) Are you able to focus both near and distant objects?

    People with normal vision can focus clearly near and distant objects. Those 
    who only focus near objects are said to be short-sighted, meaning that they 

    see nearer

    Short-sightedness is the defect whereby a person can see near objects clearly 
    but cannot focus distant objects. His far point is nearer than infinity. This is 
    because the eyeball is too long or the lens is too strong so that rays of light 

    from a distance object are focused in front of the retina.

    C

    The rays are focused in front of the retina because the focal length of the eye lens 
    is too short for the length of the eye ball. This defect can be corrected by wearing 
    a concave (diverging) spectacle lens. The rays of light from a distant object are 
    diverged so that they appear to come from a point near, and so they are focused by 

    the eye.

    C

    Rays from object at infinity appear to come from a near point F and converge to 

    the retina.

    Long-sightedness (hypermetropia) 

    This is where a person is able to see distant objects clearly but cannot focus near 
    objects. This is because either his eye ball is too short or the eye lens is too weak 
    (thin) so that rays of light from a close object are focused behind the retina.

    This eye’s near point is further than 25 cm.

    C

    The image of the near object is focused behind the retina because the focal 
    length of the eye lens is too long for the length of the eye ball. This defect can 
    be corrected by wearing a convex lens spectacle. The rays of light from a near 
    object are converged so that the rays appear to come from a point far, and so 

    are focused by the eye.

    C

    Rays from a near object O appear to come from a distant object.

    Presbyopia

    Activity 9

    (i) How many of you still have their grandparents?
    (ii) Have you ever tried to observe how grand parents observe objects?
    (iii) Discuss with your neighbour and write in your notebook results of 

    your discussion. 

    When people grow older, their eye lens become stiff and it becomes hard for 
    the ciliary muscles to adjust it. Such people have a defect called Presbyopia. 
    Presbyopia is the stiffening of the eye lens such that it is less capable of being 
    adjusted by the ciliary muscles. This means that the eye lens becomes less 
    flexible and loses its power (ability) to accommodate for objects at different 
    distances. This defect is corrected by wearing bifocals spectacles whose lenses 
    have a top part for looking at distant objects and a bottom part for close ones. 
    These bifocal spectacles have a diverging top part to correct for distant vision 

    and converging lower part for reading.

    Astigmatism 

    This is the defect that occurs if the curvature of the cornea varies in different 
    directions so that rays in different planes from an object are focused in 
    different positions by the eye and the image is distorted. A person suffering 
    from astigmatism sees one set of lines more sharply than others. This defect is 
    corrected by wearing corrected lenses. These help to bend the incoming rays 

    to correct for irregular refraction.

    Example 
    The far point of the defective eye is 1m. What lens is needed to correct 
    this lens. With this lens, at what distance from the eye is its near point, if 

    the near point is 25cm without the lens?

    C

    Formation of an image by the eye

    Light enters the eye through the transparent cornea, passes through the 
    lens and is focused on the retina. The retina is sensitive to light and sends 
    messages to the brain for interpretation. Although the image is inverted, the 

    brain interpretes it correctly.

    Visual Angle

    Activity 3

    (i) Go outside class and view the trees around.
    (ii) Are the trees of the same height?
    Notice that some trees at a distance, look shorter than the nearby trees 
    when it is not the case? Why do you think it is so?

    Discuss and write down in your notebook about your observation.

    The height of an object depends on the angle of elevation of its top from the 
    eye. The larger the angle, the taller the objects. This angle is called the visual 

    angle.

    The visual angle is the angle subtended at the eye by an object.

    C

    C

    This angle decreases when the distance D increases and increases when the 
    distance D decreases. It also increases when the length AB increases and 

    decreases when AB decreases. We call it visual angle of the object.

    Lead the learners to define the visual angle of an object as the angle between 
    two rays of light from extremities of the object and penetrating into the eye of 

    an observer.

    Activity 4

    C

    Objects that subtend the same angle at the eye appear 

    to be of the same size as viewed by the eye.

    C

    Application activity 2.1

    1. Name the part of the eye 
    a) which controls how much light enters it, 
    b) on which the image is formed, 
    c) which changes the focal length of the crystalline lens. 
    2. A farsighted person has a near point of 100 cm. Reading glasses 
    must have what lens power so that this person can read a 
    newspaper at distance of 25 cm? Assume the lens is very 
    close to the eye.
    3. A nearsighted eye has a near and far point of 12 cm and 17 cm, 
    respectively. 
    a) What lens power is needed for this person to see distant objects 
    clearly, and
    b) What then will be the near point? 

    Assume that the lens is 2.0 cm from the eye (typical for eyeglasses).

    A lens camera

    Activity 10

    (i) Make a paper box and carefully use a pin to make a tiny hole in the 
    centre of the bottom of the paper box.
    (ii) Place a piece of wax paper on the open end of the box. Hold the 
    paper in place with the rubber band.
    (iii) Turn off the room lights. Point the end of the box with a hole in a 
    bright window.
    (iv) Look at the image formed on the wax paper.

    Which kind of image have you seen? Is it upside down or right side up. Is 

    it smaller or larger than the actual object? What type of image is it?

    The image is upside down. The pin hole helps you to see the image of the 

    object. This device is called a pin hole camera.

    Activity 11

    (i) When you were going to register for Rwanda National Examinations, 
    you took some photographs.

    (ii) What device did the person that took your photograph use?

    C

    In our daily lives, we take photographs. We use a lens camera to take these 

    photographs.

    Activity 12

    (i) Enlarge the hole in the pinhole camera above at the front of the box 
    and hold convex lens over the hole.
    (ii) Adjust the position of the lens for either near or far objects to make 
    a sharp image on the screen.
    (iii) Is the image erect or inverted? If the objects are coloured, is the 

    image coloured?

    Notice that the image formed is inverted and coloured if the object is coloured. 
    By placing a lens above the hole, you are making a lens camera from a pin 

    hole camera.

    Formation of images by a lens camera

    Activity 13

    (i) Draw a ray diagram for the formation of an image of an object 
    placed at a point beyond 2F of a thin converging lens. 
    (ii) State the nature and size of the image.

    Is the image bigger or smaller?

    We have already seen that when an object is beyond 2F of a thin converging 

    lens, the image formed is smaller than the object. 

    A camera consists of a light- tight box with a convex (converging) lens at 
    one end and the film at the other end. It uses the convex lens to form a small, 

    inverted, real image on the film at the back.

    c

    C

    The lens focuses light from the object onto a light sensitive film. It is moved 
                        to and fro so that a sharp image is formed on the film. In many 
                       cameras, this happens automatically. In cheaper cameras, the lens 
                       is fixed and the photographer moves forwards and backwards to 

                       focus the object. 

    The diaphragm is a set of sliding plates between the lens and the film. It 
                                       controls the aperture (diameter) of a hole through which light 
                                       passes.
                                       In bright light, a small aperture is used to cut down the amount 
                                      of light reaching the film and in dim light, a large hole is needed.
                                     Very large apertures give blurred images because of aberrations 
                                    so the aperture has to be reduced to obtain clear images.
                                   In many cameras, the amount of light passing through the lens 
                                   can be altered by an aperture control or stop of variable width. 
                                   This size of the hole is marked in f – numbers i.e 1.4, 2, 2.8, 4, 
                                   5.6, 8, 11, 16, 22, 32. The smaller the f-number, the larger the 
                                  aperture. An f-number of 4 means the diameter d of the aperture 
                                  is ¼ the focal length, f of the lens. To widen the aperture, the f 
                                 number should therefore be decreased.
                                 The aperture also controls the depth of field of the lens camera. 
                                 The depth of field is a range of distances in which the camera can 
                                 focus objects simultaneously. This depth of field is increased by 

                                 reducing the aperture.

                                This large depth of field ensures a large depth of focus. The 
                               depth of focus is the tiny distance the film plane can be moved 
                               to or from the lens without defocusing the image. A large depth 
                               of focus means that both near and far objects appear to be in 
                              focus at the same time which is obtained by a small hole in the 

                             diaphragm.

    The shutter controls the exposure time of the film. It opens and closes quickly 

                              to let a small amount of light into the camera.

    The exposure time affects the sharpness of the image. When the exposure 
                              time is short, the image is clear (sharp) but when it is long the 

                             image becomes blurred.

    The film. This is where the image is formed. It is kept in darkness until the 
                          shutter is opened. It is coated with light sensitive chemicals 
                          which are changed by the different shades and colours in the 
                          image. When the film is processed, these changes are fixed and 

                          the developed film is used to print the photograph. 

    Note that a diminished image is always formed on the film and that the image 
    of distant object is formed on a film at distance f from the lens. For near 
    objects, the lens is moved further away from the film (closer to the object) to 
    obtain a clear image. In this case, the film is at a distance greater than f of the 
    lens. Digital cameras are similar to film cameras except that the light does not 
    form an image on photographic film. The image in a digital camera is formed 

    on a charge-coupled device (CCD).

    C

    The CCD is the light-sensitive component of the camera. In a nondigital

    C

    C

    C

    Application activity 2.2

    1. A camera gives a clear image of a distant landscape when the lens 
    is 8 cm from the film. What adjustment is required to get a 
    good photograph of a map placed 72 cm from the lens?

    2. The lens of a certain 35 mm camera (where 35 mm is the width of 
    the film strip) has a focal length of 55 mm and a speed (an 
    f-number) of f/1.8. The correct exposure time for this speed 
    under certain conditions is known to be (1/500) s.

    a) Determine the diameter of the lens.
    b) Calculate the correct exposure time if the f-number is 

    changed to f/4 under the same lighting conditions.

    The slide projector

    Activity 14

    (i) Have you ever seen an instrument called a slide projector?
    (ii) What is that instrument used for?
    (iii) Have you ever watched a cinema where the pictures are seen on the 
    white wall?
    (iv) What device were they using to throw the pictures on the screen 
    (wall or white cloth)?
    (v) Where do you think the pictures came from?

    (vi) Are the images small or large?

    C

    The pictures are thrown on the screen using a slide projector.
    A projector is a device used to throw on a screen a magnified image of a film 

    or a transparent slide. It produces a magnified real image of an object.

    C

    A slide projector is an opto-mechanical device for showing photograhic slides.
    It consists of an illumination system and a projection lens. The illumination 
    system consists of a lamp, concave reflector and the condenser. The illuminant 
    is either a carbon electric arc or a quartz lamp to give a small but very high 

    intensity source of light in order to make the image brighter.

    The lamp is situated at the centre of curvature of the mirror so that the rays are 
    reflected back along their original path. The concave mirror reflects back light 
    which would otherwise be wasted at the back of the projector housing. The 
    condenser consisting of two Plano concave lenses collects light which would 
    otherwise spread out and be wasted, and concentrates it on to the film (slide) 

    so that it is very bright and evenly illuminated.

    The light is then scattered by the film and focused by a convex projection lens 
    on to the film. The projection lens is mounted in the sliding tube so that it is 

    moved to and fro to focus a sharp image on the screen.

    C

    C

    Application activity 2.3

    1. A colour slide has a picture area 2.4 cm x 3.6 cm. Find the focal 
    length of the projection lens which will be needed to throw an image 

    1.2m x 1.8m on a screen 5m from the lens.

    2. A projector projects an image of area 1 m2
     onto a screen placed 5m 
    from the lens. If the area of the slide is 4 cm2
    , calculate;
    (i) The focal length of the projection lens.

    (ii) The distance of the slide from the lens

    Activity 15

    Make a projector on the bench using a ray box lamp, a single convex lens (focal 
    length about 5 cm) for the condenser; a slide; a convex lens (focal length 5cm or 
    10cm) as the projection lens and a sheet of paper for the screen.
    Is the image inverted?

    By how much is it magnified?

    Note that if the film is placed just after the lamp, the object would be poorly 
    illuminated. So to give a bright picture, a condenser is included. The film O 
    is placed between F and 2F of the projection lens so that the image I is real, 
    inverted and magnified. The film is put in the projector while it is upside down 

    so that the picture on the screen is upright.

    Microscope

    Simple Microscope (Magnifying Glass)

    Activity 16

    (i) Hold a hand lens at above the word Rwanda at a distance of about 
    4cm from the word.
    (ii) Move the lens farther away slowly from the word while observing 
    the word through the lens.
    (iii) What changes do you notice after observing?
    (iv) Share ideas with your neighbour and write your observation in 

    your notebook.

    C

    The word Rwanda becomes larger and larger and finally disappears. This word 

    gets larger because of the lens. We say that it is being magnified by the lens.

    Activity 17

    (i) Place your hand on a table and hold a hand lens above it and do the 
    same as in activity 16.

    (ii) What do you notice?

    C

    Notice that the hair (fur) and other small holes on the skin are seen clearly. 
    These parts of the skin are made bigger by the glass lens and this enables one 
    to see them clearly. This lens which magnifies images is called a magnifying 

    glass or a simple microscope.

    A magnifying glass consists of a thin converging lens and It is used to view 
    very small organisms or parts of organisms which cannot be easily seen by the 

    naked eye. 

    Formation of images by a magnifying glass

    Activity 18

    Using the knowledge from thin lenses, draw a ray diagram to show the 
    formation of an image by a magnifying glass.

    State the characteristics of the image formed.

    We have already seen in unit 1 that when an object is between the lens and its 
    principal focus, the image formed is magnified and upright. So, a magnifying 
    glass forms a virtual, upright, magnified image of an object placed between 

    the lens and its principal focus. 

    Activity 19

    Making a simple microscope
    (i) Use a pin or a nail to make a hole about 2 mm in diameter in a piece 
    of a kitchen foil or glass.
    (ii) Carefully let a drop of water fall on the hole so that it stays there 
    and acts as a tiny lens with short focal length.

    (iii) Use it to observe prints on a piece of paper.

    Simple microscope (magnifying glass) in normal adjustment.
    The magnification of a magnifying glass depends upon where it is placed 
    between the user’s eye and the object being viewed and the total distance 

    between them.

    Activity 20

    (i) Carefully place a magnifying glass above some prints on a piece of 
    paper and adjust it until they are seen clearly.
    (ii) Make sure that you don’t feel any strain in the eye while you are 
    observing.

    (iii) What do you think is the position of the image from the eye?

    The image is at the least distance of vision since the eyes are not strained and 

    the magnifying glass is said to be in normal adjustment.

    A microscope is in normal adjustment if the final image is formed at the near 

    point, and it is not in normal adjustment if the final image is at infinity.

    Magnifying power of a simple microscope

    We have already seen that the size of the image depends on the angle subtended 
    by the object on the eye called the visual angle. Thus, the magnifying power 

    depends on the visual angle.

    It is defined as the ratio of the angle subtended by the image to the lens to the 
    angle subtended by the object at the near point to the eye.

    a) Magnifying power of a simple microscope in normal adjustment

    C

    V

    C

    C

    C

    C

    Application activity 2.4

    1. Find the angular magnification produced by a simple microscope of 
    focal length 5cm when used not in normal adjustment.
    2. Explain why angular magnification of a simple microscope is high 
    for a lens of short local length.
    3. Why the image formed by magnifying glass is free from chromatic 

    abberation.

    Activity 21

    In groups of five, discuss why the image formed in a magnifying glass is 
    almost free of chromatic abbreviation.
    When an object is viewed through the magnifying glass, various coloured 
    images corresponding to IR, IV for red and violet rays are formed but each 
    image subtends the same angle at the eye close to the lens and therefore these 
    colours overlap. The overlap of these colours makes a virtual image seen in a 

    magnifying glass free of a chromatic abberation.

    Group Activity 22

    Provided a magnifying glass, go outside and pick different kinds of leaves. 
    Examine, with the use of a magnifying glass, the structures of the leaves. 

    Discuss in details the structural characteristics of each leaf

    Group Activity 23

    You are provided with dirty water in a glass container.
    Use the magnifying glass provided and view some living organisms in it. 

    Record what you see.

    Activity 24

    C

    (i) Observe critically and describe the activity being done in the 
    photograph.

    (ii) State other uses of a magnifying glass.

    Uses of magnifying glass: Magnifying glasses have many different uses. 
    Some people use it for fun activities such as starting fires, or use the lens to 
    help them read. You can start a fire with a magnifying glass when the sun rays 
    are concentrated on the lens. Some retail stores sell reading glasses with the 
    double convex lens. In everyday life, magnifying glasses can be used to do a 
    variety of things. The most common use for magnifying glasses would be how 

    scientists use them, they use magnifying glasses to study tiny germs

    The compound microscope

    Activity 25

    Have you ever heard or seen an instrument called a compound 
    microscope?

    What is it used for?

    C

    The compound microscope is used to detect small objects; is probably the 

    most well-known and well-used research tool in biology. 

    Activity 26

    Observe the above pictures carefully and discuss places where a compound 
    microscope is used in daily life.

    In daily life, microscopes are used in hospitals, in biology laboratories, etc.

    Activity 27

    (i) You are provided with two lenses of focal lengths 5cm and 10cm 

    together with a half meter ruler and some plasticine. 

    C

    (iii) Move the object to and fro until it appears in focus.
    What do you notice about the image? Is it distorted? Is it coloured 

    differently in any way?`

    By arranging the lenses as above, you have actually made a compound 
    microscope. We have already seen how a single lens (magnifying glass) can 
    be used to magnify objects. However, to give a higher magnifying power, two 

    lenses are needed. This arrangement of lenses makes a compound microscope. 

    It produces a magnified inverted image of an object.
    A compound microscope is used to view very small organisms that cannot be 

    seen using our naked eyes for example micro organisms.

    C

    A compound microscope consists of two convex lenses of short focal lengths 
    referred to as the objective and the eye piece. The objective is nearest to the 
    object and the eye piece is nearest to the eye of the observer. The object to be 
    viewed is placed just outside the focal point (at a distance just greater than the 
    focal length) of the objective lens. This objective lens forms a real, magnified, 
    inverted image at a point inside the principal focus of the eye piece. This 
    image acts as an object for the eye piece and it produces a magnified virtual 
    image. So the viewer, looking through the eye piece sees a magnified virtual 

    image of a picture formed by the objective i.e of the real image.

    Image formation in a compound microscope

    V

    C

    Compound microscope in normal adjustment (normal use)

    Activity 28

    You are provided with a bird's feather; observe it critically using a 
    compound microscope and draw it in a fine detail.

    Make sure you observe the features when your eyes are relaxed.

    When the eyes are relaxed, the image is at the near point and the compound 
    microscope is said to be in normal adjustment. The compound microscope is 
    in normal adjustment when the final image is formed at the near point (least 

    distance of distinct vision), D of the eye.

    Angular magnification (magnifying power) of a compound 

    microscope 

    The magnifying power of a compound microscope is the ratio of the angle 
    subtended by the final image to the eye when the microscope is used to the 

    angle subtended by the object the unaided eye.

    Angular magnification of a compound microscope in normal use 

    We have already seen that when a microscope is in normal use, the image I2

    is formed at the least distance of distinct vision, D from the eye. Thus v = D.

    C

    Consider an object of height h at a given distance slightly greater than the 

    focal length of the objective lens. 

    C

    C

    C

    V

    C

    Example 

    A compound microscope has an eye piece of focal length 2.50cm and an 
    objective of focal length 1.60cm. If the distance between the objective 
    and eye piece is 22.1cm, calculate the magnifying power produced when 

    the final image is at infinity.

    C

    Activity 29 
    Viewing specimens

    The purpose of this exercise is to view micro organisms found in pond 
    water while learning to operate a microscope.

    Equipment
    * Microscope
    * Jar of pond water
    * Slide
    * Coverslip

    * Dropper

    Procedure

    1. Collect a jar of pond water containing micro organisms. To ensure 
    that you capture the largest number of micro organisms, do not 
    simply scoop a jar of water from the centre of a pond. Instead, fill the 
    jar partway with pond water and then squeeze water into the container 

    from water plants or pond scum.

    C

    3. Set up the microscope.
    a) Remove the dust cover from the microscope.
    b) Plug in the microscope.

    c) Turn on the microscope’s light source.

    4. View the specimen with the low-power objective. Move the slide 
    around on the stage using your fingers or the control knobs until you 

    find a micro organism.

    5. View the micro organism with the high-power objective.

    6. Sketch a picture of the micro organism.

    7. Repeat steps 4, 5, and 6 until you have sketched atleast five different 

    micro organisms.

    8. Turn off the microscope.

    a) Carefully, lower the objective to its lowest position by turning 

    the coarse’ adjustment knob.

    b) Turn off the light source.

    c) Remove your slide. Clean the slide and cover slip with water.

    d) Unplug the microscope and store it under a dust cloth.

    Application activity 2.5

    A compound microscope consists of a 10× eyepiece and 50× objective 
    17.0 cm apart. Determine (a) the overall magnification, (b) the focal length 
    of each lens, (c) the position of the object when the final image is in focus 

    with the eye relaxed. Assume a normal eye, so N = 25 cm.

    Telescopes

    Activity 30

    You have heard in your early secondary that there are some heavenly and 
    distant earthly bodies that cannot be seen by our naked eyes. How did the 

    people know that there exist such bodies?

    Which instrument do you think is used to see these bodies and to observe 

    what takes place on these bodies?

    Why do you think it is difficult to see distant objects using our eyes?

    Telescopes are instruments used to view distant objects such as stars and other 
    heavenly bodies. Distant objects are difficult to see because light from them 
    has spread out by the time it reaches the eyes, and since our eyes are too small 

    to gather much light. 

    There are two kinds of telescopes; refracting telescopes and reflecting 

    telescopes.

    Refracting telescopes

    Activity 31

    (i) Hold a convex lens of focal length 5cm close to your eye.
    (ii) Hold another lens of focal length 20cm at an arm’s length. 
    (iii) Use the lens combination to view distant objects.
    (iv) Adjust the distance of the farther lens until the image is clear (take 
    care not to drop the lenses).

    What type of image do you see?

    The above lens combination is a refracting telescope. It is called a refracting 
    telescope because it forms an image of the object by refracting light. Therefore, 
    Refracting telescopes use lenses and they form images by refraction of light. 

    Below are different kinds of refracting telescopes.

    Astronomical telescope 

    The telescope made in the above activity is called an astronomical telescope. 
    It consists of two convex lenses, the objective lens of long focal length and an 

    eye piece lens of short focal length.

    An astronomical telescope in normal adjustment

    Activity 32

    Using a telescope made in activity (30) above, view a distant object by 
    moving the lenses so that the eyes are relaxed. 

    What do you think is the position of the image?

    When the eyes are relaxed, the image is at infinity and the telescope is in 
    normal adjustment. Therefore, an astronomical telescope is in normal 

    adjustment when the final image is formed at infinity.

    C

    The rays of light coming from a distant object form a parallel beam of 
    light. This parallel beam is focused by the objective lens and it forms a real, 
    diminished image at its principal focus Fo
    . The eye piece is adjusted so that 
    this image lies in its focal plane. This image acts as the object for the eye 

    piece and the eye piece produces the image at infinity. 

    Note that in normal adjustment, the eye is relaxed or un accommodated when 

    viewing the image. In this case, the eye has minimum strain.

    Magnifying power or angular magnification of an astronomical 

    telescope 

    The magnifying power of a telescope is the ratio of the angle subtended by 
    the image to the eye when the telescope is used to the angle subtended at the 
    unaided eye by the object. Since the telescope length is very small compared 
    with the distance of the object from either lens, the angle subtended at the 
    unaided eye by the object is the same as that subtended at the objective by the 

    object.

    C

    C

     Activity 33

    Discuss and give a summary of differences between a compound 

    microscope and an astronomical telescope.

    The table below shows the differences between a compound microscope and 

    an astronomical telescope.

    C

    Example 

    An astronomical telescope has an objective lens of focal length 120 
    cm and an eye piece of focal length 5 cm. If the telescope is in normal 

    adjustment, what is;

    (i) The angular magnification (magnifying power)

    (ii) The separation of the two lenses?

    C

    Application activity 2.6

    An astronomical telescope is used to view a scale that is 300 cm from the 
    objective lens. The objective lens has a focal length of 20cm and the eye 
    piece has a focal length of 2 cm. Calculate the angular magnification when 

    the telescope is adjusted for minimum eye strain.

    An astronomical telecope with the final image at the near point 
    In this case, the image is seen in detail but the telecope is not in normal 

    adjustment (use) because the eyes are strained.

    C

    C

    C

    The eye ring 

    The eye ring is the best position to place the eye in order to be able to view 
    as much of the final image as possible. The best position for an observer to 
    place the eye when using a microscope is where it gathers most light from that 
    passing through the objective. In this case, the image is brightest and the field 
    of view is greatest. In case of the telescope, all the light from a distant object 
    must pass through the eye ring after leaving the telescope. So by placing 
    the eye at the eye ring, the viewer is able to see the final image as much as 

    possible. 

    Terrestrial telescope 

    An astronomical telescope produces an inverted image, so it is not suitable 
    for viewing objects on the earth. It is suitable for viewing stars and other 
    heavenly bodies. A terrestrial telescope provides an erect image and this 

    makes it suitable to view objectives on the earth.

    C

    Activity 34

    Discuss the advantages and disadvantages of a terrestrial telescope over 

    an astronomical telescope.

    C

    Galilean Telescope 

     Activity 35

    (i) Hold a concave lens of focal length 5cm close to your eye.
    (ii) Hold another convex lens of focal length 20cm at an arm’s length. 
    (iii) Use the lens combination to view distant objects.

    (iv) What is the nature of the image?

    The above lens combination is a Galilean telescope. A Galilean telescope 
    consists of an objective lens which is a convex lens of long focal length and an 
    eye piece which is a concave lens of short focal length. It forms erect images 

    both in normal and not in normal adjustment.

    C

    C

    C

    C

    Activity 36

    Discuss the advantages and disadvantages of a Galilean telescope over an 

    astronomical telescope and write them in your notebook.

    C

    Reflecting telescopes

    Activity 37

    Take the case of a TV satellite dish in the neighborhood. Discuss and 

    explain the functioning and principle of a satellite dish

    Reflecting telescopes consist of a large concave mirror of long focal length as 
    their objective. There are three kinds of reflector telescopes, all named after 

    their inventors.

    C

    C

    Cassegrain reflecting telescope

    C

    This is the type used in most observatories It consists of a concave mirror 
    which acts as an objective, a small convex mirror and the eye piece lens. Light 
    from a distant object is reflected by the concave mirror to the convex mirror 
    which reflects it back to the centre of the concave mirror where there is a small 
    hole to allow the light through. So the convex mirror forms the final image 

    (real) at the pole of the objective.

    Coude Reflector Telescope

    This is a combination of Newtonian and cassegrain reflector telescopes.

    C

    The plane and convex mirrors used in reflecting telescopes are used to bring 
    the light to a more convenient focus where the image can be photographed and 

    magnified several times by the eye piece for observation.

    Activity 38

    Discuss and explain the advantages of reflecting telescopes over refracting 

    telescopes.

    The reflecting telescopes are free from chromatic aberration since no refraction 
    occurs. The image formed is brighter than in refracting telescopes where there 

    is some loss of light during refraction at the lens surfaces.

    Spherical aberration can be eliminated by using a parabolic mirror instead of a 
    spherical mirror as an objective. They have a power because of higher ability 
    to distinguish two closely related objects because of the large diameter of 
    the parabolic mirror. We say that they have a high resolving power. They are 

    easier to construct since only one surface requires to be grounded.

    Critical Thinking Exercise

    What is meant by the resolving power of an optical instrument? Explain its 
    usefulness.
    Explain why astronomers use reflecting telescopes rather than refracting 

    telescopes?

    Prism binoculars

    Activity 39

    Have you ever asked yourself how tourists and scientists are able to see 
    distant animals and birds in a forest or any hidden places?

    Discuss with your neighbour and write in your notebook the observation.

    Tourists and scientists use prism binoculars to view wild animals and birds in 

    hidden places such as caves and forests. 

    These consist of a pair of refracting astronomical telescopes with two totally 
    reflecting prisms between each objective and eyepiece. The prisms use total 
    internal reflection to invert rays of light so that the final image is seen the 
    correct way. These prisms reflect up and down the light and by doing so, they 

    shorten the length of the instrument.

    C

    Prism A causes lateral inversion and prism B inverts vertically so that the 
    final image is the same way round and same way up as the object. Each prism 
    reflects light through 180o
    . This makes the effective length of each telescope 
    three times shorter than the distance between the objective and the eye piece. 

    So good magnifying power is obtained with compactness.

    END UNIT ASSESSMENT

    1. A certain nearsighted person cannot see distinctly objects beyond 80 
    cm from the eye. What is the power in diopters of the spectacle lenses 

    that that will enable him to see distant objects clearly?

    2. Explain the difference between the terms magnifying power and 
    magnification, as used about optical systems. Illustrate this, by 
    calculating both, in the case of an object placed 5.0 cm from a simple 
    magnifying glass of focal length 6.0 cm, assuming that the minimum 

    distance of distinct vision for the observer is 25 cm.

    3. An eyepiece is made of two positive thin lenses, each of focal length f 
    = 20 mm, separated by a distance of 16 mm. 
    (a) Where must a small object viewed by the eyepiece be placed so 
    that the eye receives parallel light from the eyepiece? 
    (b)Does the eye see an erect image relative to the object? Is it 
    magnified? 

    (c) Use a ray-trace diagram to answer these questions by inspection.

    4. A common telephoto lens for a 35 mm camera has a focal length of 
    200 mm; its range from to (a)What is the corresponding range of 
    aperture diameters? (b)What is the corresponding range of image 

    intensities on the film?

    5. What is the maximum stop rating of a camera lens having a focal 
    length of and a diameter of ? If the correct exposure at , what 

    exposure is needed when the diaphragm setting is changed to ? 

  • Topic 4

  • Topic 5

  • Unit 3 : Moments and Equilibrium of Bodies

    Key Unit Competence 

    Analyse the principle of moments and equilibrium of bodies. .

    My goals

    By the end of this unit, I should be able to:

    * Explain the principle of moments and apply it to equilibrium of a 
    body.
    * Come out with the effects of forces when applied onto a body.

    * Know the effects of forces.

    INTRODUCTORY ACTIVITY

    When a patient goes to hospital having a headache and fever, a doctor may 
    require a blood test for malaria. When a sample of blood is taken, it is not 
    possible to check whether a patient has malaria or not. But a laboratory 
    technician may need to test the blood using some instrument and decide 

    whether the patient has malaria or not.

    Questions:
    (i) Which instrument do you think may be used to test malaria from 
    blood sample?
    (ii) In summary, discuss how that instrument function.

    (iii) What other instrument do you think can be used for such purpose?

    Introduction

    In here, we shall majorly concetrate on the turning effect of force. As you 
    know, it is very hard to close a door when you apply force near its turning 
    point. That’s why door handles are always put at the end of the door so that 
    the distance from the turning point to where force is applied increases. This 
    increases the turning effect of the force applied. Which is the effect of forces 

    on bodies one of our interest in this unit.

    Scalar and vector quantities

    Activity 1

    Try to stand bricks in a line behind one another. Push one brick.
    (i) What happens to other bricks?

    (ii) What if in the process one brick stops, what would happen?

    In daily life, we normally pull the objects from one place to another. When 
    pulling a goat that is to be tethered, obviously it will take the direction of the 
    pull. We can call this a force. This is a quantity that changes body’s state of 

    rest or uniform motion.

    You noticed that after pushing your friend he/she changed position and 
    direction. Hence, a force has both magnitude and direction. This quantity can 
    be termed as a vector quantity. This is a quantity with both magnitude and 

    direction.

    Activity 2

    (i) Using the above example, discuss in groups or as a class other 
    vector Quantities.
    (ii) Analyse the effects of these physical quantities.
    (iii) In daily life, how are these quantities utilised? 

    (iv) Ask your friend what time is it? 

    You will realise that he/she will tell the exact time not even indicating direction. 

    Such a quantity is termed to be a scalar quantity.

    A scalar quantity is a physical quantity that is defined by only magnitude 

    (size).

    Other examples of scalar quantities are volume, mass, speed, and time 
    intervals. The rules of ordinary arithmetic are used to manipulate scalar 

    quantities.

    Application activity 3.1

    Of the following physical quantities, group them in different 
    sets of scalar and vector quantities: mass, energy, power, weight, 
    acceleration, velocity, momentum, time, impulse, magnetic flux density, 

    pressure, displacement.

    Force as vector

    Activity 3

    1. As an individual or a group push the desk.
    2. What happens to it?
    3. What causes the change in position?
    a) Let as a class move to:
    (i) Football pitch.
    (ii)Net ball pitch.
    (iii) Basket ball play ground.
    Try to kick a ball. What happens to it? What causes it to change its 
    position?

    Note what you observe.

    Also, as you sit reading this book, you eventually feel tired. This is because of 

    gravitational force acting on your body and yet you remain stationary.

    From the above examples, we can define the “quantity force”. We have to 
    know the direction and the magnitude. For that matter, we conclude that the 
    force is vector quantity. We can think of force as that which causes an object 

    to accelerate.

    What happens when several forces act simultaneously on an object? In this case, 
    the object accelerates only if the net force acting on it is not equal to zero. The 
    net force acting on an object is defined as the vector sum of all forces acting on 
    the object. (We sometimes refer to the net force as the total force, the resultant 

    force, or the unbalanced force.)

    Application activity 3.2

    1. ............................ is an example of a scalar quantity
    a) Velocity.
    b) Force.
    c) Volume.

    d) Acceleration.

    2. ............................ is an example of a vector quantity
    a) Mass.
    b) Force.
    c) Volume.

    d) Density.

    3. A scalar quantity:
    a) always has mass.
    b) is a quantity that is completely specified by its magnitude.
    c) shows direction.

    d) does not have units.

    4. A vector quantity
    a) can be a dimensionless quantity.
    b) specifies only magnitude.
    c) specifies only direction.

    d) specifies both a magnitude and a direction.

    5. A boy pushes against the wall with 50 kilogrammes of force. The wall 
    does not move. The resultant force is:
    a) -50 kilogrammes.
    b) 100 kilogrammes.
    c) 0 kilogrammes.

    d) -75 kilogrammes.

    6. A man walks 3 miles north then turns right and walks 4 miles east. The 
    resultant displacement is:
    a) 1 kilometre SW
    b) 7 kilometres NE
    c) 5 kilometres NE

    d) 5 kilometres E

    7. A plane flying 500km/hr due north has a tail wind of 45 mi/hr the 
    resultant velocity is:
    a) 545 kilometres/hour due south.
    b) 455 kilometres/hour north.
    c) 545 kilometres/hour due north.

    d) 455 kilometres/hour due south.

    8. The difference between speed and velocity is:
    a) Speed has no units.
    b) Speed shows only magnitude, while velocity represents both 
    magnitude (strength) and direction.
    c) They use different units to represent their magnitude.

    d) Velocity has a higher magnitude.

    9. The resultant magnitude of two vectors
    a) Is always positive.
    b) Can never be zero.
    c) Can never be negative.

    d) Is usually zero.

    10.Which of the following is not true.
    a) Velocity can be negative.
    b) Velocity is a vector.
    c) Speed is a scalar.

    d) Speed can be negative.

    Table summarising Scalar and vector Quantities

    V

    Turning effect of force

    Moment of a force about a point
    Every time we open a door, turn on a tap or tighten up a nut with a spanner, 
    we exert a turning force. The combined effect of the force and distance which 
    determines the magnitude of the turning force is called the moment of the 

    force or torque and is defined as follows: 

    “The moment (turning effect) of a force about a point is the force multiplied 
    by the perpendicular distance from the place where the force is applied to that 

    point.” Fig. a

    C

    Or Moment is force times lever arm where is the lever arm, and the 
    perpendicular symbol egg reminds us that we must use the distance from the 
    axis of rotation that is perpendicular to the line of action of the force (Fig.3.1 

    a). The SI unit for moment is N m.

    A lever arm or moment arm is the perpendicular distance from the axis of 

    rotation to a line drawn along the direction of the force.

    C

    An equivalent way of determining the torque associated with a force is to 
    resolve the force into components parallel and perpendicular to the line that 
    connects the axis to the point of application of the force, as shown in Fig. 3.1b. 
    The component exerts no torque since it is directed at the rotation axis (its 
    lever arm is zero). Hence the torque will be equal to times the distance r from 

    the axis to the point of application of the force:

    C

    Activity 4

    (i) Suspend a meter rule at its middle point, either by passing a string 
    through a hole or a knife edge. If necessary stick plasticine one one 
    and to make it balance exactly.

    (ii) Tie loops of thread to several 

    (iii) Hang a 0.5 N weight A (the load) one the left hand side of the ruler 
    on the 34 cm mark (16 cm ) from the fulcrum)

    (iv) Place another weight 0.2 N (the effort) on the other side and move 

    it until the ruler balances.

    C

    (v) Note the distances of the weights from the fulcrum, i.e from the 
    midpoint.

    (vi) Record the results in a suitable as shown in the table.

    (vii) Reapeat several times with (a) the same weights in different 

    position, and (b) different weights.

    C

    The table suggests that when the turning effects of forces acting 
    on an object are balanced the sum of clock wise moments is the 
    same as the sum of anticlockwise moments. This is the principle of 

    moment

    Principle of moment of force

    When a body is in equilibrium (balanced), the sum of the anticlockwise 
    moments about any point is equal to the sum of the clockwise moments about 

    the same point.

    C

    Example 

    In kinesiology (the study of human motion), it is often useful to know the 
    location of the center of mass of a person. This can be determined with 
    the arrangement shown fig A. A plank of weight 40 N is placed on two 
    scales separated by 2.0 m. A person lies on the plank and the left scale 
    reads 3214 N and the right scale reads 216 N as shown in fig B. What is 

    the distance from the left scale to the person’s center of mass?

    C

    C

    C

    A torque is a quantity that measures the ability of a force to rotate an 
    object around some axis. Net torque produces rotation. A torque is positive 
    or negative; depending on the direction the force tends to rotate an object. 

    Torques that produce counterclockwise rotation are defined to be positive.

    Example 

    A basketball is being pushed by two basketball players during tip-off. 
    Assuming each force acts perpendicular to the axis of rotation through 

    the centre of ball the ball; find the net torque acting on the ball.

    C

    C

    Torque should not be confused with work
    Torque (τ = Fd sin θ ) and work (W = Fd cos θ ) can both be expressed in 
    units of N m, so be careful to distinguish torque and work. The components of 
    a force that produces work is parallel to a distance (the displacement), while 
    the component of force that produces torque is perpendicular to a distance (the 

    lever arm). 

    Torque should not be confused with force. Forces can cause a change in 
    linear motion, as described by Newton’s second law. Forces can also cause a 
    change in rotational motion, but the effectiveness of the forces in causing this 
    change depends on both the forces and the moment arms of the forces, in the 

    combination that we call torque. 

    If two or more forces are acting on a rigid object, as shown in Fig.3.3, each 
    tends to produce rotation about the pivot at O. We use the convention that the 
    sign of the torque resulting from a force is positive if the turning tendency 
    of the force is counterclockwise and is negative if the turning tendency is 

    clockwise.

    C

    Equilibrium of a body

    Conditions for equilibrium

    Objects in daily life have at least one force acting on them (gravity). If they are 
    at rest, then there must be other forces acting on them as well so that the net 
    force is zero. A book at rest on a table, for example, has two forces acting on 
    it, the downward force of gravity and the normal force the table exerts upward 

    on it (Fig. 3.4).

    C

    Because the book is at rest, Newton’s second law tells us that the net force 
    on it is zero. Thus the upward force exerted by the table on the book must be 
    equal in magnitude to the force of gravity acting downward on the book. Such 
    an object is said to be in equilibrium (Latin for “equal forces” or “balance”) 
    under the action of these two forces. This is often called the first condition 

    for equilibrium (Fig.3.5a).

    Force not only push or pull but have a turning-effect or moment about an 
    axis.     In cases of equilibrium the moments have also to be considered. If forces 
    act at different points on an extended body an additional requirement must 
    be satisfied to ensure that the body has no tendency to rotate: The sum of the 
    torques about any point must be zero (Fig.3.5.b). This is called the second 

    condition for equilibrium.

    C

    C

    A rigid body is in mechanical equilibrium when the sum of all forces on all 
    particles of the system is zero (i.e. when all the particles of the system are 
    at rest or that its center of mass moves with constant velocity relative to the 
    observer and the total force on each particle is permanently zero)., and also 
    the sum of all torques on all particles of the system is zero so that its state of 
    rotational motion remains constant. The bodies are rigid if they do not deform 

    under the action of applied forces.

    We will apply the first and second conditions for equilibrium to situations 
    in which a rigid body is at rest (no translation or rotation). Such a body is 
    said to be in static equilibrium (Fig. 3.6). But the same conditions apply to 
    a rigid body in uniform translational motion (without rotation), such as an 
    airplane in flight with constant speed, direction, and altitude. Such a body is in 

    equilibrium but is not static.

    C

    The above conditions of equilibrium are also used to determine the resultant 
    of non-parallel, non-concurrent systems of forces i.e. all of the lines of 
    action of the forces in this system do not meet at one point. The parallel force 
    system was a special case of this type. Since all of these forces are not entirely 
    parallel, the position of the resultant can be established using the graphical or 

    algebraic methods of resolving co-planar forces 

    There are a number of ways in which one could resolve the force system that 
    is shown. One graphical method would be to resolve a pair of forces using the 
    parallelogram or triangle method into a resultant. The resultant would then be 
    combined with one of the remaining forces and a new resultant determined, 
    and so on until all of the forces had been accounted for. This could prove to 
    be very complex if there are a great number of forces. The algebraic solution 
    to this system would potentially be simpler if the forces that are applied to the 

    system are easy to break into components.

    ■ Addition of Forces in a Plane (Stevinus law)
    If all the forces acting on a body act in a plane, they are called coplanar forces. 
    If they have a common point of application they are called concurrent forces

    Consider a body that is subjected to two forces F1 and F2, whose lines of 
    action intersect at point A (Fig. 3.7). It is postulated that the two forces can 
    be replaced by a statically equivalent force R. This postulate is an axiom; it is 
    known as the parallelogram law of forces. The force R is called the resultant 
    of F1 and F2 . It is the diagonal of the parallelogram for which F1 and F2 are 

    adjacent sides. 

    C

    Now consider a system of n forces that all lie in a plane and whose lines of 
    action intersect at point A (Fig. 3.8). Such a system is called a coplanar system 
    of concurrent forces. The resultant can be obtained through successive 
    application of the parallelogram law of forces. Mathematically, the summation 

    may be written in the form of the following vector equation:

    C

    C

    ■ Representation in Cartesian Coordinates

    It is usually convenient to resolve forces into two components that are 
    perpendicular to each other. The directions of the components may then be 
    given by the axes x and y of a Cartesian coordinate system (Fig. 3.8). The 
    quantities Fx and Fy are called the coordinates of the vector F or components 

    of F.

    C

    Examples

    1. A hiker begins a trip by first walking 25.0 km southeast from her car. 
    She stops and sets up her tent for the night. On the second day, she 
    walks 40.0 km in a direction 60.0° north of east, at which point she 
    discovers a forest ranger’s tower.
    (a) Determine the components of the hiker’s displacement for each 
    day.
    (b)Determine the components of the hiker’s resultant displacement 
    R for the trip. Find an expression for R in terms of unit vectors.

    (c) Determine the magnitude and direction of the total displacement.

    Solution

    (a) If we denote the displacement vectors on the first and second days by 
    A and B, respectively, and use the car as the origin of coordinates, we 

    obtain the vectors shown in Figure below.

    C

    C

    2. Under what circumstances would a nonzero vector lying in the xy 

    plane have components that are equal in magnitude?

    Solution

    Any vector that points along a line at 45° to the x and y axes has components 

    equal in magnitude.

    3. In what circumstance is the x component of a vector given by the 

    magnitude of the vector times the sine of its direction angle?

    Solution

    If the direction of a vector is specified by giving the angle of the vector 
    measured clockwise from the positive y-axis, then the x-component of the 
    vector is equal to the sine of the angle multiplied by the magnitude of the 

    vector.

    4. If A = B, what can you conclude about the components of A and B?

    Solution

    Any vector that points along a line at 45° to the x and y axes has components 

    equal in magnitude.

    ■ Lami’s theorem

    Lami’s theorem gives the conditions of equilibrium for three forces acting at 
    a point O. Lami’s theorem states that if three forces acting at a point are in 
    equilibrium, then each of the force is directly proportional to the sine of the 

    angle between the remaining two forces.

    C

    Branch of mechanics which deals with state of equilibrium is called statics. 
    Statics is the branch of mechanics concerned with the analysis of loads 
    (force, torque/moment) on physical systems in static equilibrium, that is, in 
    a state where the relative positions of subsystems do not vary over time, or 
    where components and structures are at a constant velocity. When in static 
    equilibrium, the system is either at rest, or its center of mass moves at constant 
    velocity. The study of moving bodies is known as dynamics, and in fact the 

    entire field of statics is a special case of dynamics.

    Stability and Balance

    An object in static equilibrium, if left undisturbed, will undergo no translational 
    or rotational acceleration since the sum of all the forces and the sum of all the 
    torques acting on it are zero. However, if the object is displaced slightly, three 

    outcomes are possible:

    • Equilibrium is said to be stable if small, externally induced 
    displacements from that state produce forces that tend to oppose the 
    displacement and return the body or particle to the equilibrium state. 
    Examples include a weight suspended by a spring or a brick lying on a 

    level surface. 

    • Equilibrium is unstable if the least departure produces forces that tend 
    to increase the displacement. An example is a ball bearing balanced on 

    the edge of a razor blade.

    • Static Equilibrium (neutral equilibrium) is equilibrium where 
    all forces are balanced, but it also applies to bodies in uniform or 
    accelerated motion. For example, a book resting on a table applies a 
    downward force equal to its weight on the table. According to the third 
    law, the table applies an equal and opposite force to the book. This 
    force occurs because the weight of the book causes the table to deform 

    slightly so that it pushes back on the book like a coiled spring.

    By summing up

    • A system is said to be in stable equilibrium if, when displaced from 
    equilibrium, it experiences a net force or torque in a direction opposite 

    the direction of the displacement.

    • A system is in unstable equilibrium if, when displaced from equilibrium, 
    it experiences a net force or torque in the same direction as the 

    displacement from equilibrium.

    • A system is in neutral equilibrium if its equilibrium is independent of 

    displacements from its original position.

    C

    Centre of gravity and center of mass

    Concept of centre of gravity and center of mass

    The center of gravity is the average location of the weight of an object. The 
    centre of gravity is defined as the point of application of the resultant force 
    due to the earth’s attraction on it. The center of gravity is a geometric property 

    of any object.

    The centre of gravity of a body also coincides with its centre of mass. The 
    center of mass of an object may be defined as the point at which an applied 

    force produces acceleration but no rotation

    C

    Centre of gravity and base of support of a body

    1. For balance to exist, the line of gravity must intersect the base of support.

    2. If the area of the base of support of an object is increased, this tends to 
    increase the stability of the object.

    3. The lower the center of gravity is above the base of support the more 
    stable the object tends to be. (This is true even though the size of the 
    base of support is unchanged.)

    4. Objects that are more massive tend to be more stable.

    5. For an object, the farther the line of gravity’s intersection is from the 
    edge of its base of support the more stable the object tends to be in that 
    direction.

    Determining the center of gravity

    Determining the center of gravity is very important for any flying object. In 
    general, determining the center of gravity (cg) is a complicated procedure 
    because the mass (and weight) may not be uniformly distributed throughout the 
    object. If the mass is uniformly distributed, the problem is greatly simplified. 
    If the object has a line (or plane) of symmetry, the center of gravity lies on the 

    line of symmetry. 

    For a solid block of uniform material, the center of gravity is simply at the 

    average location of the physical dimensions. 

    Example

    C

    Activity 5

    To determine the centre of gravity of different regular shapes

    Apparatus

    * Manila paper, scissors, a pencil and a ruler

    Procedure

    • Make a number of shapes from a manila paper, as below:

    C

    • Find the centre of gravity of those different figures.

    Conclusion
    The point of intersection of diagonals (a,d and e), bisectors (c) , 

    diameters(b) is the centre of gravity of those figures

    For a general shaped object, there is a simple mechanical way to determine 

    the center of gravity: 

    In Step 1, you hang the object from any point and you drop a weighted string 

    (plumb line) from the same point. Draw a line on the object along the string.

    C

    For Step 2, repeat the procedure from another point on the object you now 
    have two lines drawn on the object which intersect. The center of gravity is 
    the point where the lines intersect. This procedure works well for irregularly 

    shaped objects that are hard to balance.

    Activity 6

    To determine the centre of gravity of an irregularly shaped lamina
    Apparatus

    * A plumb line, a thread, a stand and a cardboard

    C

    Procedure

    • Make 3 holes, A, B and C on the edges the cardboard.
    • Suspend it by a rod through the hole A as shown in Figure2.4 
    • Tie a plumb line on the rod beside the cardboard.
    • After the cardboard and plumb line have stopped swinging, draw 
    a vertical line on the cardboard as set by plumb line.
    • Repeat the experiment using other holes B and C. The lines 
    intersect at a point noted G
    • Balance the cardboard with the tip of a pencil at the point of 

    intersection of the three lines. What do you observe?

    Observation

    The suspended object will always rest with its centre of gravity vertically 
    below the point of support. The object balances on the tip of the pencil if 

    placed at its centre of gravit.

    C

    C

    C

    Stability and center of gravity

    Consider a ball suspended freely from a string is in stable equilibrium, for if it 
    is displaced to one side, it will return to its original position (Fig. 3.14a) due 
    to the net force and torque exerted on it. On the other hand, a pencil standing 
    on its point is in unstable equilibrium. If its center of gravity is directly over 
    its tip (Fig. 3.14b), the net force and net torque on it will be zero. But if it 
    is displaced ever so slightly as shown—say, by a slight vibration or tiny air 
    current—there will be a torque on it, and this torque acts to make the pencil 
    continue to fall in the direction of the original displacement. 
    Finally, an example of an object in neutral equilibrium is a sphere resting on a 
    horizontal tabletop. If it is moved slightly to one side, it will remain in its new 

    position—no net torque acts on it.

    C

    In general, an object whose center of gravity (CG) is below its point of support, 
    such as a ball on a string, will be in stable equilibrium. Consider a standing 
    refrigerator (Fig. 3.15a). If it is tipped slightly, it will return to its original 
    position due to the torque on it as shown in Fig. 3.15b. But if it is tipped too 
    far, Fig. 3.15c, it will fall over. The critical point is reached when the CG shifts 
    from one side of the pivot point to the other.When the CG is on one side, the 
    torque pulls the object back onto its original base of support, Fig. 3.15b. If the 
    object is tipped further, the CG goes past the pivot point and the torque causes 

    the object to topple, Fig. 3.15c.

    C

    In general, 

    • an object whose center of gravity is above its base of support will be 
    stable if a vertical line projected downward from the CG falls within the 
    base of support. This is because the normal force upward on the object 
    (which balances out gravity) can be exerted only within the area of 
    contact, so if the force of gravity acts beyond this area, a net torque will 
    act to topple the object. 
    • the larger the base and the lower the CG, the more stable the object.
    • an object tends to fall when its center of gravity is away from the base 

    that supports it. 

    Applications of equlilibrium

    Tower Crane – Method of Joints

    The tower crane shown in the figure below consists of tower DCE fixed at the 
    ground and two jibs AC and CB. The jibs are supported by tie bars AD and 
    DB, and are assumed to be attached to the tower by pinned connections. The 

    counterweight WC weighs and the crane has a lifting capacity of W.

    C

    Beam balance

    It consist of pivoted horizontal lever of equal length arms called the beam, 
    with a weighing pan also called scale, scale pan or boson, suspended from 

    each arm.

    C

    The unknown mass is placed on one pan and standard masses are added on the 
    other pan until is as close to equilibrium as possible. In precision balances a 
    slider mass is moved along a graduated scale. The slider position gives a fine 

    collection to the mass value.

    Example

    A uniform meter stick supported at the 25 cm mark is in equilibrium 
    when a 1 kg rock is suspended at the 0 cm end (as shown in Fig.). Is the 
    mass of the meter stick greater than, equal to, or less than the mass of the 

    rock? Explain your reasoning.

    C

    Solution

    C

    Application Activity 3.3

    A friend of yours that has been sick comes from the hospital where he 
    had gone for medication telling you that he is 62 kg and that is what is 
    indicated on his medical papers.
    Imagine you have the following
    * A strong wood bar of a bout 5m (or any length as long as you know 
    its actual length) and has a mass of 10kg.
    * A wood that you can use as a pivot

    a)Since you know the exact mass of your friend and that of the 
    wood. Explain all the procedures you can follow to know your 
    mass. In your procedures include all the necessary equations and 
    diagrams you may need to use.

    b)Do you think the mass (your mass) you obtained is accurate? 
    Explain your reasoning.

    c)What do you think may be the source of errors in determining 
    your mass? Or any unknown mass?

    d)Basing on your knowledge and understanding you have obtained 
    from this unit, how can this unit be applied in your career you are 

    pursuing?

    END UNIT ASSESSMENT

    1. The uniform bar shown below weighs 40 N and is subjected to the 
    forces shown. Find the magnitude, location, and direction of the force 

    needed to keep the bar in equilibrium.

    C

    2. System given below is in equilibrium. If the potential energies of 
    objects A and B are equal, find the mass of object A in terms of G. 

    (Rod is homogeneous and weight of it is G.)

    C

    A 172 cm tall person lies on a light (massless) board which is sup-ported by 
    two scales, one under the top of her head and one beneath 
    the bottom of her feet see Fig. The two scales read respectively 35.1 
    kg and 31.6 kg. What distance is the center of gravity of this person 

    from the bottom of her feet?

    C

    3. A) A seesaw consisting of a uniform board of mass M =10kg and 
    length l=2m supports a father and daughter with masses mf and md, 50 

    and 20kg respectively as shown in the Figure below.

    C

    The support (called the fulcrum) is under the center of gravity of the 
    board, the father is a distance d from the center, and the daughter is a 

    distance l / 2 from the center.

    a. Determine the magnitude of the upward force n exerted by the 

    support on the board.

    b. Determine where the father should sit to balance the system.

    B) Three children are trying to balance on a seesaw, which consists of 
    a fulcrum rock, acting as a pivot at the center, and a very light board 

    3.6 m long (see fig.).

    c

    Two playmates are already on either end. Boy A has a mass of 45 kg, and 
    girl B a mass of 35 kg. Where should girl C, whose mass is 25 kg, place 

    herself so as to balance the seesaw?

    4. A uniform 1500kg beam, 20m long, supports a 15,000kg printing press 
    5 from the right support column, see the figure. Calculate the force on 

    each of the vertical support columns.

    c

    5. A horizontal rod AB is suspended at its ends by two strings. (See the 
    figure below). The rod is 0.6m long and its weight of 3N acts at G 

    where AG is 0.4m and BG is 0.2m. Find the tensions X and Y

    c

    C

    C

    C

  • Unit 4 Work, Energy and Power

    Key Unit Competence 
    Evaluate the relation between work, energy and power and the resulting 
    phenomena.
    My goals 
    By the end of this unit, I will be able to: 
    * define work done, energy and power.
    * state the formulae of work, energy and power.
    * explain how power depends on energy.
    * explain how gravitational potential energy.
    * identify the difference between potential energy and kinetic energy
    * describe strain and work done in deforming materials

    INTRODUCTORY ACTIVITY
    Kaliza and Kalisa are classmates in senior four sciences. When Kaliza 
    mops the class alone; she can do it in only 30 minutes. But when Kalisa 
    is given the same task of mopping the class, he can do it in 45 minutes.
    Questions
    1. Discuss the work being done as in the above case study. In your own 
    words define the term work.
    2. Sometimes people do confuse the term work and energy. Are they 
    different? Are they the same? Explain your choice.
    3. Between Kaliza and Kalisa who is more powerful? Why?
    4. By research find the meaning of the terms: Work, energy and power
    Introduction
    In real life, we always use the term work. Which means “task to be 
    accomplished. But before the task to be done, one must have energy. Then if 
    a given work is done in a given time, we say that one has power i.e work done 
    in a given time.
    Work
    Review of the idea of work
    Activity 1
    Study and interprete the diagram below
    CASE I
    Work is done when a force moves its point of application along the 
    direction of its action.
    C
    * How is the force applied onto the body?
    * Why does it change its position?
    * What if the body is 10 times the mass of the boy. Would the body 
    change its position? Why?
    * State the direction of application of force.
    From the fig. 4.1 and your deductions, how can you define Work?
    CASE II
    C
    Activity 2 
    Aim; To relate distance, force and work
    Let us as a class visit any where people are constructing a house,a bride, 
    road.
    Ask them why they are paid?
    Ask them how they measure what they do.
    C
    C
    Expressions of some kinds of work
    Work of the gravitational force 
    Activity 3
    a) Hold a book in your hands at a height say h.
    b) Leave it to fall vertically onto the ground.
    V
    From the deductions, it can be noted that:
    The work done by the gravitational force does not depend on the path 
    followed but on the change of the height.
    Work done by the force of pressure
    Activity 4 
    Requirements
    * A syringe with a piston.
    Aim: To determine work done by a piston.
    * Pull the piston through a small distance ∆x as shown in the figure below.
    Assuming you applied a force F, What is the work done?
    C
    V
    C
    Energy
    Activity 5 
    Ask yourself why some times you feel like not working or bored. What 
    do you normally say when you are asked why you are not performing any 
    duty? Use what comes into your mind to define energy.
    Normally we say that Energy is ability of a body to do work.
    It’s measured in Joules like work. When an interchange of energy occurs 
    between two bodies, we can consider the work done as a measure of the 
    quantity of energy transferred between them. It has the same units as work 
    and heat i.e. Joule.
    The displacement is that of the point of application of the force. If the 
    force is applied to a particle or a non-deformable, non-rotating system, this 
    displacement is the same as the displacement of the particle or system. For 
    deformable systems, however, these two displacements are often not the same.
    Force is not necessarily the cause of the object’s displacement. For example, 
    if you lift an object, work is done by the gravitational force, although gravity 
    is not the cause of the object moving upward! An important consideration 
    is that work is an energy transfer. If W is the work done on a system and W
    is positive, energy is transferred to the system; if W is negative, energy is 
    transferred from the system. Whenever work is done energy is transferred or 
    converted from one form to another. Work is performed not only in motion and
    displacement (mechanical work); it is done also by fire flame and electricity 
    in electric lamps.
    Example
    The gravitational force exerted by the Sun on the Earth holds the Earth in 
    an orbit around the Sun. Let us assume that the orbit is perfectly circular. 
    The work done by this gravitational force during a short time interval in 
    which the Earth moves through a displacement in its orbital path is
    (a) zero (b) positive (c) negative (d) impossible to determine.
    Solution
    (a). The force does no work on the Earth because the force is pointed 
    toward the center of the circle and is therefore perpendicular to the 
    direction of the displacement.
    Categories of Energy in Our Environment
    There are several forms of energy in our environment e.g. mechanical energy, 
    heat energy, light energy, electromagnetic energy, electric energy, nuclear 
    energy, sound energy, chemical energy stored in petrol, food and other 
    materials, moving matter such as water, wind, falling rocks, etc.
    Furthermore, one form of energy can be converted to another. For example, 
    when an electric motor is connected to a battery, the chemical energy in the 
    battery is converted to electrical energy in the motor, which in turn is converted 
    to mechanical energy as the motor turns some device. 
    Scientists classify forms of energy into two major categories: Potential energy 
    and Kinetic energy.
    Potential energy
    Activity 6
    How do we know that things have energy just because of their height? 
    Well, let’s think about the following process: 
    1. You lift a ball off the ground until it is above your head. 
    2. You drop it. 
    3. It is moving fast right before it hits the ground.
    4. Draw a conclusion. 
    Potential energy may be defined as the energy possessed by an objects or 
    bodies due to their position or state of strain or the position of their parts. 
    Potential energy is energy deriving from position. Potential energy is referred 
    to as stored energy because it can be looked at as energy which will be used 
    when time comes for it to be used. Thus a stretched rubber band has elastic 
    potential energy.

    Kinds of potential energy:
    a) Chemical potential energy
    Activities such as tug of war or riding a bicycle, we use energy provided by 
    the food we eat. In cars or motorcycles, petrol is used to provide energy. Petrol 
    contains energy which makes these vehicles move. Food and petrol contain 
    energy called chemical potential energy. It is called chemical energy because 
    it is from the chemical bonds found in the food or petrol it is potentially 
    available for use when it is needed.
    b) Elastic potential energy
    Restoring force
    A block on a horizontal, frictionless surface is connected to a spring. If the 
    spring is either stretched or compressed a small distance from its unstretched 
    (equilibrium) configuration, it exerts on the block a force that can be expressed 
    as F = −k x
    x is the position of the block relative to its equilibrium (x = 0) position
    • k is a positive constant called the force constant or the spring 
    constant of the spring. The units of k are N/m.
    • The negative sign in Equation signifies that the force exerted by 
    the spring is always directed opposite to the displacement from 
    equilibrium.
    This force law for springs is known as Hooke’s law. Because the spring force 
    always acts toward the equilibrium position (x = 0), it is sometimes called a 
    restoring force.
    C
    C

    Elastic Potential Energy
    Activity 7
    Aim; To find out whether there is energy stored in elastic Materials
    In laboratory,
    * Try to perform experiment arranging your apparatus as shown in the 
    figure below.
    * What do you observe after putting a mass on the spring.
    * What would happen if the mass of the body is given a small 
    displacement downwards?
    C

    The elastic potential energy of the system can be thought of as the energy 
    stored in the deformed spring (one that is either compressed or stretched from 
    its equilibrium position.
    Consider Figure below, which shows a spring on a frictionless, horizontal 
    surface.
    C
    C
    C
    C
    C
    Kinetic energy
    Kinetic energy is the form of energy possessed by moving bodies. Such bodies 
    have the ability to do work e.g. a flying bullet can kill a dangerous wild animal. 
    Wind (a moving mass of air) flowing streams, falling rocks, heat flowing from 
    a body at high temperature to one at a lower temperature, electricity (flowing 
    electrons), moving cars, lorries, busses, etc, all have kinetic energy. Kinetic 
    energy of a body is dependent upon both the body’s mass and speed.
    C
    C
    C
    C
    Total Mechanical energy
    Mechanical energy can be either kinetic energy (energy of motion) or potential 
    energy (stored energy of position). Objects have mechanical energy if they are 
    in motion and/or if they are at some position relative to a zero potential energy 
    position (for example, a brick held at a vertical position above the ground or 
    zero height position).
    We call Mechanical energy is the sum of kinetic energy and all forms of potential 
    energy associated with an object or group of objects i.e. E PE KE = +


    Conservation of mechanical energy

    If a body of mass m is thrown vertically upwards with an initial velocity v0 at 

    A, it has to do work against the constant force of gravity.

    C


    Example

    A 6.0 kg block initially at rest is pulled to the right along a horizontal by 
    a constant horizontal force of 12 N; Find the speed of the block after it 
    has moved 3.0 m.

    a) If the surfaces in contact is frictionless surface.

    b) If the surfaces in contact have a coefficient of kinetic friction of 
    0.15.

    c) Suppose the force F is applied at an angle as shown in( Fig.b). 
    At what angle should the force be applied to achieve the largest 
    possible speed after the block has moved 3.0 m to the right?

    C

    Power

    Activity 8

    Interpret the diagrams below

    C

    C

    C

    C

    C

    Solution

    The force is perpendicular to every increment of displacement. Therefore,
    p mv =
    2. Discuss whether any work is being done by each of the following 
    agents and, if so, whether the work is positive or negative: (a) a 
    chicken scratching the ground, (b) a person studying, (c) a crane 
    lifting a bucket of concrete, (d) the gravitational force on the bucket 
    in part (c), (e) the leg muscles of a person in the act of sitting down.



    Solution

    (a) Positive work is done by the chicken on the dirt.(b) No work is done, 
    although it may seem like there is. (c) Positive work is done on the bucket. 
    (d) Negative work is done on the bucket. (e) Negative work is done on the 

    person’s torso.

    3. When a punter kicks a football, is he doing any work on the ball 
    while his toe is in contact with it? Is he doing any work on the ball 
    after it loses contact with his toe? Are any forces doing work on the 

    ball while it is in flight?

    Solution

    Yes. Force times distance over which the toe is in contact with the ball. 
    No, he is no longer applying a force. Yes, both air friction and gravity do 

    work.

    4. Cite two examples in which a force is exerted on an object without 

    doing any work on the object.

    Solution

    Force of tension on a ball rotating on the end of a string. Normal force 

    and gravitational force on an object at rest or moving across a level floor.

    5. As a simple pendulum swings back and forth, the forces acting on 
    the suspended object are the gravitational force, the tension in the 
    supporting cord, and air resistance. (a) Which of these forces, if 
    any, does no work on the pendulum? (b) Which of these forces does 
    negative work at all times during its motion? (c) Describe the work 

    done by the gravitational force while the pendulum is swinging.

    Solution

    a) Tension (b) Air resistance (c) Positive in increasing velocity on the 

    downswing.

    Negative in decreasing velocity on the upswing.

    C

    Everyday usage of the term momentum is in accord with the definition 
    above. According to the equation of momentum, a fast-moving car has more 
    momentum than a slow-moving car of the same mass; a heavy truck has 
    more momentum than a small car moving with the same speed. The more 
    momentum an object has, the harder it is to stop it, and the greater effect it will 
    have if it is brought to rest by striking another object. A football player is more 
    likely to be stunned if tackled by a heavy opponent running at top speed than 
    by a lighter or slower-moving tackler. A heavy fast-moving truck can do more 

    damage than a slow-moving motorcycle.

    Momentum (symbol: p) of an object is the product of the mass and velocity

    of a moving body.

    Example

    Calculate the linear momentum of the car in the figure below.

    C

    C

    C

    Example

    A 2250 kg pickup truck has a velocity of 25 m/s to the east. What is the 

    momentum of the truck?

    Solution

    C


    Conservation of momentum

    Activity 9: Field work

    As a class, visit a place with a pool table.
    Let each and every body try to hit the ball using the playing stick.
    What happens when one ball hits another?
    State and observe what you notice.

    Draw a conclusion.

    C

    Activity 10: Fieldwork

    * As a class, visit a place where there is billiard.
    * Try to arrange the balls with the help of your teacher or any of the 
    learners who have ever played it.
    * Let one of you hit the white ball to strike/hit the rest.
    * State what you observe after the white ball has hit the balls.
    * Draw your conclusion from your observations.

    * Repeat the same procedures using balls in the play grounds.

    C

    C

    C

    In a head-on collision:

    1. Which truck will experience the greatest force?
    2. Which truck will experience the greatest change in momentum?
    3. Which truck will experience the greatest change in velocity?
    4. Which truck will experience the greatest acceleration?

    5. Which truck would you rather be in during the collision?

    Impulse

    Activity 11

    * Move out the class to the play ground.

    * In pairs (one pair at a time), kick a ball so that it is moving with a 
    low speed. Let your friend stop it. Ask him/her what he/she felt. Let 
    your friend do the same.

    * For the second time, make sure that you kick the ball by applying a 
    strong force so that it moves faster. Let your friend try to stop it. Ask 
    him/her what this time he/she has felt?

    * Go back in class and summarise what you observed and felt while in 

    the play ground. 

    C

    C

    C

    C

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    C

    Application activity 4.5

    1. Two bodies of mass 3kg and 5kg travelling in opposite directions 
    on a horizontal surface collide. The velocities of the bodies before 
    collision are 6m/s and 5m/s respectively. Given that after collision 
    the two separate and move in the same direction in which the 5kg 
    body was moving before collision, and the velocity of the 5kg mass 
    is 1m/s, find the speed of the 3kg body after impact. Find also the 

    loss in the energy.

    2. A body of mass 2kg initially moving with a velocity of 1m/s is 
    acted upon by a horizontal force of 6N for 3seconds. Find 
    (i) Impulse given to the body 

    (ii) Final speed of the body

    Collisions

    Activity 12

    To know what happens to bodies after impact/colliding.

    To know the effect of collision on the velocities and masses of bodies after 

    colliding.

    Observe the diagram below carefully assuming the black car to have a 

    larger mass than a white one and answer the questions that follow.

    C

    Questions about the picture above:

    a) Do you think after collision , the two cars continue moving? 
    Explain why?
    b) From what you observed, what is the effect of collision?
    c) After separating the cars, do you think the masses of the 

    cars changed? Explain why.

    Note: We can define collision as an interaction between bodies in 
    which the time intervals during which the bodies interaction 

    is small relative to the time for which we can observe them.

    In collision the total momentum of colliding objects is always conserved. 
    Usually, however, their total kinetic energy is not conserved; some of it is 
    changed to heat or sound energy, which is recoverable. Such collisions are 

    said to be inelastic.

    For example, when a lump of putty falls to the ground, the total momentum of 
    putty and earth is conserved, that is, the putty loses momentum and the earth 
    gains an equal amount of momentum. But all the kinetic energy of putty is 

    changed to heat and sound on collision.

    An Inelastic collision is the collision where the total kinetic energy is not 
    conserved     (total momentum always conserved in any type of collision). If the 
    total kinetic energy is conserved, the collision is said to be elastic. For example, 

    the collision between two smooth smoker balls is approximately elastic.

    Elastic collision

    In here, we shall consider objects colliding in a straight line and thereafter 

    they move with different speeds in the same direction.

    C

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    Special cases:

    • Consider the case when the mass of one body, is equal to that of the 
    other. And the particle 2 is initially at rest, when they collide the velocity 
    of the first become zero and the velocity of the second is equal to the 

    velocity of the first one before collision.

    C

    That is a heavy particle collides with head-on with alight one that is at 
    rest , heavy particle continue its motion unaltered after the collision and 
    the light particle rebounds with a speed equal to about twice the initial 
    speed of the heavy particle. an example of such collision would be that 
    of the moving heavy atom such as uranium stricking a light atom such as 

    hydrogen.

    C

    That is a light particle collides with head-on with a heavy one that is at 
    rest, heavy particle remain approximatively at rest , the light particle has 

    a velocity reversed.

    Example is that a golf ball hitting a brick wall. The wall remains at rest, 

    and the ball bounces back with its speed unchanged.

    Example

    A Ball of 0.1kg makes an elastic head-on collision with a ball of 
    unknown mass that is initially at rest. If the 0.1 kg ball rebounds at one 

    third of its original speed. what is the mass of the other ball?

    Solution

    C

    Elastic collision in two or three dimensions

    To understand this, use billiards as shown in the previous lesson

    C

    Some times after hitting the balls, they do not move in a straight line most 
    especially (those who know how to play it) when you want to score in the 
    centre hole. You must make sure that you hit the ball in target so that it moves 

    at a certain angle.

    Activity 13

    • Try to hit a billiard ball as shown in figure above.
    • Observe what happens when one ball hits another.
    • Note down your observations.

    • Present your findings/observation to the whole class.

    On Striking the balls ,energy and momentum is conserved.

    Conservation of momentum and energy can also be applied to collisions in 
    two or three dimensions and in this case the vector nature of momentum is 

    important. 

    One common type of non-head-on collision is one for which one particle 
    (called the “projectile”) strikes a second particle initially at rest (the “tangent” 

    particle). This is the common situation in games such as billiards.

    C

    C

    C

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    Inelastic collision

    Collisions in which kinetic energy is not conserved are called inelastic 
    collisions. Some of the initial kinetic energy in such collisions is transformed 

    into other types of energy, such as thermal or potential energy.

    A common example of a perfectly inelastic collision is when two objects 
    collide and then stick together afterwards and move with a common velocity 

    after colliding.

    C

    C

    C

    2. An Object A of mass 2 kg is moving with a velocity of 3 m/s and 
    collides head on with an object B of mass 1 kg moving in the opposite 
    direction with a velocity of 4 m/s. After collision both objects sticks, 

    so that they move with a common velocity v. calculate v.

    C

    Solution
    The phrase “become entangled” tell us that this is a perfectly inelastic 
    collision.

    Answer: v=6.67m/s


    Other examples of collision

    The Ballistic Pendulum

    1. The ballistic pendulum (seen figure below) is an apparatus used to 
    measure the speed of a fast-moving projectile, such as a bullet. A 
    bullet of mass mB is fired into a large block of wood of mass mw 
    suspended from some light wires and makes a completely inelastic 
    collision with it. The bullet embeds in the block, and the entire system 
    swings through a height h. How can we determine the speed of the 

    bullet from a measurement of h??

    Solution

    Let configuration (A) be the bullet and block before the collision, and 
    configuration (B) be the bullet and block immediately after colliding.
    The bullet and the block form an isolated system, so we can categorize the 

    collision between them as a conservation of momentum problem.

    C

    C

    C

    An automobile collision

    A 1000Kg compact car is travelling North at 15 m/s when it collides with 
    a 2000 kg truck travelling East at 10m/s. all occupants are wearing seat 
    belts are there are no injuries, but the two vehicles are thoroughly tangled 
    and move away from the impact point as one mass. The insurance adjustor 
    has asked you to find the velocity of the wreckage just after the impact. 

    What do you tell her? The figure below show the situation

    C

    C

    If the velocities of the two objects make a certain angle before collision 

    and after collision they stick together, analytically we have this situation: 

    C

    C

    END UNIT ASSESSMENT

    “Work, Energy and Power”

    1. A worker lifts up a stone of 3.5kg to a height of 1.80m each 30s. Find 

    the work done in one hour.

    2. Calculate the kinetic energy and the velocity required for a 70kg pole 
    vaulter to pass over a 5.0m high bar. Assume the vaulter’s centre of 
    mass is initially 0.90m off the ground and reaches its maximum height 

    at the level of the bar itself.

    3. Calculate the power required of a 1400 kg car under the following 
    circumstances
    a) The car climbs a 10° hill at a steady 80km/h and
    b) The car accelerates from 90 to 110km/h in 6.0s to pass another 
    car on a level road. Assume the force of friction on the car is 

    700N in both parts of the problem.

    4. A woman of mass 75kg walks up a mountain of height 20m.
    a) What is the work done?
    b) The walking up being done in 1.5 min, find the power,
    c) What time will be taken by this woman to walk up the 20m in 

    order to develop a power of 73W?

    C

    C

    C

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    C

    C


  • Unit 5 : Kirchhoff ’s Laws and Electric Circuits

    Key Unit Competence
    Analyse the complex electric circuits using Kirchhoff’s laws 
    My goals
    By the end of this unit, I will be able to:
    * analyse complex electric circuits using Kirchhoff’s laws.
    * identify sources of electric current.
    * describe components of simple electric circuits. 
    * state Kirchhoff’s laws and apply them to solve problems in electric 
    circuits.
    * acquire practical skills to manipulate apparatus and evaluate 
    experimental producers.
    * explain the differences between the potential difference and 
    electromotive forces.
    Introductory activity 
    Provided two bulbs, cell holder, bulb holder, voltmeter, ammeter, connecting 
    wires and a switch.
    C
    1. Make a simple electric circuit (as in the above figure) and comment 
    on your observation
    2. On the circuit made in (1), add a voltmeter and an ammeter.
    3. What would happen if you remove one bulb from the circuit?
    4. Would there be another type of connection of the circuit? Try it and 
    comment your observation.
    5. Discuss on the function of each element in provided list.
    Introduction
    This unit is one of the most interesting units in Physics. Even if you 
    ask someone who did not have enough studies in Physics he or she 
    will tell you that People studying physics will be engineers specifically 
    electricians. This Unit addresses the principles those electricians use in 
    their career.
    Review of elements of simple electric circuits 
    and their respective role
    An electric current consists of moving electric charges. Electric current must 
    flow in electric devices connected by conductors (wires). The motion of 
    electrons in a conductor is compared to water flow in a pipe. To move electrons, 
    there must be a source of electric current, a cell, a battery, a generator which 
    acts as a pump of water.
    Making a simple circuit
    Activity 1 
    Making a simple electric circuit with a bulb, a battery and wires
    Materials:
    * 2 pieces of copper wire
    * 1 bulb
    * 1 battery
    Procedure
    1. Examine diagrams A-J below. Predict whether the circuit will be 
    complete, and record your prediction on the chart below.
    2. Demonstrate the arrangements to test your predictions..
    PREDICTION CHART
    C
    Activity 1
    C
    What makes the bulb light?
    You may already understand an electrical circuit, or this may seem 
    like magic to you. Give what your teacher demonstrated some thought. 
    Why do you think the bulb in the diagram lights?
    C
    It would be useful here to summarize some basic electricity points which 
    you may know already.

    a) A current flows along a metal or wire when a battery is 
    connected to it.

    b) The current is due to free electrons moving along the metal.

    c) The battery has a potential difference p.d or voltage 
    between its poles due to chemical changes inside the 

    battery. The p.d. pushes the electrons along the metal.

    One pole of the battery is called the positive (+) pole, the other is called the 
    negative (-) pole. The “conventional” current, shown by an arrow, flows 
    in a circuit connected from the + to the – pole. The electrons carrying the 
    current along the circuit wires actually move in the opposite direction to the 
    conventional current but this need not to be taken in account in calculations 
    or circuit formulae.
    C

    Any path along which electrons can flow is a circuit. For a continuous flow 
    of electrons, there must be a complete circuit with no gaps. A gap is usually 
    provided by an electric switch that can be opened or closed to either cut off or 

    allow energy flow.

    Most circuits contain more than one device that receives electric energy 
    from the circuit. These devices are commonly connected in a circuit in one 
    of two ways, series or parallel. When connected in series, the devices and 
    wires connecting them form a single pathway for electron flow between the 
    terminals of the battery, generator or wall socket. When connected in parallel, 
    the devices and wires connecting them form branches, each of which is a 

    separate path for the flow of electrons. 

    Making a series and parallel circuit

    Activity 2 

    Making a series circuit

    Materials:
    * 1 Battery .
    * 3 Bulbs.
    * 3 bulb holders .
    * Assembled battery holder.

    * 4 Pieces of copper wire (as needed).

    Procedure
    1. Construct a complete circuit with a battery and a bulb.
    2. Using another wire, add a second bulb as shown on the picture 

    below.

    C

    3. What did you notice happened to the first bulb when the second 
    bulb was added?
    _____________________________________________________
    _____________________________________________________
    _____________________________________________________

    4. Look carefully at how the series circuit is set up. Write a prediction 
    of what you think will happen if you unscrew one of the bulbs.
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    Why did you make this prediction? 
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    5. Unscrew bulb “X”. Describe what happens to bulb “Y”.
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    6. Tighten bulb “X”, and unscrew bulb “Y”. Describe what happens 
    to bulb “X”.
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    7. Add a third bulb to your series circuit. What happens to the brightness 
    of the bulbs each time another bulb is added to the series? 
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    8. Add a third bulb to your series circuit. What happens to the brightness 
    of the bulbs each time another bulb is added to the series? 
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    9. Draw a schematic diagram of the circuit you constructed with 

    three bulbs.

    Activity 3

    Making a parallel circuit

    Materials:
    * 1 battery 
    * 3 bulbs
    * Assembled battery holder 3 bulb holders 

    * 6 pieces of copper wire

    Procedure
    1. Construct a complete circuit with one battery and one bulb. 
    2. Using another two wires, add a second bulb as shown in the figure 

    below.

    C

    3. What do you notice happened to the first bulb when the second 
    bulb was added?
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    4. Look carefully at how a parallel circuit is set up. Write a 
    prediction of what you think will happen if you unscrew one of 
    the bulbs in the parallel circuit.
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    Why did you make this prediction? 
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    5. Unscrew bulb “X”. Describe what happens to bulb “Y”.
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    6. Tighten bulb “X” and unscrew bulb “Y”. Describe what happens 
    to bulb “X”.
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    After carrying out experiments for series and parallel circuits, 
    * What advantages and disadvantages can you note for the two cases?
    * What are the characteristics of a series connection and a parallel 

    connection?

    Application activity 5.1
    1. If a battery provides a high voltage, it can ____.
    a. do a lot of work over the course of its lifetime 
    b. do a lot of work on each charge it encounters
    c. push a lot of charge through a circuit
    d. last a long time
    2. Which of the following is true about the electrical circuit in your 
    flashlight?
    a) Charge moves around the circuit very fast - nearly as fast as 
    the speed of light. 
    b) The battery supplies the charge (electrons) which move 
    through the wires.
    c) The battery supplies the charge (protons) which move through 
    the wires.
    d) The charge becomes used up as it passes through the light 
    bulb.
    e) The battery supplies energy which raises charge from low to 
    high voltage.

    f) Nonsense! None of these are true.

    series connections of resistors

    For a series combination of two resistors, the currents are the same in both 
    resistors because the amount of charge that passes through R1 must also pass 

    through R2 and R3 in the same interval of time.

    The potential difference at the generators is equal to the sum of potential 
    difference at each resistor mean that The potential difference applied across 

    the series combination of resistors will divide between the resistors.

    C

    C

    Parallel connection of resistors

    The total current equal to the sum of the current pass through the separate 

    branches.

    The potential difference is the same at each resistor and at the generator or 
    cell.

    C
    C

    Conclusion

    Series and parallel connections each have their own distinctive characteristics.

    In a series circuit, the current is the same at all points; it is not used up. In a parallel 

    circuit the total current equals the sum of the currents in the separate branches.

    C

    Application activity 5.2

    1. As the number of resistors in a series circuit increases, the overall 
    resistance ______ (increases, decreases, remains the same) 
    and the current in the circuit _______ (increases, decreases, 
    remains the same).+
    2. Three resistors are connected in series. If placed in a circuit with 
    a 12 V power supply. Determine the equivalent resistance, the 
    total circuit current, and the voltage drop across and current at 
    each resistor.

    C

    3. Determine the values of the current at and electric potential difference 

    across each of the resistors in a parallel circuit.

    C

    Generators and receptors 

    Electromotive force

    Activity 4

    Electromotive force of a generator

    Materials
    * Battery of 6V,
    * Rheostat
    * Voltmeter
    * Ammeter

    * Connecting wires

    Procedure

    1. Make the connection as shown in the following figure.

    C

    2. Write down the voltage and the current indicated by the voltmeter 

    and the ammeter when the switch is open.

    3. Close the switch and vary the current in the circuit by varying the 
    values of the rheostat and every time write down values of voltage 

    and current indicated respectively by voltmeter and Ammeter.

    4. Fill in the following table the obtained data:

    C

    5. When you vary the value of the resistance of the rheostat, does the 

    intensity of current remain constant? Why?

    6. Does the voltage remain constant?

    7. What is the maximum voltage that you have got? How is this 

    voltage called?

    8. In general, if a charge Q (in coulombs) passes through a source of 
    emf E (in volts) which relation will give the electrical energy W

    supplied by the source (in joules)?

    9. Which relation will give the total power of the source?

    Interpretation

    A voltmeter connected to terminals of a battery measures the voltage between 
    terminals of battery. When the switch was closed, we have noticed that there 

    was a current across the circuit and the value of the voltage has been changed.

    By varying the value of the resistance of the rheostat, current in the circuit is 
    changed; voltage indicated by the voltmeter changes also, it decreases when 
    current increases. Its maximum value is reached when the switch is open. 

    Such voltage is called electromotive force E (emf) of the battery.

    The electromotive force emf E of a source (a battery, generator, etc) is the 
    energy transferred to electrical energy when unit charge passes through it. 
    In other words, we can say that the emf of a source of electrical energy is its 

    terminal p.d. on open circuit. 

    The emf of a battery is the maximum possible voltage that the battery can 

    provide between its terminals.

    C

    Internal resistance

    Activity 5 

    Existence of internal resistance in a generator
    * Consider a certain number of cells which you put in an electric 

    apparatus, like a radio…

    * With your cheek, feel their temperatures before use.

    * Put the cells in your apparatus and let them work for a certain time.

    * Remove the cells and again with your cheek, feel the new 

    temperature of the cells then answer the following questions:

    • Are the two temperatures of the cells equal? (Before and after 
    use)
    • If not, what do you think is the cause of different temperatures?
    • Is one part of the current produced by the generator consumed by 

    it? Why? 

    * The same observations can be made by feeling the temperature of 
    the battery of a telephone before a call and after a call of about 10 
    minutes. Have you felt the increasing of temperature of a phone after 

    using it? If yes, you think it’s due to what? 

    C

    The term internal resistance refers to the resistance within an emf. The terminal 

    p.d. of a cell on closed circuit is also the p.d. applied to the external circuit.

    In an external circuit electrical energy is changed onto other forms of energy 
    and we regard the terminal p.d. of a cell on closed circuit as being the number 

    of joules of electrical energy changed by each coulomb in the external circuit.

    Not all the electrical energy supplied by a cell to each coulomb is changed 
    in the external circuit. The “lost” energy per coulomb is due to the cell itself 
    having resistance. Each coulomb has to “waste” some energy to get through 
    the cell itself and so less is available for the external circuit. The resistance of 
    a cell is called its internal resistance [r] and depends among other things on 

    its size.

    The electric power dissipated as heat in a cell is given by: Pi = I2 r

    Examples 

    1. The power dissipated as heat in a cell is of 7W, find its internal 
    resistance if a current of 2A flows through it.
    2. Find the power dissipated as heat in a generator of internal 

    resistance 0.6Ω crossed by a current of 3A. 

    Solution

    C

    Remark: Any electrical generator, then, has two important properties, 
    an emf E and an internal resistance r. E and r may be 
    represented separately in a diagram, though in practice they 
    are together between the terminals. To represent a cell, we 
    can write (E, r). So we can think of the battery as an “electric 
    pump”, with its emf E pushing the current round the circuit 
    through both the external (outside) resistor R and internal 

    resistance r

    In an electric circuit, a generator is then represented by the following symbol:

    C

    Activity 6 

    Experiment to find the emf (E) and the internal resistance (r) of a 

    cell

    Materials
    * 1.5V (approx) cell, 
    * Resistance box,
    * Push switch, 

    * Digital Ammeter (0-1A).

    C

    C

    Relationship between the P.d and the emf at terminals of 

    a cell of a closed circuit

    Activity 7

    Relation between the Emf and the P.d 

    Materials 
    * Dry cell, 
    * Analogy multimetres (2), 

    * Rheostat and switch.

    Procedure

    1. Set up the circuit as shown. 

    C

    2. Set the resistance of the rheostat to a large value to protect the 
    circuit before switch on the circuit. 
    3. Set the milliammeter to the range 0-1 A or suitable range. 
    4. Set the voltmeter to the range 0-5V or suitable range. 
    5. Switch on the circuit. Record down the readings of the Ammeter 
    and voltmeter. Slide the rider of the rheostat to another position. 
    Record the readings of the ammeter and voltmeter again. Tabulate 

    the results.

    C

    C

    Examples

    1. What is the voltage at terminals of a battery of emf 3V and internal 
    resistance 0.3 Ω when sending a current of 1.5 A in a circuit?

    2. Knowing that the voltage at terminals of a cell is 1.5V and the 
    current crossing the circuit is 1.2 A. Find its emf if its internal 

    resistance is of 0.4 Ω. 

    Solution

    C

    Efficiency of a cell

    Activity 8 

    C


    Since the relation above involves the total power in the circuit, which is the 
    sum of the power supplied in the external circuit and the power dissipated in 
    the cell itself, this one is useful. It’s used to find the efficiency of the cell and 

    how to calculate the voltage at terminals of a cell or battery.

    To calculate the efficiency of the cell.

    C

    Ohm’s law for a circuit having a cell and a 

    resistor

    Activity 9

    From the observation of the following diagram and analysing 

    different elements deduce a relation 

    Questions:

    a) State the Ohm’s law.

    b) Observe the following diagram and list constituting elements.

    C

    c) Do these elements make an electric circuit? If yes, why?
    d) Are these elements connected in series or in parallel? Why?
    e) How can you find the total resistance of a series connection?
    f) Having the total power supplied by a cell, the power supplied by a 
    cell in an external circuit and the power distributed by the internal 
    resistance. Write the relation between them.
    g) Write down relations for each type of power and substitute them 
    in the relation above. (Powers dissipated in internal and external 
    resistances must be written in terms of resistance)
    h) From the relation found, deduce the emf E. The relation found 
    expresses the Ohm’s law for a circuit having a cell and a resistor.

    i) Express the intensity of the current for this specific case. 

    Pouillet’s law
    We can obtain a form of Ohm’s law by considering a segment of straight 
    wire of uniform cross-sectional area A and length , as shown in Figure 5.8. 
    A potential difference is maintained across the wire, creating in the wire an 

    electric field and a current. 

    C

    The equation representing the dependency of the resistance (R) of a 
    cylindrically shaped conductor (e.g., a wire) upon the variables which affect 

    it, is called Pouillet’s law,

    C


    where 

    • L represents the length of the wire (in m). The longer the wire, the more 
    resistance that there will be

    • A represents the cross-sectional area of the wire (in m2). The wider the wire, 
    the less resistance that there will be to the flow of electric charge. When all 
    other variables are the same, charge will flow at higher rates through wider 
    wires with greater cross-sectional areas than through thinner wires.

    •represents the resistivity of the material (in ). Some materials are better 
    conductors than others and offer less resistance to the flow of charge; they 
    are better conductors. Silver is one of the best conductors but is never used 
    in wires of household circuits due to its cost. Copper and aluminum are 
    among the least expensive materials with suitable conducting ability to 
    permit their use in wires of household circuits. 

    Resistivity of a material is numerically the resistance of a sample of unit 
    length and unit cross-section area, at a certain temperature. The resistivity 
    of a material is dependent upon the material's electronic structure and its 
    temperature. For most (but not all) materials, resistivity increases with 

    increasing temperature. 

    C

    C

    Application activity 5.3

    1. The total power of a battery is of 9V and its internal resistance 3Ω. 
    Knowing that the current crossed is of 0.4A. Find the efficiency of 
    the battery.

    2. A generator of internal resistance 2Ω sends a current of 4A in a 
    resistor of resistance 10Ω. Calculate its power.

    3. An external resistance of 4Ω is connected to an electric cell of 
    emf 1.5V and internal resistance 2Ω. Calculate the intensity of the 
    current flowing the external resistance.

    4. An electric cell of emf 1.5V and internal resistance 2Ω is 
    connected in series with a resistance of 28Ω. Calculate the power 
    dissipated as heat in the cell. 

    Combination of cells

    Activity 10

    Cells wired in parallel and in series

    Materials:
    * 3 batteries 
    * 2 bulbs 
    * 3 assembled battery holders 
    * 2 bulb holders 
    * 6 pieces of copper wire

    Procedure

    1. Construct a complete circuit with one battery and one bulb.
    2. Observe the brightness of the bulb. 
    3. Construct the circuit below. Are these batteries in series or parallel?
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    How can you tell? 
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

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    4. Observe the brightness of this bulb. Is the bulb brighter than it was 
    with one battery?
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    5. If you added a third battery to this circuit in series, what do you 
    think would happen to the brightness of the bulb? 
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    Why do you think this?
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    6. Add a third battery to this circuit. Describe what happens to the bulb 
    as this battery is added to this circuit in series and why you think the 
    bulb is acting in this way.
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    7. Construct another complete circuit with one battery and one 
    bulb. Record again what the brightness of the bulb is using your 

    brightness metre. 

    8. Look at the pictures below, are the batteries in the picture in series 
    or parallel?
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    How can you tell?
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    C

    Construct the circuit in 8. Is the bulb brighter with two batteries 
    than it was with one battery?
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    9. Add one more battery to this circuit in parallel. Describe what 
    happens to the bulb as one more battery is added to this circuit in 
    parallel and why you think the bulb is acting this way.
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    10. Connect then two batteries in opposition to mean the positive 
    (negative) terminals of batteries are connected together and the 
    two free negative (positive) terminals are connected to the bulb. 
    What happens to the bulb?
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    11. Connect two batteries in series in opposition with one battery. 
    When the two free ends of the combination are connected to 
    terminals of the bulb, what happens to the brightness of the bulb?
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    Interpretation

    Combination in series

    C

    When two or more cells are arranged in series, the total emf is the algebraic 
    sum of their emfs and the total internal resistance is the algebraic sum of their 

    internal resistances

    C

    Note: A series arrangement is used to increase the voltage, also the 
                total internal resistance of the circuit, so the energy loss due to 

                internal resistance is greater than for a single cell.

    C

    Examples

    1. Four 1.5V cells are connected in series to a 12Ω lightbulb. If the 
    resulting current is 0.45A, what is the internal resistance of each 

    cell, assuming they are identical and neglecting wires?

    2. A certain number of cells of emf 1.5V and internal resistance 
    2Ω are connected in series. When connected this combination to 
    an external resistance of 10Ω, a current of 500mA flows in this 

    resistance. Find the number of cells used. 

    C

    Combination in opposition

    Let us see also what the result could be if the cells were associated in opposition 

    C

    Let us combine two cells in opposition. Two terminals of same sign are 
    connected together. The direction of the current in the circuit will be determined 
    by the direction of the current produced by the cell having the higher emf. For 

    internal resistances they are in series. So we write: 

    C

    Note: You might think that connecting batteries in opposition would 
    be wasteful. For more purposes, that will be true. But such 
    an opposition arrangement is precisely how a battery charger 

    works.

    Example

    A cell of emf 2V and internal resistance 0.2Ω is associated in opposition 
    with another cell of emf 1.5V and internal resistance 1.2Ω. Calculate the 

    intensity of the current knowing that the external resistance is 1.1Ω. 

    Solution

    C

    Combination in parallel

    C

    C

    Note: The parallel arrangement is useful normally only if the emfs are 
    the same. A parallel arrangement is not used to increase the 
    voltage, but rather to provide large currents. Each of the cells 
    in parallel has to produce only a fraction of the total current, 
    so the energy loss due to internal resistance is less than for a 

    single cell; the batteries will be exhausted less quickly.

    Examples

    1. We have 8 cells of emf 1.5V and internal resistance 2Ω. Calculate 
    the intensity of the current flowing in an external resistance of 1Ω 

    connected to the terminals of the 8 cells combined in parallel.

    2. Six cells of unknown emf and internal resistance of 2Ω are 
    associated in parallel. When an external resistance of 1Ω is 
    connected to this combination a current of 1.5A is produced. 

    Calculate the emf.

    Solution

    C

    Mixing series and parallel combinations

    C

    C

    Example

    Four cells of emf 4.5V each and internal resistance 2Ω are combined in 
    series. The combination is connected to an external resistance of 24Ω
    a) What is the intensity of the current?
    b) Same question if the cells are combined in parallel.
    c) Same question if the combination has two parallel series of two 

    cells each. 

    Solution

    C

    Receptors

    Activity 11

    Distinguishing a receptor from a passive resistor

    a) Observe the following devices and name them

    C

    b) What is the use of each one?
    c) The flowing of the current in them produces the same 
    effect? Explain.
    d) Among them, which ones transform the whole electric 
    energy consumed in heat and which ones transform a part of 
    electric energy consumed in another kind of energy which is 
    not heat? 
    e) As we had in the case of generators, what are characteristics 

    of these apparatuses?

    Conclusion: Among the apparatuses above, there are some which transform 
    the total electric energy consumed into heat and some transform just a part 
    into heat, other part transformed into another type of energy which is not heat. 
    Those which transform the whole quantity of electric energy consumed into 
    heat are passive resistors or passive receptors and those transforming a part of 
    the consumed electric energy in another form of energy which is not heat are 

    called receptors or active receptors. 

    The main characteristics are back electromotive force and internal resistance. 

    Back electromotive force

    Back electromotive force (emf) is normally used to refer to the voltage that 

    is developed in electric motors. This is due to the relative motion between the 

    magnetic field from the motor's field windings and the armature of the motor!

    Internal resistance

    The internal resistance of a receptor r’ is its ability to oppose electric current. 
    When a receptor is traversed by an electric current, part of the energy consumed 
    is transformed into heat. The power dissipated in the receptor by joule effect 
    is: PJ = I 2r

    The p.d at terminals of a receptor

    Activity 12

    Find the P.d at terminals of a motor

    Materials
    * Electric motor
    * Ammeter
    * Voltmeter

    * Power supply

    Procedure

    1. Make the connection as shown in the figure below.

    C

    Measure the voltage (V) between terminals of the motor (M) and 

    the current I in the circuit. 

    Questions
    a) What is the net electrical power received by the motor?
    b) What becomes this power and how is it transformed?
    c) What is the relation between the voltage and the back 
    electromotive force?
    d) From the relation found, how do you calculate the intensity 

    of the current flowing?

    Application activity 5.4

    1. A circuit has in series a generator of emf 6V and internal 
    resistance 0.1Ω, a receptor of back emf 1.5V and internal 
    resistance 0.4Ω and a passive resistor of 8.5Ω. Calculate:
    a) The intensity of the current flowing in the circuit.
    b) The power supply by the generator.

    c) The quantity of heat produced in the resistor in one minute.

    2. A battery has an emf of 12.0V and an internal resistance of 0.05Ω. 
    Its terminals are connected to a load resistance of 3.00Ω. (a) Find 
    the current in the circuit and the terminal voltage of the battery. 
    (b) Calculate the power delivered to the load resistor, the power 
    delivered to the internal resistance of the battery, and the power 

    delivered by the battery.

    3. Calculate the terminal voltage for a battery with an internal 
    resistance of 0.9Ω and an emf of 8.5V when the battery is connected 

    in series with (a) an 81Ω resistor, and (b) 810Ω.

    4. A 9V battery whose internal resistance r is 0.5Ω is connected in 

    the circuit shown in the figure. 

    C

    a) How much current is drawn from the source?
    b) What is the terminal voltage of the battery? 

    c) What is the current in the 6Ω resistor?

    5. What is the internal resistance of a 12V car battery whose terminal 
    voltage drops to 8.4V when the starter draws 75A? What is the 

    resistance of the starter?

    6. A 1.5V dry cell can be tested by connecting it to a low-resistance 
    Ammeter. It should be able to supply atleast 22A. What is the 
    internal resistance of the cell in this case, assuming it is much 

    greater than that of the Ammeter?

    7. A cell whose terminals are connected to a wire in nickel silver of 
    resistivity 30 x 10 -6 Ωcm and cross sectional area 0.25mm2
     and 
    length 5m sends a current of 160mA. When the length is reduced 
    to a half, the intensity of the current is of 300mA. Calculate:
    a) The internal resistance.

    b) The emf of the cell.

    8. A cell (E = 1.5V, r = 1.3Ω) sends a current in an external 
    resistance of 3Ω. Calculate: 
    a) The intensity of the current in the circuit.
    b) The p.d at terminals of the cell.
    c) The power of generator.

    d) The efficiency of the cell.

    9. A battery is composed by 120 cells in series. Each element has an 
    emf of 2V and an internal resistance of 0.001Ω. The combination 
    is connected to an external resistance of 4.8Ω. Calculate: 
    a) The intensity of the current in the circuit.
    b) The voltage at terminals of the battery.
    c) The energy dissipated by joule effect when the current 

    flows in the circuit in one hour.

    Kirchhoff’s rules 

    Activity 13

    Find the equivalent Resistance

    C

    In this experiment you will be using a digital multimeter (DMM) which 
    can function as either a voltmeter or an Ammeter. 
    The voltmeter must always be wired in parallel with the resistor whose 
    voltage you are measuring. The ammeter, used to measure current, must 
    always be wired in series. Disconnect the meter from the circuit before 
    you change the function setting. Failure to follow these procedures can 
    result in serious damage to the meter. Be sure that you use the correct units 

    with your data.

    Materials
    * 1 multimeter.
    * 1 330 Ω or 240 Ω resistor.
    * 1 1000 Ω (1KΩ) resistor.
    * 1 2000 Ω (2KΩ) resistor.
    * 1 3000 Ω (3KΩ) resistor or 1 3300 Ω (3.3KΩ) resistor. 
    * 1 0- 10K resistor substitution box.
    * 2 spade lugs.
    * 2 2’ red banana wires.
    * 2 2’ black banana wires. 

    * 4 4” black banana wires. 

    C

    C

    C

    Simple circuits can be analysed using the expression V = IR and the rules 
    for series and parallel combinations of resistors. Very often, however, it is 
    not possible to reduce a circuit to a single loop. The procedure for analysing 
    more complex circuits is greatly simplified if we use two principles called 

    Kirchhoff's rules:

    C

    Kirchhoff’s second rule follows from the law of conservation of energy. Let us 
    imagine moving a charge around a closed loop of a circuit. When the charge 
    returns to the starting point, the charge–circuit system must have the same 
    total energy as it had before the charge was moved. The sum of the increases in 
    energy as the charge passes through some circuit elements must equal the sum 
    of the decreases in energy as it passes through other elements. The potential 
    energy decreases whenever the charge moves through a potential drop -IR 
    across a resistor or whenever it moves in the reverse direction through a source 
    of emf. The potential energy increases whenever the charge passes through a 

    battery from the negative terminal to the positive terminal.

    C

    Examples

    1. A single-loop circuit contains two resistors and two batteries, as 
    shown in figure 5.29 (neglect the internal resistances of the batteries). 
    (a) Find the current in the circuit. (b) What power is delivered to 

    each resistor? What power is delivered by the 12V battery?

    C

    C

    C

    C

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    We therefore need only two loop equations to determine the unknown 

    currents. (The third loop equation would give no new information).

    Applying Kirchhoff’s loop rule to loops ABCDA and BEFCB and 

    traversing these loops clockwise, we obtain the expressions

    C

    C

    C

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    END UNIT ASSESSMENT

    C

    C

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    V

    C

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    13. A dead battery is charged by connecting it to the live battery of 
    another car with jumper cables as shown in the figure. Determine 

    the current in the starter and in the dead battery.

    C

  • Unit 6 : Sources of Energy in the World

    Key unit Competence

    Evaluate the energy sources in the world

    My goals

    By the end of this unit, I will be able: 
    * identify sources of energy in Rwanda.

    * outline the basic features of renewable and non renewable energy 
    sources.

    * evaluate energy uses and availability in Rwanda.

    * identify various advantages and disadvantages of various energy 
    sources.

    * be aware of the moral and ethical uses associated with use of energy.

    Introductory activity 
    At hospital, during patient treatment, doctors recommend to patients to take 
    a balanced diet. That is true because as a patient takes medicines and eat 

    very well, it yields a quick recovery and one gets strong.

    Questions:
    1.What do you think is contained in food that we eat?

    2.Discuss why it is very important to take a sunbath when a person is sick?

    3.Discuss other different sources of energy in environment?

    Introduction

    Origins of the power used for transportation, for heat and light in dwelling 
    and working areas, and for the manufacture of goods of all kinds, among other 
    applications. The development of science and civilization is closely linked 
    to the availability of energy in useful forms. Modern society consumes vast 
    amounts of energy in all forms: light, heat, electrical, mechanical, chemical, 
    and nuclear. The rate at which energy is produced or consumed is called power, 
    although this term is sometimes used in common speech synonymously with 
    energy.

    Activity 1
    Answer these questions.
    a) What do you think when you hear the word “energy”? Give 
    its definition and that of the term “energy source”.
    b) Among scientists and energy professionals, a standard list 
    of current energy sources would include: biomass (plant 
    matter), nuclear, coal, oil, geothermal, solar, hydro (rivers), 
    wave or tidal, natural gas, wind. Add other sources of 
    energy which you may know.
    c) From the list given in (b) what is the major category of 
    renewable energy? 
    d) d) Between renewable and non-renewable energy which 
    one produces a little or no pollution or hazardous waste and 
    pose few risks to public safety? How the other produces it? 
    e) e) Discuss in groups this consequence above.
    f) f) List as many as you can uses of renewable energy 

    sources.

    Read carefully these key terms in the table below then give answers to related 

    questions.

    Key Terms 

    Biomass energy Energy released from plants (wood, corn, etc) 

                                      through combustion or other chemical process 

    Fossil fuel  A non-renewable energy resource that began to 
                            form millions of years ago from the remains of 
                            once living plants and animals. Its current forms 

                           include petroleum, coal and natural gas. 

    Geothermal energy Heat energy from the earth.

    Hydropower Transformation of the energy stored in a depth of 

                                water into electricity.

    Non renewable energy 
    Resources, such as fossil fuels that cannot be 

    replaced by natural processes at the same rate it is consumed.

    Photovoltaic A chemical process that releases electrons from a 
                                semi-conductor material in the presence of sunlight 

                               to generate electricity.

    Renewable energy Resources, such as wind and water that can be 
                                             recycled or replaced at a rate faster than they are 

                                            consumed.

    Solar Energy Energy from the sun; often captured directly 
                                 as heat or as electricity through a photovoltaic 

                                 process. 

    Uranium An element that releases heat as it undergoes 
                        radioactive decay.

    Wind energy Energy transferred with the motion of air in the 
                                lower atmosphere that arises from differential 
                                heating of the earth. The energy in the wind can be 
                                extracted as mechanical energy to do work such as 
                                grind grains (a wind mill) or generate electricity 

                               (wind turbine). 

    Wave energy Wave power captures the energy of ocean surface 
                                waves, and tidal power. Converting the energy 
                                of tides, are two forms of hydropower with 
                               future potential; however, they are not yet widely 

                               employed commercially. 

    Worldwide, wood is the largest source of biomass for non-food energy, but 
    other sources are also used, including municipal wastes and crop wastes. 
    Crops such as sugar cane are used to make alcohol for transportation fuel. 
    In many developing countries, wood is the most important energy source. 
    Global resources of geothermal energy (the heat contained below Earth’s 
    surface) are so immense that they are usually considered to be renewable. 
    But this classification is not strictly correct, since the heat stored in any given 
    volume of rock or underground water is depletable. In addition, the most 
    easily accessed geothermal resources, natural hot springs and geysers, will 
    not last for more than a few decades if exploited for energy on a large scale. 
    Estimates vary widely as to how long fossil fuels, oil, coal, and natural gas 
    will last. These estimates depend on assumptions about how much fossil fuel 
    remains in the ground, how fast it will be used, and how much money and 
    effort will be spent to recover it. However, most estimates agree that, if present 
    rates of consumption continue, proven oil and natural gas reserves will run 
    out in this century, while coal reserves will last more than 200 years. Once 

    they are used, these energy sources cannot be replaced.

    Fossil fuel

    Fossil fuels are fuels formed by natural processes such as anaerobic 
    decomposition of buried dead organisms. The age of the organisms and their 
    resulting fossil fuels is typically millions of years, and sometimes exceeds 
    650 million years. Fossil fuels contain high percentages of carbon and include 

    coal, petroleum and natural gas. 

    C

    C

    Rwanda’s main fossil fuel resource is methane gas. It is estimated that there 
    are 50 billion cubic metres of exploitable methane, which is the equivalent 
    of 40 million tons of petrol (TOE) laying at the bottom of the Lake Kivu 
    under 250 metres of water. Of the 55 billion cubic metres (cum) Standard 
    Temperature and Pressure, STP) of methane gas reserves, 39 billion cum 
    (STP) are potentially extractable. This represents a market value of USD 16 
    billion, equivalent to 31 million Ton Oil Equivalent (TOE). The technical and 
    economic feasibility of methane gas exploitation has been clearly demonstrated 
    since 1963 by the small methane extraction pilot unit at Cape Rubona with a 
    capacity of 5000 cum of methane per day at 80 % purity. The resource is 
    estimated to be sufficient to generate 700 mW of electricity for 55 years with 

    Rwanda’s share being 350 mW.

    C

    Nuclear fuel

    Nuclear fuel is a material that can be ‘burned’ by nuclear fission or fusion to 

    derive nuclear energy. 

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    Most nuclear fuels contain heavy fissile elements that are capable of nuclear 
    fission. When these fuels are struck by neutrons, they are in turn capable of 
    emitting neutrons when they break apart. This makes possible a self-sustaining 
    chain reaction that releases energy with a controlled rate in a nuclear reactor or 

    with a very rapid uncontrolled rate in a nuclear weapon.

    “…Rwanda should choose a path to renewable energy—although nuclear is 
    the best other alternative; Rwanda does not have the technology to generate 

    nuclear energy.

    Even if Rwanda was ready to develop it despite the international laws and 
    regulations, nuclear energy poses a great danger especially, Rwanda being 
    located in a volcanic region. Nuclear energy for Rwanda in my opinion is a no 

    go option”. New times May 21, 2015

    Renewable sources

    Renewable energy is generally defined as energy that comes from resourcesthat 
    are not significantly depleted by their use, such as sunlight, wind, rain, 
    tides, waves and geothermal heat. Renewable energy is gradually replacing 
    conventional fuels in four distinct areas: electricity generation, hot water/

    space heating, motor fuels, and rural (off-grid) energy services.

    C

    Generally, Rwanda is well endowed with renewable energy resources, but most 
    potential still remains untapped. Micro hydro-power in particular constitutes 
    a significant potential for rural power supply with many areas having ample 
    rainfall and most streams and rivers unexploited. Solar irradiation is high - 
    between 4-6 kWh/m2/day - but diffusion is hampered by high initial cost and 
    limitations on high load usage. Biogas is promising for thermal energy needs 
    for farms and small institutions, especially considering the large number of 

    households that own cows and other livestock.

    Geothermal
    Geothermal energy is from thermal energy generated and stored in the Earth. 

    Thermal energy is the energy that determines the temperature of matter. 

    C

    According to a study by Geothermal Energy Association, geothermal potential 
    in Rwanda ranges from 170 - 340 MW. In Rwanda geothermal is a main
    energy policy priority and forms a significant part of the 7-year electricity 
    development strategy including a very ambitious action plan targeting 150 
    MW of generation capacity by 2017 (which represents up to 50% of total 
    generation). A Geothermal Act and a geothermal exploration and development 
    paper have been drafted although a proposal for a feed-in-tariff for geothermal 
    still needs to be developed. Three sites (Rubavu, Karongi and Rusizi) were 
    identified already in the 1980’s with resource temperatures in excess of 150°C 
    which could be suitable for geothermal power generation. In early 2012, test 
    drilling commenced to explore possibilities to harness energy in Rubavu, 
    Karisimbi, Kinigi located in western region as well as Bugarama in southern 
    region. The Government has self-financed and contracted the first exploratory 

    drilling in 2013.

    Biomass and biofuels 
    Biomass is biological material derived from living, or recently living 
    organisms. It most often refers to plants or plant-derived materials which 
    are specifically called lignocellulosic biomass. As an energy source, biomass 
    can either be used directly via combustion to produce heat, or indirectly after 
    converting it to various forms of biofuel. Conversion of biomass to biofuel can 
    be achieved by different methods which are broadly classified into: thermal, 

    chemical, and biochemical methods.

    C

    In Rwanda, It has been observed that if an average household used 1.8 tonnes 
    of firewood in a year to satisfy its cooking needs with a traditional stove, the 
    same household would use 3.5 tonnes of wood if it were to switch to charcoal 
    with an improved stove. The use of charcoal in urban areas, in combination 
    with high urban growth rates, therefore is a worrisome phenomenon that 
    accelerates pressure on wood resources. Peat is also a resource the government 
    intends to promote use of. It is estimated that there exists in Rwanda estimated 
    reserves of 155 million tons of dry peat spread over an area of about 50,000
    hectares. About 77% of peat reserves are near Akanyaru and Nyabarongo 
    rivers and the Rwabusoro plains Potential for Peat-to-Power Generation. Peat 
    in the Rwabusoro marshland and around the Akanyaru river can fuel 450 mW 
    of electricity generation for 25 years. Currently, a cement plant and some 

    prisons utilize peat for cooking

    C

    Biogas has been introduced in the country many years ago and Rwanda 
    has gained international recognition for its program in prisons and large 
    institutions. The Government in 2008 announced a policy to introduce biogas 
    digesters in all boarding schools (estimated at around 600 schools), large 
    health centres and institutions with canteens to reduce the consumption of 
    firewood. This process started in 2010 but until today the focus has been 
    mainly on installations for schools. In total, about 50 large biogas digesters 
    have been constructed in institutions in Rwanda and the biogas systems that 
    have been installed in the prisons over the last decade have reduced firewood 

    consumption by up to 40% and improved hygienic conditions.

    C

    Activities in the domestic biogas sector started much later. It is estimated that 
    over 120,000 households have dairy cows that are kept under zero grazing 
    conditions to reduce soil erosion and also due to lack of grazing areas. These 
    numbers are increasing due to the governments programs to increase the 

    number of families with dairy cows.

    C

    Solar energy (photovoltaic cells and solar heating panels)

    Photovoltaic Cells

    Solar energy, radiant light and heat from the sun, is harnessed using a range of 
    ever-evolving technologies such as solar heating, photovoltaic, concentrated 

    solar power, solar architecture and artificial photosynthesis. 

    C

    The Rwandan government is set to commission the first utility-scale solar 
    photovoltaic (PV) plant at eastern Rwanda’s Rwamagana district in August 
    2014 The project, with a production capacity of 8.5 mW, has commenced 
    testing, stated local reports. Dutch company Gigawatt Global is the developer 
    of the project, while Norwegian firm Scatec Solar has agreed to operate and 

    maintain the plant.

    C

    Solar Heating Panels

    A solar thermal collector collects heat by absorbing sunlight. A collector is 
    a device for capturing solar radiation. Solar radiation is energy in the form 
    of electromagnetic radiation from the infrared (long) to the ultraviolet (short) 

    wavelengths. 

    C

    The term “solar collector” commonly refers to solar hot water panels, but may 
    refer to installations such as solar parabolic troughs and solar towers; or basic 

    installations such as solar air heaters.

    Hydroelectric power, wind power and wave power

    Hydroelectricity

    C

    Energy in water can be harnessed and used. Since water is about 800 times 
    denser than air, even a slow flowing stream of water, or moderate sea swell, 
    can yield considerable amounts of energy. Hydroelectricity is the term 
    referring to electricity generated by hydropower; the production of electrical 
    power through the use of the kinetic energy of falling or flowing water. It is 
    the most widely used form of renewable energy, accounting for 16% of global 

    electricity consumption.

    C

    The country currently has about 57 MW installed hydropower generating 
    capacity. Hydroelectric power is mainly from the northern and southern parts 
    of the country (Musanze , Rubavu and Rusizi) namely from the following 
    power plants: Ntaruka, Mukungwa , Rubavu, Gihira as well as Rusizi 1 and 
    Rusizi 2. A number of new sources are supposed to come on line within 
    the coming years adding a capacity of 232 MW by 2013. This includes the 
    hydropower site Nyaborongo with 27.5 MW in Muhanga and Ngororero 
    Districts planned to come on line by February 2013 but currently experiencing 
    delays, and numerous mini/micro hydro plants adding up to 35 MW. The 
    new hydropower plant, Rukarara located in Nyamagabe district, Southern 
    Province, with 9.5 MW and costs of US$ 23.5 million was commissioned in 

    January 2011. Construction for this plant had already started in 2007. 

    Wind Power 

    Airflows can be used to run wind turbines. Modern utility-scale wind turbines 
    range from around 600 kW to 5 MW of rated power, although turbines with 
    rated output of 1.5–3 MW have become the most common for commercial 
    use; the power available from the wind is a function of the cube of the wind 
    speed, so as wind speed increases, power output increases up to the maximum 

    output for the particular turbine.

    C

    Wind Potential in Rwanda has not been fully exploited for power generation 
    although potential wind power that Rwanda has in some areas may provide 
    with possible solutions such as water pumping, windmill and electricity 
    generation. A study of wind speed distribution has been made. (In this study, 
    the results have been found for the average wind speeds and directions for 3 

    stations (Kigali, Rubavu and Huye) from 1985 to 1993.

    These results can be summarised as follows:

    • Direction of wind varies from 11 to 16°.

    • Wind speed varies from 2 to 5.5m/s

    The analysis of the wind energy possible solution for energy supply in rural 
    areas of Rwanda, was undertaken to estimate the wind power potential. In 
    total data from 4 stations (Kamembe, Huye, Nyagatare and Rubavu) have 
    been analysed by the National Meteorological Division in 1989. Once again, 
    the data from 3 synoptic sites (Kigali, Huye and Rubavu) are analysed by the 
    Weibull function. The considered data has been used to evaluate the annual 
    frequency of wind speed and the direction of wind, yearly variation of the 
    monthly average, annual and daily variation, and vertical profile of wind 
    energy potential. Nevertheless more detailed data is still required. In 2010 a 
    wind system was put in place to serve the Rwanda office of information RBA 
    on Mount Jali overlooking Kigali. This is the same site for the 250KW solar 
    system feeding to the grid. There is need for more thorough assessment of the 

    wind potential in the country.

    Wave Power 
    Wave power captures the energy of ocean surface waves, and tidal power, 
    converting the energy of tides, are the two forms of hydropower with future 

    potential; however, they are not yet widely employed commercially. 

    Activity 2

    Energy Source research 

    Purpose

    Although most of the energy consumed in Rwanda comes from fossil fuel 
    sources, there are many other potential sources of energy available. In all 
    cases, there are pros and cons (advantages and disadvantages) to our use of 
    these sources. Some of the energy sources are limited by their availability 
    or environmental impact; others need technological improvements before 
    they can become widely used. For scientists and engineers, research is the 

    best way to learn about unknown topics. 

    In this section, we will examine information about energy sources and 
    how those sources are used to produce electrical energy. We can use this 
    information to help us understand the various pros and cons that affect 
    our use of different energy sources. In this activity, each group of students 
    will begin to become an expert on one aspect of each source of energy and 

    report their findings back to the class. 

    Procedure 
    1. Break into a group of 2-3 learners. 
    2. Choose or accept an assignment to research one particular question 
    about each source of energy. 
    3. Using the provided information packet, find the answer to your 
    question for all seven energy sources.
    4. Once you have answered your question for all seven sources, answer 
    the two conclusion questions. 

    As a class, we will fill in the energy sources chart based on your findings.

    Research Questions 
    1. What is this energy source? Where can we find it in Rwanda? 
    2. How do we harness the energy? (How does it work?) 
    3. Are there different types or uses of this source? If yes, what are 
    the differences? 
    4. How is this energy source currently used? For example: At farms, 
    in industry etc. Could this source be used in a family home? 

    Note: Prepare a report summarizing your research and present the report 

    to the class.

    Primary energy sources take many forms, including nuclear energy, fossil 
    energy-like oil, coal and natural gas - and renewable sources 
    like wind, solar, geothermal and hydropower. These primary sources are 
    converted to electricity, a secondary energy source, which flows through 
    power lines and other transmission infrastructure to your home and business.

    Activity 3

    Discussion Questions 

    1. If you had to choose an energy system to tell your community about 
    based only on the aspect you researched, which system would you 
    choose? Why? 
    2. Why do we as a nation depend so much on fossil fuels? AND what do 

    you think we could do to reduce this dependence on fossil fuels?

    Note: Prepare a report summarizing your research and present the report 

    to the class.

    C

    While listening to the other groups in your class present their information, list 
    some “pros” and “cons” (advantages and disadvantages) of using their energy 
    source to solve your problem. While listening to the students in your group 
    present their information, list some “pros” and “cons” of using that energy 

    source to solve the energy problem. 

    Advantages and disadvantages of renewable and non-renewable energies

    Activity 6 

    Do research in the library or internet and complete the task below

    1. Complete the chart below about the basic types of renewable energy 

    resources.

    C

    2. List those energy sources that are fossil fuels.
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    3. What main advantage do fossil fuels have over the renewable 
    energy resources? 
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    4. What are two main disadvantages of fossil fuels compared to 
    renewable energy?
    _____________________________________________________
    _____________________________________________________

    _____________________________________________________

    The sun, prime source of world energy

    Solar energy comes from thermonuclear fusion; 30% of solar energy arriving 
    on higher layers of atmosphere are reflected in space. 47% of that energy are 
    absorbed by the ground and oceans during daytime and become the Earth’s 
    internal energy. The remaining 23% of solar energy are used in evaporation 
    of water of oceans. When it rains, a part of energy is transformed into 
    potential gravitational energy, stocked in mountains, lakes, which are the 
    sources of hydroelectric power. About 0.2% is used by convection currents 
    in atmosphere and creates wind energy. Finally 0.02% is absorbed by plants 

    during photosynthesis and is stocked by them in form of chemical energy. 

    Plants are sources of biomass. Photovoltaic cells transform solar energy in 

    electrical energy.

    The table below show the summary of energy sources

    C

    C

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    Extraction and creation of renewable and non-renewable energy sources 

    Activity 5

    Creation of renewable and non renewable energy
    From what you have already learned, you’ll do also research and tell 
    how these energies are created: Solar energy, hydropower, wind energy, 
    geothermal energy, and biomass. Try to find or to formulate how they are 

    extracted.

    C

    Creation

    Non-renewable resources 

    A non-renewable resource (also called a finite resource) is a resource that 
    does not renew itself at a sufficient rate for sustainable economic extraction 
    in meaningful human time-frames. An example is carbon-based, organically-
    derived fuel. The original organic material, with the aid of heat and pressure, 

    becomes a fuel such as oil or gas.

    Earth minerals and metal ores, fossil fuels(such as coal, petroleum, and natural 
    gas), nuclear fuels, and groundwater in certain aquifers are all non-renewable 

    resources.

    Natural resources such as coal, petroleum (crude oil) and natural gas take 
    thousands of years to form naturally and cannot be replaced as fast as they are 
    being consumed. Eventually it is considered that fossil-based resources will 
    become too costly to harvest and humanity will need to shift its reliance to 

    other sources of energy. These resources are yet to be named. 

    Renewable resources

    Natural resources, known as renewable resources, are replaced by natural 
    processes and forces persistent in the natural environment. There 
    are intermittent and reoccurring renewable and recyclable materials, which are 
    utilized during a cycle across a certain amount of time, and can be harnessed 

    for any number of cycles.

    The production of goods and services by manufacturing products in
    economic systems creates many types of waste during production and after 
    the consumer has made use of it. The material is then incinerated, buried in 
    a landfill or recycled for reuse. Recycling turns materials of value that would 

    otherwise become waste into valuable resources again.

    The natural environment, with soil, water, forests, plants and animals are 
    all renewable resources, as long as they are adequately monitored, protected 
    and conserved. Sustainable agriculture is the cultivation of plant and animal 
    materials in a manner that preserves plant and animal ecosystems over the 
    long term. The overfishing of the oceans is one example of where an industry 
    practice or method can threaten an ecosystem, endanger species and possibly 
    even determine whether or not a fishery is sustainable for use by humans. 
    An unregulated industry practice or method can lead to complete resource 

    depletion.

    Extraction

    Resource extraction involves any activity that withdraws resources from 
    nature. This can range in scale from the traditional use of preindustrial societies, 
    to global industry. Extractive industries are, along with agriculture, the basis 
    of the primary sector of the economy. Extraction produces raw material 
    which is then processed to add value. Examples of extractive industries are 
    hunting, trapping, mining, oil and gas drilling, and forestry. Natural resources 
    can add substantial amounts to a country’s wealth, however a sudden inflow 
    of money caused by a resource boom can create social problems including 
    inflation harming other industries (“Dutch disease”) and corruption, leading 

    to inequality and underdevelopment, this is known as the “resource curse”.

    C

    C

    C

    END UNIT ASSESSMENT

    1. Differentiate between renewable and non-renewable energy 

    resources?

    2. Using a table to distinguish renewable and nonrenewable resources: 

    Sun, coal, water, natural gas, wood; petroleum; wind; nuclear 

    fission; biomass

    3. Which instrument is used to measure a wind energy? 

    4. What kind of energy will people be using in the future? Why?

    5. What are benefits of renewable energy?

    6. Why don’t people use more renewable energy now?

    7. Are there reasons to use more renewables now rather than wait until 

    the non-renewables run out?

  • Unit 7 : Projectile and uniform circular motion

    Key Unit Competence

    Analyse and solve problems related to projectile and circular motion.

    My goal

    By the end of this unit, I will be able to:

    * define and explain terms used in projectile motion.

    * discuss the different applications of projectile motion.

    * apply concepts of projectile and circular motion in real life.

    * differentiate between projectile motion and circular motion. 

    Introductory activity 

    Take the case of a basketball player shooting the ball into the basket’s net 
    from a given distance at a given angle as shown in the figure below then 

    answer to the following questions.

    C

    1.Discuss on the motion of the ball after being shot
    2.Discuss other situation where one can observe such a king of motion.
    3.Explain how this kind of motion is important in normal life activities.
    4.Contrast between projectile motion and circular motion using typical 

    examples.

    Introduction

    We have different kinds of sports, for examples; football, netball, tennis 
    amongst others. A lot of reasoning is needed while playing football to score 
    one of which is to kick a ball at a certain angle (i.e. to move above the ground). 
    We say that the ball is projected. This also applies to basket ball; the ball to 
    enter the round ring for a score it has to be thrown at a certain angle. Hence, 
    projected. The same principle is used by the military in shooting and launching 

    their missiles.

    Projectile Motion

    Activity 1: Field study

    Aim; To study motion of bodies in free space
    a) Out of class, (in pitch,or in school compound), throw a 
    ball,a stone or any body upward.
    b) State what happens.
    c) Hold a ball in your hands and release it to fall.
    d) Is the motion of the ball same as in the first case?
    e) Note down your observation.

    f) Relate your observations for bodies moving linearly. 

    Caution
    While throwing a stone or any body, take care so that it does not harm you.
    We can define a projectile as any body thrown into space/air. The path 
    taken is called a trajectory. The motion of a projectile unless taken otherwise 
    is a free motion under gravity. We assume that air resistance is negligible in 
    this kind of motion. 
    We have three cases: oblique projection, vertical projection and horizontal 

    projection. 

    Projection at an angle above the horizontal
    • Study the picture below carefully.
    • Go outside class and try to kick the ball so that it does not roll on the 
    ground.
    • State when will the ball cover a long horizontal range. (State down 

    the conditions for that to occur).

    C

    From the figure above, if the ball is kicked so that it does not roll on the 

    ground, it will move at certain angle relative to the ground. 

    Activity 2
    a) In the ground, kick the football individually.
    b) By observing, the flight of the ball state whether it will cover a 
    longer horizontal distance when it is projected at a large angle or a 
    small angle.
    c) Explain your observation and note down any key points in your 
    book.
    C
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    Upward projection

    From the figure above, a football player can kick the ball and it takes the 

    motion of a projectile.

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    Horizontal projection

    Activity 3

    Place a stone on top of a table.
    Displace it so that its motion takes the shape below.
    Try to observe the motion carefully.

    Note down what you observe and share it with your class members.

    Take care
    In throwing the stone / displacing it, you should take care so that it does not 

    hit you because it may harm you. 

    C

    C

    Activity 4
    Using the information given above;
    a) Derive the equations for the motion.
    b) Study the picture below and do the same.

    c) State the condition when the body attains maximum height.

    C

    Examples of application of projectile motions are as follow:

    • Football kicked in a game

    • A cannonball fired from a cannon.

    • A bullet fired from a gun

    • A disc thrown in the sport of discus throw.

    • The flight of golf ball.

    • A jet of water escaping a hose.

    • Motorcycles and cars jumping in extreme sports

    Application activity 7.1

    1. A gun has a muzzle velocity of (i.e. a shell leaves the gun with 
    an initial speed of ). Find the horizontal range of the gun when 
    the angle of projection is 300. Find also the maximum horizontal 

    range of the gun.

    2. A stone is thrown from the top of a cliff (as shown in figure below) 
    70 m high at an angle of 300 below the horizontal and the sea 20 
    m from the bottom of the cliff. Find the initial speed of the stone 

    and the direction in which it is moving when it hits the sea.

    C

    3. A man fires a rock out of a slingshot directly upward. The rock has 
    an initial velocity of . How long will it take for the rock to return 

    to the level he fired it at?

    Circular motion

    Activity 5

    Study carefully the motion of the ball shown below.
    State what would happen if at any point the thread holding the ball breaks?

    Note and record your observation.

    C

    A motion is said to be circular if the trajectory is a circle of constant 

    radius R.

    The motion is uniform if the body describes in equal angular displacements 
    in equal times.. Even if the motion is uniform, it has an acceleration because 

    the velocity changes after every moment since its direction keeps changing. 

    Angular displacement θ

    Definition of key terms in circular motion

    C

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    Centripetal acceleration

    As we said, in a circular uniform motion, there is acceleration. This acceleration 
    is called centripetal acceleration. The easiest way of proofing this formula is 

    as follow: 

    Consider an object moving with a constant speed (a scalar which has no 

    direction) round a circle of radius r.

    C

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    Periodic time, frequency

    Activity 6

    * Go to the play ground.

    * Make sure you round the playground two times.

    * Note and observe the time taken to make one complete revolution.

    * What do you call the time taken to move around the play ground.

    C

    From the activity 6, you made two rounds in a given time. The number of 

    rounds made in a Unit time is called frequency

    C

    In summary

    C

    Distance time-graph of a uniform circular motion

    When an object executes a circular motion of constant radius R, its projection 

    on an axis executes a motion of amplitude a that repeats itself back and forth, 

    over the same path.

    C

    Considering the displacement and the time, we find the following graph

    C

    (b) The speed of the child is most nearly 

    (A) 4 m/s

    (B) 12 m/s

    (C) 24 m/s

    (D) 120 m/s

    (E) 360 m/s

    Solution

    C

    2.A 150 g ball at the end of a string is revolving uniformly in horizontal 
    circle of radius 0.600 m; the ball makes 2.00 revolutions in a second. What 

    is its centripetal acceleration?

    Solution

    If the ball makes two complete revolutions per second then the Period is 

    C

    Centripetal force
    If you try to move / run in a circular path, you will finally notice that you keep 
    moving in a circle even when you try to stop. There is a force that keeps you 

    more in a circular path called centripetal force.

    Since a body moving in a circle (or a circular arc) is accelerating, it follows 
    from Newton’s first law of motion that there must be force acting on it to cause 

    the acceleration. 

    This force, like the acceleration, will also be directed toward the centre and 
    is called the centripetal force. The value F of the centripetal force is given by 

    Newton’s second law, that is:

    C

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    Example
    1. A 40 kg child sits on the edge of a carnival ride at a radius of 4 m. The 
    ride makes 3 revolutions in 6 s. The force which is holding the child on the 

    ride is most nearly

    (A) 30 N
    (B) 160 N
    (C) 320 N
    (D) 1440 N
    (E) 2880 N

    Solution

    C

    Application of circular motion

    Vertical and horizontal circle

    Vertical circle

    Taking the approach that the ball moves in a vertical circle and is not 
    undergoing uniform circular motion, the radius is assumed constant, but the 
    speed v changes because of gravity. Nonetheless, the formula of centripetal 

    acceleration is valid at each point along the circle, and we use it at point 1 and 

    2. The free-body diagram is shown in the figure 8.10 for the positions 1 and 2.

    C

    a) At the top (point 1), two forces act on the ball: the force of 
    gravity and the tension force the cord exerts at point 1. Both 
    act downward and their vector sum acts to give the ball its 
    centripetal acceleration. We apply Newton’s second law, for t 
    a vertical direction, choosing downward as positive since the 

    acceleration is downward (toward the centre):

    C

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    Example

    A puck of mass 0.500 kg is attached to the end of a cord 1.50 m long. The 
    puck moves in a horizontal circle as shown in the figure. If the cord can 
    withstand a maximum tension of 50.0 N, what is the maximum speed at 
    which the puck can

    move before the cord breaks?

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    Conical pendulum

    Activity 7

    DO THIS!

    * Tie a thread of about 50cm on retort stand. 

    * On a thread, tie a pendulum bob.

    * Displace the bob through a certain angle. 

    * Displace the bob through a certain angle. What do you observe.

    * Release the bob to move through a certain angle so that it moves in a 

    horizontal circle.

    * Try to investigate forces acting in the bob.

    * Relate your findings to fig. 8.13.

    A small object of massm is suspended from a string of length L. The object 
    revolves with constant speed v in a horizontal circle of radius r, as shown in 
    Figure 8.13. Because the string sweeps out the surface of a cone, the system is 

    known as a conical pendulum. 

    Let us find an expression for v.

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    Road banking 

    Circular motion on JOB

    Activity 8

    A car negotiating a corner

    C

    The successful negotiation of a bend on a flat road therefore depends on the 
    tyres and the road surface being in a condition that enables them to provide a 
    sufficiently high frictional force, otherwise skidding occurs. Safe cornering 

    that does not rely on friction is achieved by “banking” the road.

    C

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    The equation shows that for a given radius of bend, the angle of banking is 

    only correct for one speed.

    Spinning dryer is also known as tumble dryer is a powered household 
    appliance that is used to remove moisture from a load of clothing, bedding 

    another textures, usually shortly after they are washed in a washing machine.

    Example
    1. The car has a mass m and a speed v as it moves around the track 
    of radius R. Which of the following expressions can be used to find 

    the value of the coefficient of friction between the tires and the road?

    C

    Career centre 
    Learn more about careers in physics where projectile and circular motion are 

    applied.

    END UNIT ASSESSMENT

    1. A body is projected upwards from the level ground at an angle of 
    500 with the horizontal has an initial speed of 40 m/s. how long will 

    it be before it hits the ground?

    2. A body is projected downwards at an angle of 300 with the 
    horizontal from the top of a building 170m high. Its initial speed is 
    40m/s. 
    a) How long will it take before striking the ground?
    b) Find out how far from the foot of the building the body will 
    strike,

    c) What the angle with the horizontal?

    3. A body is projected from the ground at the angle of 300 with 
    the horizontal at an initial speed of 128m/s, ignoring air friction, 
    determine: 
    a) In how may seconds, it will strike the ground?
    b) How high it will go?

    c) What is its range will be?

    4. A ball is thrown upwards at an angle of 300 to the horizontal and 
    lands on the top edge of a building that is 20m away, the top edge is 

    5m above the throwing point. How fast was the ball thrown?

    5. A projectile is fired with initial velocity v0=95m/s at an angle . 
    After five seconds it strikes the top of hill. What is the elevation of 
    the hill above the point of firing? At what horizontal distance from 

    the gun does the projectile lands?

    6. 6. A ball is thrown from the top of the one building towards a tall 
    building 50m away. The initial velocity of the ball is 20m/s at 400 
    above the horizontal. how far above all below its original level, will 

    the ball strikes the opposite wall?

    7. A projectile is fired with horizontal velocity of 330m/s from the top 
    of a cliff 80m high. 
    a) How long will it take for the projectile to strike the level 
    ground at the base of the cliff?
    b) How far from the foot of the cliff will strike?

    c) With what velocity will it strike?

    8. A 0.3kg mass attached to 1.5m long string is whirled around the 
    horizontal circle at a speed of 6m/s.
    a) What is the centripetal acceleration of the mass?

    b) What is the tension of the string?

    9. (Moderate), a race car, moving at a constant tangential speed 
    of 60m/s, takes one lap around a circular track in 50 seconds. 

    Determine the magnitude of the acceleration of the car.

    10. An object that moves in uniform circular motion has a centripetal 
    acceleration of 13m/s2. If the radius of the motion is 0.02m, what is 

    the frequency of the motion?

    11. Find the centripetal acceleration for a n object on the surface of a 

    planet with the following characteristics: radius and 1day seconds.

    12. An 8.0g cork is swung in a horizontal circle with a radius of 35 cm. 
    it makes 30 revolutions in 12 seconds. What is the tension in the 

    string? (Assume the string in nearly horizontal).

    13. A 15g stopper is swung in a horizontal circle with a radius of 
    0.80meters. The tension in the string is 1.5Newtons. Find the speed 
    of the stopper and determine how long it takes to complete 30 

    revolutions. (assume the string is very nearly horizontally)

    14. A brass ball with a mass of 120 grams is suspended from that is 
    60.0 cm long. The ball is given a push and it moves in a horizontal 
    circle. The string is not nearly horizontal. It forms an angle of just 
    22.6 degrees from the vertical. (this is sometimes called a conical 
    pendulum because the string sweeps out the surface of a cone.
    a) Draw free body diagram indicating the forces acting on the 
    ball.
    b) What is the y-component of the tension force equal to? how 
    do you know?
    c) Use trigonometry to find the x-component of the tension 
    force.
    d) What is the radius of the ball’s motion?

    e) Use your answer to c and d to find the speed of the ball 

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  • Unit 8 :Universal gravitational field and potential

    Key Unit Competence

    Explain the gravitational field, gravitational potential and their applications in 

    planetary motion.

    My goals

    By the end of this unit, I will be able to: 

    * explain universal gravitation field.

    * describe the factors affecting force of gravity.

    * state and explain Kepler’s laws of planetary motion.

    * investigate planetory motion using computer simulation.

    Link to other subjects

    Geography and Astronomy (Landslides, motion of planets and satellites) 

    Chemistry (Electrons orbiting the nucleus).

    Introductory activity

    In observatory, people can see different heavenly bodies such as planets, 
    stars, moons, asteroids, comets and others moving in the universe. The 
    most question that people ask themselves is to know what hold those bodies 

    in their positions in the universe.

    Questions

    1. Discuss on what you think maintain planets in their position when 
    revolving around the Sun.
    2. Discuss the factors that the motion planets depend on.
    3. Give other examples of bodies that use same properties in the case 

    study above.

    Introduction

    The Universe is composed of different planets one of which is the earth.

    All objects on the earth remain on it. They cannot move away unless acted on 
    by external forces. This shows that there is a region around it that provides a 

    force that attracts these earthly objects.

    Since the earth is part of the universe it follows that a round the universe there 

    is attracting field.

    This is called universal gravitational field.

    Universal gravitational field potential

    To have potential is to have energy, therefore gravitational field potential is the 

    ability of gravity to attract other objects.

    Gravitational field

    Questions to think about!

    1. What force that unites us as Banyarwanda?
    2. How do you feel if you come close to a fellow munyarwanda when 
    you find him/her outside our country?
     Relate the situation to the force around the earth.

    3. What makes you feel attracted to your fellow munyarwanda?

    A field is a region of space where forces are exerted on objects with certain 

    properties.

    The diagram represents the Earth’s gravitational field. The lines show the 

    direction of the force that acts on a mass that is within the field.

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    This diagram shows that:

    • Gravitational forces are always attractive – the Earth cannot repel 
    any objects.
    • The Earth’s gravitational pull acts towards the centre of the Earth.
    • The Earth’s gravitational field is radial; the field lines become less 

    concentrated with increasing distance from the Earth.

    The force exerted on an object in a gravitational field depends on its position.

    The less concentrated the field lines, the smaller the force. If the gravitational 

    field strength at any point is known, then the size of the force can be calculated.

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    Gravitational potential energy

    Potential and potential energy

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    Question about fig. 8.3

    • The car at the top of the hill has more potential energy than the 
    one at the bottom, but relative to ground level they both have zero. 

    why?

    • Note and record in your notebook your analysis.

    Using this reference point:

    • All objects at infinity have the same amount of potential energy; zero.
    • Any object closer than infinity has a negative amount of potential 
    energy, since it would need to acquire energy in order to reach 

    infinity and have zero energy.

    The gravitation potential energy is defined as the energy possessed by object 

    because of its position in a gravitational field.

    The gravitational potential at a point in a gravitational field is the potential 

    energy per unit mass placed at that point, measured relative to infinity.

    Calculating potential and potential energy

    When an object is within the gravitational field of a planet, it has a negative 
    amount of potential energy measured relative to infinity. The amount of 
    potential energy depends on:
    • The mass of the object.
    • The mass of the planet.
    • The distance between the centres of mass of the object and the 

    planet.

    The Centre of mass of a planet is normally taken to be at its centre.

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    Kepler’s Laws

    Activity1: Field work

    As a class, let us visit one of the roundabouts (where three roads meet).
    Try to see/check how cars, motorcycles, bicycles move around it.

    Qn i) Does the features on a roundabout move?

    Assuming a roundabout to be a sun and vehicles to be planets, what can 
    you say?
    1. Discuss your findings in groups of 5 members.
    2. Present your findings to the whole class.
    3. Note down the observation.

    4. Present your work to the teacher for marking.

    Activity 2

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    We can relate the movement of the minute hand as the movement of 

    planets about the sun.

    Kepler’s first law: The path of each planet about the sun is an ellipse with the 

    sun at one focus(or planets describe ellipse about the sun as one focus).

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    Kepler’s second law: The line joining the sun to the moving planet sweeps 

    out equal areas in equal times.

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    Proof of Kepler’s third law

    Activity 3

    • Using Newton’s law of gravitation (Formula) and the formula that 
    keeps the planet in circular paths (Formula for centripetal force), 

    Derive expression for Kepler’s third law of planetary motion

    • Put your derivation in your notebook after discussing it with your 

    friends.

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    Examples

    1. Calculate the force of gravity between two bowling balls each 

    having a mass of 8.0kg, when they are 0.50m apart.

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  • Unit 9 : Electric field and electric potential

    Key Unit Competence

    Analyze the electric field and electric potential. 

    My goals

    By the end of this unit, I will be able to:
    * define electric field and electric potential.
    * explain the relationship between electric potential and electric field 
    intensity.
    * describe functioning of lightening arrestors.

    * identify the dangers of lightening and how to avoid them.

    Introductory activity

    When you pass nearby an electric power station, you may notice some sign 

    post as shown in figure below.

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    This is to show that, the high voltage on the grid may affect people who 

    dares to approach it.

    Questions

    1. Discuss about the reasons behind the information given by the sign post.
    2. Is it possible that electricity can affect someone without touching the 
    wires (cables)?
    3. Discuss other areas where you can get such sign post and explain the 

    reason behind.

    Introduction

    Have you ever heard sound due to lightening? If yes, what do you think was 
    the cause?

    If not, ask your friend in your class, at home, or neighbour about lightening.

    Scientifically, lightening and thunder are effects of electric charges created in 

    space (will be discussed later).

    Attraction and repulsion of charges

    Activity 1

    In this section, you will observe the characteristics of the two types of 
    charges, and verify experimentally that opposite charges attract and like 

    charges repel.

    Equipment
    * Two lucite rods
    * One rough plastic rod
    * Stand with stirrup holder
    * Silk cloth

    * Cat’s fur

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    Procedure

    1. Charge one lucite rod by rubbing it vigorously with silk. Place the 
    rod into the stirrup holder as shown in Figure.

    2. Rub the second lucite rod with silk, and bring it close to the first 

    rod

    3. What happens? Record the observations in your notes.

    4. Rub the rough plastic rod with cat’s fur, and bring this rod near the 

    lucite rod in the stirrup. Record your observations.

    5. What do you conclude?

    6. Note down observation in your notebook.

    For reference purposes, according to the convention originally chosen by 
    Benjamin Franklin, the lucite rods rubbed with silk become positively charged, 
    and the rough plastic rods rubbed with cat’s fur become negatively charged. 

    Hard rubber rods, which are also commonly used, become negatively charged.

    Coulomb’s law

    Activity 2 

    Materials
    * Coulomb’s Law apparatus
    * Electrophorus (The electrophorus is a simple electrostatic induction 
    device. It’s an inexhaustible source of charge”).
    * Silk cloth.

    * A computer for the graph and quick calculation.

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    Procedure

    1. Take a moment to check to position of the hanging ball in your 
    Coulomb apparatus. Look in through the side plastic window. The 
    hanging ball should be at the same height as the sliding ball (i.e. 
    the top of the mirrored scale should pass behind the centre of the 
    hanging pith ball, as in Figure 10.3 below). Lift off the top cover 
    and look down on the ball. The hanging ball should be centred on 
    a line with the sliding balls. If necessary, adjust carefully the fine 

    threads that hold the hanging ball to position it properly.

    2. Charge the metal plate of the electrophorus in the usual way by 
    rubbing the plastic base with silk, placing the metal plate on the 

    base, and touching it with your finger.

    3. Lift off the metal plate by its insulating handle, and touch it 

    carefully to the ball on the left sliding block.

    4. Slide the block into the Coulomb apparatus without touching the 
    sides of the box with the ball. Slide the block in until it is close to 
    the hanging ball. The hanging ball will be attracted by polarization, 
    as in Section III of this lab. After it touches the sliding ball, the 
    hanging ball will pick up half the charge and be repelled away. 
    Repeat the procedure if necessary, pushing the sliding ball up until 

    it touches the hanging ball.

    5. Recharge the sliding ball so it produces the maximum force, and 
    experiment with pushing it towards the hanging ball. The hanging 

    ball should be repelled strongly.

    6. You are going to measure the displacement of the hanging ball. You 
    do not need to measure the position of its centre, but will record 
    the position of its inside edge. Remove the sliding ball and record 
    the equilibrium position of its inside edge that faces the sliding 
    ball, which you will subtract from all the other measurements to 

    determine the displacement d.

    7. Put the sliding ball in, and make trial measurements of the inside 
    edge of the sliding ball and the inside edge of the hanging ball. The 
    difference between these two measurements, plus the diameter of 
    one of the balls, is the distance r between their centres. Practice 
    taking measurements and compare your readings with those of your 
    lab partner until you are sure you can do them accurately. Try to 

    estimate measurements to 0.2 mm.

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    8. Take measurements, and record the diameter of the balls (by 

    sighting on the scale).

    9. Remove the sliding ball, and recheck the equilibrium position of 

    the inside edge of the hanging ball.

    10. You can record and graph data in Excel or by hand (although if 
    you work by hand, you will lose the opportunity for 2 mills of 
    additional credit below). Recharge the balls as in steps 1 – 4, 
    and record a series of measurements of the inside edges of the 
    balls. Move the sliding ball in steps of 0.5 cm for each new 

    measurement.

    11. Compute columns of displacements d (position of the hanging ball 
    minus the equilibrium position) and the separations r (difference 
    between the two recorded measurements plus the diameter of one 

    ball).

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    Knowledge of the forces that exist between charged particles is necessary for 
    an understanding of the structure of the atom and of matter. The magnitude of 
    the forces between point charges was first investigated quantitatively in 1785 

    by Coulomb, a French scientist. The law he discovered is stated as follows:

    “The force between two point charges is directly proportional to the 

    product of charges divided by the square of their distance apart”.

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    Electric field

    Notions and definitions

    Questions to think about 

    a) You have learned about Coulomb’s law and you have seen 
    that when an electric charge is brought near to another, there 
    is an attractive or a repulsive force. Does that force acts when 

    charges are in contact or it acts even at a certain distance?

    b) If so, what can be the reason?

    c) Does that force increase or decrease when the distance 

    between charges increases? 

    After responding to those questions, you’ll see that around an electric charge 
    is a region so that when another charge is placed in it, it undergoes an electric 

    force. That region is called electric field created by the first charge. 

    An electric field can be defined as a region where an electric force is obtained. 

    It’s a region where an electric charge experiences a force.

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    Field lines (lines of force)

    Activity 4: Lab zone 

    Existence of field lines 

    This shows the shape of electric fields, in much the same way that magnetic 

    fields are demonstrated with iron filings.

    Materials
    * Power supply, EHT, 0-5kV.          * Electric fields apparatus.

    * Semolina.                                          * Castor oil.

    Procedure

    a) Fill the electrode unit with a layer of castor oil to a depth 
    of about 0.5cm. Sprinkle a thin layer of semolina over the 
    surface. (A thin piece of glass tubing drawn out to give a 
    fine pointed stirrer is helpful so that the semolina is evenly 
    distributed.) It is better to start with too little semolina than 
    to start with too much. You can always increase the quantity 

    later.

    b) Place the electrodes in the castor oil. Connect the positive 
    and negative terminals of the EHT power supply to the 
    electrodes. Adjust the supply to give 3,000 to 4,000 volts. 
    When the voltage is switched on, the field lines will be 

    clearly visible.

    c) Try electrodes of different shapes. For example, one can be 
    a ‘point’ electrode whilst the other is a plate, or two point 
    electrodes can be used. A wire circular electrode with a 
    point electrode at the centre will show a radial field. The 
    field with two plates quite close together should also be 

    shown.

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    A line of force or field lines is defined as a line such that the tangent to it at a 

    point is in the direction of force on a small positive charge placed the point. 

    Arrows on the lines of force show the direction of the force on a positive 

    charge; the force on a negative charge. 

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    Electric field due to a distribution of electric charges

    Activity 5

    Electric field due to a distribution of charges

    Materials
    * A sheet of paper
    * A pen

    * A ruler

    Procedure
    1. Represent a distribution of charges where you have charges of 
    different signs.
    2. Represent a point A where you want to find the total electric field.
    3. At the point, A represents directions of electric fields vectors 
    produced by each charge.
    4. Do the sum of electric fields. Remember that an electric field is 
    a vector. When they make a certain angle between them, use the 
    method of parallelogram. When they have the same direction or 
    opposite directions, use the appropriate method. 
    5. Establish a mathematical relation of the total electric field due to the 

    distribution of charges.

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    Potential difference

    Work of electric force

    Activity 6

    Find the expression of the work done by an electric force

    a) When can we say that we have a uniform electric field?

    b) Draw a diagram showing two plates of opposite signs (the left 

    plate is positive and the right one is negative) between which the 

    electric field is uniform.

    c) Show the direction of field lines in the electric field.

    d) Between the two plates, put a positive charge at a point A which 

    has to travel toward a point B in the field.

    e) Represent the direction of the vector force on the line joining A and B.

    f) Write down the expression of the force undergone by the charge.

    g) What is the expression of the work done if the charge has to move 

    from A to B (in the final formula)?

    Particles that are free to move, if positively charged, normally tend towards 
    regions of lower voltage (net negative charge), while if negatively charged 

    they tend to shift towards regions of higher voltage (net positive charge).

    However, any movement of a positive charge into a region of higher voltage 
    requires external work to be done against the field of the electric force, work 
    equal to that electric field would do in moving that positive charge the same 
    distance in the opposite direction. Similarly, it requires positive external work 
    to transfer a negatively charged particle from a region of higher voltage to a 

    region of lower voltage.

    The electric force is a conservative force: work done by a static electric field is 
    independent of the path taken by the charge. There is no change in the voltage 
    (electric potential) around any closed path; when returning to the starting 

    point in a closed path, the net of the external work done is zero. 

    Potential in a field

    Activity 7

    Understanding the potential in a field
    1. What kind of energy has a body when it’s held above the earth? If 
    the body has to move under the force of gravity, does it move from 

    a point of great height to one of less or it’s the inverse?

    2. Do you agree or not that points in the earth’s gravitational field 

    have potential values depending on their heights?

    3. According to you, can this theory be similar to the one established 

    for electric field? Explain.

    4. For charges, instead of saying gravitational potential for 
    gravitational field, can we say electric potential for the case of 

    electric field? Explain.

    5. Can points around the charge be said to have electric potential?

    6. How can we define the electric potential at a point? 
    Potential generally refers to a currently unrealized ability. The term is used in 
    a wide variety of fields, from physics to the social sciences to indicate things 
    that are in a state where they are able to change in ways ranging from the 

    simple release of energy by objects to the realization of abilities in people.

    Although the concept of electric potential is useful in understanding electrical 
    phenomena, only differences in potential energy are measurable. If an electric 
    field is defined as the force per unit charge, then by analogy an electric potential 
    can be thought of as the potential energy per unit charge. Therefore, the work 
    done in moving a unit charge from one point to another (e.g., within an electric 

    circuit) is equal to the difference in potential energies at each point.

    Potential difference, work, energy of charges

    Activity 8

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    Electric potential is a location-dependent quantity that expresses the amount 
    of potential energy per unit of charge at a specified location. When a Coulomb 
    of charge (or any given amount of charge) possesses a relatively large quantity 
    of potential energy at a given location, then that location is said to be a location 
    of high electric potential. And similarly, if a Coulomb of charge (or any given 
    amount of charge) possesses a relatively small quantity of potential energy 
    at a given location, then that location is said to be a location of low electric 
    potential. As we begin to apply our concepts of potential energy and electric 
    potential to circuits, we will begin to refer to the difference in electric potential 

    between two points.

    Relation between E and V

    Activity 9

    Relation between E and V

    1. What is the relation to find the work done by an electric force to 

    move a charge from A to B, knowing that the distance between A 

    and B is d?

    2. What is the relation of the work using the potential difference?

    3. Equalize the two relations and deduce the value of E. The relation 

    found is the one between E and V.

    4. From the expression found, deduce the new unit of the electric field E.

    5. Write down the relation between E and V found, express in 
    equation of V, write the electric field produced by a charge at a 
    point deduce the electric potential created by a charge at a point 

    situated at a distance d from it. 

    The effect of any charge distribution can be described either in terms of 
    electric field or in terms of electric potential. Electric potential is often easier 
    to use since it is a scalar whereas electric field is a vector. There is an intimate 
    connection between the potential and the electric field. Let us consider the 
    case of a uniform electric field, such as that between the parallel plates (fig.) 

    whose difference of potential is Vba.

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    Examples

    In figure below, the potential difference between the metal plates is 40v. 
    a) Which plate is at the higher potential?
    b) How much work must be done to carry a +3.0C charge from B to 
    A? from A to B?
    c) How do we know that the electric the electric field is in the 
    direction indicated? 

    d) If the plate separation is 5.0mm. what is the magnitude of E?

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    Motion of electric charges in an electric field

    Activity 10

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    There are so many applications of cathode ray tube which is a practical example 
    of the motion of electrons in an electric field in daily life. For example TV 

    sets, oscilloscope, etc. use cathode ray tubes. 

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    Lightening and lightening arrestor 

    Activity 11 

    Lightening and lightening arrestor

    a) Surely, you have heard a thunder before the rainfall. What do 
    you observe in the sky during it?
    b) According to you, this is due to what?
    c) Is the fact observed dangerous?
    d) If yes, do you know some consequences which you have 
    observed or heard?
    e) If yes, is there a way to be protected from it?
    f) Do some research on internet to know more about it and 

    submit the result of your research to the teacher.

    Some explanation

    What you observe is called Lightening which is a sudden electrostatic 
    discharge (the sudden flow of electricity between two electrically charged 
    objects caused by contact, an electrical short, or dielectric breakdown) during 
    an electrical storm between electrically charged regions of a cloud (called intra-
    cloud lightening or IC), between that cloud and another cloud (CC lightening), 
    or between a cloud and the ground (CG lightening). The charged regions in the 
    atmosphere temporarily equalise themselves through this discharge referred 
    to as a strike if it hits an object on the ground. Although lightening is always 
    accompanied by the sound of thunder, distant lightening may be seen but be 
    too far away for the thunder to be heard. Lightening strikes can be damaging 

    to buildings and equipment, as well as dangerous to people.

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    Buildings often use a lightening protection or lightening rod system consisting 
    of a lightening rod (also called a lightening conductor) and metal cables to 
    divert and conduct the electrical charges safely into the ground. Another form 
    of lightening protection system creates a short circuit to prevent damage to 
    equipment. The electrically conducting metal skin of commercial aircraft is 

    isolated from the interior of to protect passengers and equipment.

    Often, the lightening protection is mounted on top of an elevated structure, 
    such as a building, a ship, or even a tree, electrically bonded using a wire 
    or electrical conductor to interface with ground or “earth” through an electrode, 
    engineered to protect the structure in the event of lightening strike. If lightening 
    hits the structure, it will preferentially strike the rod and be conducted to the 
    ground through the wire, instead of passing through the structure, where it 
    could start a fire or cause electrocution. Lightening rods are also called finials, 

    air terminals or strike termination devices.

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    In a lightening protection system, a lightening rod is a single component of 
    the system. The lightening rod requires a connection to earth to perform its 
    protective function. Lightening rods come in many different forms, including 
    hollow, solid, pointed, rounded, flat strips or even bristle brush-like. The main 
    attribute common to all lightening rods is that they are all made of conductive 
    materials, such as copper and aluminum. Copper and its alloys are the most 

    common materials used in lightening protection.

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  • Unit 10 :Applications of thermodynamics laws

    Key unit Competence

    Evaluate the applications of first and second laws of thermodynamics in real 

    life.

    Unit goals

    By the end of this unit, I will be able to:

    * differentiate between Internal energy and total energy of a system.
    * explain the work done by the expanding gas.
    * state the first law of thermodynamics.
    * state the second law of thermodynamics.

    * explain thermodynamic processes in heat engines.

    Introductory activity

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    Mutesi is a parent of two children at a certain school. Before she takes them 
    to school, she first makes sure that she prepares food and drinks for them 
    and packs some in flasks so that her children can eat and drink during lunch 

    time. 

    She then drives them to school before she reports to her working place and 
    then from the school she then diverts to her working place which is about 5 

    km away from the school.

    The parking yard at her work place is a plain place without any shade but 
    she makes sure that her car is parked near a tree that is near the parking yard 
    to prevent it from different damages among which is destruction of tyres of 

    the car.

    a) Explain why Mutesi makes use of flasks not normal utensils like metallic 

    bowels while parking foods and drinks for her children.

    b) Is there heat exchange inside the flasks? Explain your reasoning.

    c) Imagine on a certain day these two children only eat food and leaves, 
    the drink in the flask and by mistake they forget flask in the store and 
    the mother come to pick it the next day. Do you think the contents in the 
    flask will be at the same temperatures? Explain all scientific phenomena 

    that may lead to either loss or gain in energy of the contents in the flask.

    d) Explain why in most cases the outer covering of a flask is always made 
    of a poor conductor? Explain how quality and efficiency of these flasks 

    can be improved by manufactures.

    e) Based on statements above, Mutesi normally parks her car under a shade 
    to prevent her car from being exposed to sunshine. Explain how during 

    hot days the tyres of a car may burst.

    f) Her Car uses petrol in operation. During operation of her car, the engine 
    draws fuel (Petrol) air mixture from the tank into the engine, explain all 

    the processes that take place in the engine.

    Introduction

    Before, you learnt that:
    • Heat is a form of energy.

    • Heat can be changed / transformed from one form to another.

    So, if in a system heat changes from one form to another, its called thermal 

    dynamic system.

    The systems to discuss in this unit include refrigerators, heat pumps, car engines. 
    Remember that heat is the measure of total internal energy of a body. This means 

    that particles of a body vibrate because of energy they have.

    Thermal energy and internal energy

    Activity 1
    Have you ever boiled water on a sauce pan with a cover?

    Describe what happens to the cover when water boils?

    When water boils, the vapour pushes the cover off the sauce pan. You have 
    already seen in your early secondary that heat is a form of energy. Therefore, 
    when this saucepan is heated, the heat gained is used to boil off the water and 
    extra work is done to push the sauce pan cover. This total heat energy supplied 

    is called thermal energy.

    Science in action! Discover
    • Explain why an inflated bicycle tube bursts when it is left on sunshine for 
    a very long time.
    • Similarly explain why a balloon full of air bursts as it rises in the 
    atmosphere.

    • Note down your observation in your exercise books.

    You already know the characteristics of the three states of matter that is; solids, 
    liquids and gases. In this unit, we shall be interested in studying the behaviour 

    of molecules in matter.

    When the bicycle tube is left exposed to sunshine, it gets heated and the 
    molecules in the gas gain energy and hence its kinetic energy increases. As 
    a result, they collide frequently with the walls of the tube and therefore exert 

    high pressure on the walls and the tube bursts.

    C

    The same thing happens with the balloon in air.

    The energy possessed by the molecules of the gas is called internal energy 
    of the gas. This energy depends on the temperature of the gas. When a gas is 
    heated its temperature increases and hence the average speed of molecules 
    also increases increasing the internal energy of the gas. Further increase of 
    heat supplied means that extra energy is absorbed by the molecules of the gas, 

    hence expanding and pushing the tyre. As a result the tyre bursts.

    The internal energy is defined as energy associated with random disordered 

    motion of particles.

    Activity 2

    List down three utensils used for cooking food in your homes.
    Describe how these utensils are used to cook the food. Are they always left 

    open while cooking?

    C

    In all the above, there exists energy exchanges and such things are called 
    systems. Systems can either be closed or open. When water is being boiled in 
    an open sauce pan, vapour is allowed to escape. It is an example of an open 
    system. When someone cooks meat using a closed container, no gas is allowed 

    to escape. Its an example of a closed system.

    Whenever heat flows to or from a system, or work is done on or by a system, 
    there is a change in the energy of this system. The study of the processes that 

    cause these energy changes is termed thermodynamics.

    Thermodynamic systems

    Heat is the energy that flows by conduction, convection or radiation from 
    one body to another because of a temperature difference between them. These 
    bodies where exchange of heat to other forms of energy occurs are called 

    thermodynamic systems.

    A thermodynamic system consists of a fixed mass of matter, often a gas, 
    separated from its surroundings, perhaps by a cylinder and a piston. For example 
    heat engines such as a petrol engine, a steam turbine and jet engine all contain 
    thermodynamic systems designed to convert heat into mechanical work. Head 
    pumps and refrigerators are thermodynamic devices for transferring heat from 

    a cold body to a hotter one.

    A thermodynamic system is any object or set of objects that we wish to 
    consider. Everything else in the universe we will refer to as its environment or 
    the surroundings. A system is separated from the remainder of the universe by 

    a boundary. The boundary may be at rest or in motion.

    C

    A system can be homogeneous or heterogeneous. It can be gaseous, liquid or 
    solid state. A system is in equilibrium when its properties do not change with 

    time.

    A closed system is one for which no mass enters or leaves but may enter heat 
    exchanged with the environment. A closed system is sometimes referred to as 
    a control mass because the matter composing the system is assumed known 
    for all time. A closed system is said to be isolated if no energy in any form 

    passes across its boundaries, otherwise it is not isolated.

    In an open system, mass may enter or leave as well as energy. An open system 
    is sometimes referred to as a control volume because the location composing 
    the system is assumed known for all time. The surface surrounding the control 
    volume is sometimes known as a control surface. The control surface can be 
    along a real surface of the system or it can be an imaginary surface chosen for
    convenience. In general, a control mass can change shape and volume, but a 

    control volume cannot. 

    In such devices, energy is transferred from one system to another by a force 

    moving its point of application in its own direction.

    The energy of a system, whether transferred to it as heat or work is termed as 

    the internal energy of the system.

    When there is no heat transfer between two systems, that is, the two are at the 

    same temperature, they are said to be in thermal equilibrium.

    Activity 3

    Have you ever observed smoke moving in the atmosphere.
    Move outside class and go towards the kitchen and observe how smoke is 
    moving. Describe briefly how it moves.

    Why does it move like that?

    You have already seen in your early secondary that molecules in a gas are 
    more further apart and are always in constant random motion while moving at 
    high speed colliding with one another and the walls of the container, and when 

    the gas is heated their speed increases.

    Smoke particles are always in random motion and when they are moving in 
    air, they collide with air molecules and a zigzag pattern is seen.
    Similarly, when smoke is put in a container and then closed, the particles are 

    seen to be in a random motion. Smoke is an example of a real gas.

    In thermodynamics, we are mainly interested in ideal gas. At higher 

    temperatures, a real gas behaves like an ideal gas.

    Activity 4

    Have you ever heard of an ideal gas?

    What are the differences between a real gas and an ideal gas?

    When a gas is heated, molecules move further apart and the forces of attractions 

    between them become negligible and the gas becomes ideal.

    When the molecules become further apart, the gas expands and the volume of 
    the individual molecule becomes so small compared to the entire volume of
    the gas. It therefore becomes negligible compared to the volume of the gas and 

    the gas becomes ideal.

    When the molecules are colliding with one another, collisions are assumed to 
    be perfectly elastic. In this case, the gas becomes ideal because for a real gas 

    we expect to have time between approach and separation during collision.

    Work done by an expanding gas

    Activity 5: Discover 

    Explain why a pump gets hot when one pumps air into a tyre.

    When you compress air in a bicycle pump, your muscles transfer energy 
    to the handle, which in turn transfers energy to the molecules of air in the 
    pump. This additional energy makes the molecules move faster. As they are 
    compressed into a smaller space, they also collide more often with the wall of 

    the pump, so they transfer more energy to the metal wall and it becomes hot.

    We have already seen how heat can be transferred, so you probably have a 
    good idea what Q means. Work is simply a force Multiplied by distance in the 

    direction of force.

    A gas can be heated by compressing it, for example with a bicycle pump. 
    Hence the temperature of the gas can be raised either by doing work in 
    compressing it or by heating it. Likewise the temperature can be lowered by 

    either making the gas do work in expanding or by extracting heat from it.

    C

    C

    Example

    Steam to drive an old-fashioned steam locomotive is supplied at a constant 
    gauge pressure of 1.75×106N/m2 about 250 psi) to a piston with a 0.200-

    m radius.

    Find the work done by the steam when the piston moves 0.800 m.

    Solution

    C

    Specific heat capacities of gases

    Gases are considered to have a number of specific heat capacities. A change in 
    temperature of a gas is likely to cause large changes in pressure and volume of 

    the gas but for solids or liquids, the change in pressure is neglected.

    c

    C

    The reason as to why this is so can be done by considering cylinders in (a) and 
    (b) each initially containing one mole of gas at temperature T and pressure, 
    P. The piston in (a) is fixed and that in (b) is frictionless and can move freely 
    but has a constant force applied to it. If heat is supplied to each until the 
    temperature has risen by one Kelvin, the increase of internal energy must be 

    the same in each case (Since the temperature rise is the same).

    All the heat supplied in case (a) is used to increase the internal energy of the 
    gas. In (b), however, the gas expands and work is done by it on the piston; the 
    heat supplied in this case equals the increase of internal energy plus the work 

    done in the expansion of the gas. 

    Application activity 10.1

    1. Which statement best describes the concept of a system?
    A.the subject of the analysis                               C. a fluid-solid mixture
    B.an object with fixed set of molecules         D. an object that radiates heat 

    into its environment

    2. Consider a refrigerator in a kitchen. Take the refrigerator and everything 
    in it to be our system. Which best describes the system's surroundings?
    A.All of the air in the kitchen.                        C. The air inside the refrigerator
    B.Any one standing in the kitchen.             D. Everything in the universe 

     external to the system.

    3. Consider a refrigerator in a kitchen. Take the refrigerator and everything 
    in it to be our system. Which best describes the system's boundary?
    A.All of the air in the kitchen.
    B.The thin region separating the system from everything else.
    C.The air inside the refrigerator.

    D.The outer walls of the refrigerator.

    4. Which of the following statements are true?
    A.A closed system is necessarily an isolated system
    B.An isolated system is necessarily a closed system
    C.A system cannot be both closed and isolated.
    D.The concepts of closed and isolated in regards to a system are independent 

    concepts.

    5. True or False: A system is in steady state if its properties are independent 

    of time.

    The first law of the thermodynamics 

    The first law of thermodynamics states that” in a closed system the energy 
    is conserved in any transfer of energy from one form to another. Means that 
    the change in internal energy of a system is equal to the heat added to the 
    system minus the work done by the system”     this law is known as the law of 

    energy conservation.

    C

    C

    Example

    Compute the internal energy change and temperature change for the two 

    processes involving 1 mole of an ideal monatomic gas.

    a) 1500 J of heat are added to the gas and the gas does no work and 

    no work is done on the gas 

    b) 1500 J of work are done on the gas and the gas does no work and 

    no heat is added or taken away from the gas

    Solution

    C

    C

    C

    Applications of first law of thermodynamics in particular 

    gas changes 

    The first law of thermodynamics that we discussed relates the changes in 
    internal energy of a system to transfers of energy by work or heat. In this case, 
    we consider applications of the first law in processes through which a gas is 

    taken as a model.

    Isovolumetric process (Isochoric process)

    Activity 6

    (i) Have you ever heard of an isovolumetric or isochoric process?

    Study the Figure and answer questions that follow;

    C

    (ii) What substance is likely to be getting cooked using the can on the 
    left? Give a reason for your answer.
    (iii) Why is the can covered and not open?
    (iv) If one tried to open it while its on fire, what do you think would 

    happen? 

    A process that takes place at constant volume is called an isovolumetric

    process.

    C

    From the figure 10.7, process AB takes place at a constant volume (volume 

    doesn’t change).

    In such a process, the value of the work done is zero because the volume does 

    not change. Hence, from the first law we see that in an isovolumetric process, 

    W = 0 and ΔU = Q (isovolumetric process) 

    Note:

    • This expression specifies that if energy is added by heat to a system 
    at constant volume, then all of the transferred energy remains in the 

    system as an increase in its internal energy. 

    • For example, when a can of spray paint is thrown into a fire, energy 
    enters the system (the gas in the can) by heat through the metal walls 
    of the can. Consequently, the temperature, and thus the pressure in 

    the can increases until the can possibly explodes.

    Isobaric process

    Activity 7

    C

    Boiling liquids in open containers is very safe for example if the container is 
    closed, pressure may build up in the container and force it to burst. Boiling in 
    open containers imply that the pressure of the substance is kept constant. This 
    process is called an isobaric process. An isobaric process is the one that occurs 

    at constant pressure.

     Heating of water in an open vessel and the expansion of a gas in a cylinder 
    with a freely moving piston are typical examples of isobaric processes. In both 
    cases, the pressure is equal to atmospheric pressure. For example when water
    is being heated, its volume increases and the pressure inside the container is 
    constant since the number of collisions between water molecules and the walls 

    of the container is constant.

    The same process occurs when a gas enclosed in a cylinder with a frictionless 
    piston is heated such that at any time, the gas pressure equals the external 

    pressure. 

    C

    In this process, the energy supplied is used to increase the internal energy 

    since the internal energy is independent of the volume.

    C

    Isothermal change (constant temperature)

    Activity 8

    (i) Get a polythene bag and fill it with air.

    (ii) Insert a thermometre in the bag and place it in the ice-water mixture.

    (iii) Note what happens.

    Do you notice that the gas condenses and the volume decreases?

    What happens to the temperature recorded by the thermometer?

    You can notice that the temperature remains constant. This change is called 

    Condensation and is an example of isothermal process. 

    Do you think this process is reversible?

    An isothermal change can be reversible. An isothermal change is the change 
    that occurs at constant temperature. It is either a compression or expansion of 

    a gas at a constant temperature. 

    If the volume increases, the pressure must decrease and if the volume 

    decreases, the pressure must increase

    C

    C

    Conditions necessary for an isothermal process to occur

    Activity 9: Discover

    (i) On a cold day, how do you keep yourself warm?

    (ii) In groups of five, describe how you can keep the temperature of the 

    system constant.

    For an isothermal process to take place, the gas must be contained in a thin 
    –walled heat conducting vessel/container in good thermal contact with a 

    constant temperature. 

    The process must be carried out slowly to allow time for heat exchange to take 

    place.

    Work done in Isothermal Change

    Activity 10: Science at work

    (i) Have you ever tried to boil water in a closed sauce pan?

    (ii) What happens to the cover when the vapour starts to come off the water?

    (iii) Notice that this vapour pushes the cover off the pan.

    C

    C

    C

    Comment.
    The answer has a negative value. This shows that the work is done 

    on to the gas (compressed).

    Adiabatic change 

    Activity 11

    (i) Pump a bicycle tyre using a pump until it is full.
    (ii) Open the tube slowly while placing your other hand in its path.
    (iii) Do you notice that the the air coming out of the tyre is hotter than 

    the surrounding air? 

    As one pumps, the air molecules are compressed into a smaller space. They 
    also collide more often with the wall of the pump, so they transfer more energy 
    to one another and become hot. No heat has been supplied to the system. It is 

    called an adiabatic compression.

    Activity:12

    (i) Now pump the tyre and leave it standing for sometime.
    (ii) Make sure you don’t expose it to sun shine.
    (iii) Open the valve after two hours while your hand is placed in the path 

    of air from it.

    Do you notice that the air is colder than its surrounding?
    Heat has been lost but not to the surroundings. When the air is left standing, 
    expansion occurs. This is associated with a decrease in temperature. It is called 

    an adiabatic expansion.

    An adiabatic change is process in which no heat enters or leaves the gas 

    system. It is either an expansion or a compression.

    C

    If the gas expands, it does work, its internal energy is reduced and hence the 

    temperature is lowered.

    If the gas is compressed, work is done on the gas, its internal energy will 

    increase and therefore its temperature rises.

    C

    Conditions that are necessary for an adiabatic change to 

    occur

    Activity 13

    How do you always protect yourself from a bad weather?

    On a cold day, we always wear woolen jackets to protect ourselves from 
    coldness. Therefore no heat is either lost to the surrounding and or gained. In 

    this case, an adiabatic process is achieved.

    For an adiabatic process to be achieved, the gas must be contained in a thick 

    –walled and perfectly insulated isolated container.

    The process must be carried out rapidly to avoid any possible heat exchanges 

    between the gas system and the surroundings.

    C

    C

    C

    C

    Activity 14

    C

    Example I

    C

    Solution

    C

    Example 2

    C

    C

    Application activity 10.2

    1. A total of 135 J of work is done on a gaseous refrigerant as it undergoes 
    compression. If the internal energy of the gas increases by 114 J during 
    the process, what is the total amount of energy transferred by heat? Has 

    energy been added or removed from the refrigerant by heat?

    2. The internal energy of a system is initially 27 J. A total of 33 J of energy 
    is added to the system by heat while the system does 26 J of work. What 

    is the system’s final internal energy?

    3. Heat of 90 cal is supplied to a system and it is observed that no work is 

    done. Calculate the change in internal energy of this system. 

    4. One mole of an ideal gas at 250 C is allowed to expand reversibly at 
    constant temperature from a volume of 10 L to 20 L; calculate the work 

    done by the gas in joules. 

    5. A certain volume of dry air at N.T.P is expanded reversibly to three times 
    its volume 
    a) isothermally, 

    b) adiabatically. 

    Calculate the final temperature in each case, assuming ideal behaviour. 

    γ =1.40

    Second Law of thermodynamics

    Since the first law of thermodynamics states that energy is conserved. There 
    are, however, many processes we can imagine that conserve energy but are not 
    observed to occur in nature. Lets consider an example below of the first law to 

    introduce the second law.

    For example, when a hot object is placed in contact with a cold object, heat 
    flows from the hotter one to the colder one, never spontaneously the reverse. 
    If heat were to leave the colder object and pass to the hotter one, energy could 

    still be conserved. Yet it doesn’t happen spontaneously the reverse.

    There are many other examples of processes that occur in nature but whose 
    reverse does not. To explain this lack of reversibility, scientists in the latter 
    half of the nineteenth century formulated a new principle known as the second 

    law of thermodynamics.

    The second law of thermodynamics is a statement about which processes 
    occur in nature and which do not. It can be stated in a variety of ways, all of 
    which are equivalent. One statement is that: “Heat can flow spontaneously 
    from a hot object to cold object; heat will not flow spontaneously from a 

    cold object to a hot object.”

    The development of a general statement of the second law of thermodynamics 
    was based partly on the study of heat engines. A heat engine is any device 
    that changes thermal energy into mechanical work, such as steam engines and 

    automobile engines.

    Applications of the second law of thermodynamics

    Heat engines

    Activity 15
    * Have you ever heard of an engine? 
    * Where exactly do we find engines?
    * What do you think an engine is?
    * How do you think the engine operates?

    Any device that transforms heat into work or mechanical energy is called 
    heat engine. All heat engines absorb heat from a source at high temperature, 

    perform some mechanical work, and discard heat at a lower temperature.

    C

    C

    C

    Impact of heat engines on climate

    Most of air pollution is caused by the burning of fuels such as oil, natural gas 
    etc. The air pollution has an adverse effect on the climate. Climate change is 
    the greatest environmental threat of our time endangering our health. When 
    a heat engine is running, several different types of gases and particles are 

    emitted that can have detrimental effects on the environment.

    Of concern to the environment are carbon dioxide, a greenhouse gas; and 
    hydrocarbons. Engines emit greenhouse gases, such as carbon dioxide, which 
    contribute to global warming. Fuels used in heat engines contain carbon. The 

    carbon burns in air to form carbon dioxide.

    The Carbon dioxide and other global warming pollutants collect in the 
    atmosphere and act like a thickening blanket and destroy the ozone layer. 
    Therefore, the sun’s heat from the sun is received direct on the earth surface 

    and causes the planet to warm up. 

    As a result of global warming, the vegetation is destroyed, ice melts and water 
    tables are reduced. Heat engines especially diesel engines produce Soot which 

    contributes to global warming and its influence on climate. 

    The findings show that soot, also called black carbon, has a warming effect. 
    It contains black carbon particles which affect atmospheric temperatures in a 
    variety of ways. The dark particles absorb incoming and scattered heat from 
    the sun; they can promote the formation of clouds that can have either cooling 
    or warming impact. Therefore soot emissions have significant impact on 

    climate change.

     Similarly, some engines leak, for example, old car engines and oil spills all 
    over. When it rains, this oil is transported by rain water to lakes and rivers. 
    The oils then create a layer on top of the water and prevent free evaporation 

    of the water.

    Carnot cycle and Carnot engine

    In 1824 a French engineer named Sadi Carnot described a theoretical engine, 
    now called a Carnot engine, which is of great importance from both practical 
    and theoretical viewpoints. He showed that a heat engine operating in an ideal, 
    reversible cycle—called a Carnot cycle—between two energy reservoirs is 

    the most efficient engine possible.

    An ideal engine establishes an upper limit on the efficiencies of all other 
    engines. That is, the net work done by a working substance taken through the 
    Carnot cycle is the greatest amount of work possible for a given amount of 

    energy supplied to the substance at the higher temperature. 

    Carnot’s theorem can be stated that no real heat engine operating between 
    two energy reservoirs can be more efficient than a Carnot engine operating 

    between the same two reservoirs.

    Note: No Carnot engine actually exists, but as a theoretical idea it played an 

    important role in the development of thermodynamics.

    The idealized Carnot engine consisted of four processes done in a cycle, two 
    of which are adiabatic (Q = 0) and two are isothermal (ΔT = 0). This idealized 

    cycle is shown in figure 1.8. 

    c

    Each of the processes was considered to be done reversibly. That is, each of 
    the processes (say, during expansion of the gases against a piston) was done so 
    slowly that the process could be considered a series of equilibrium states, and 
    the whole process could be done in reverse with no change in the magnitude 

    of work done or heat exchanged. 

    A real process, on the other hand, would occur more quickly; there would 
    be turbulence in the gas, friction would be present, and so on. Because of 
    these factors, a real process cannot be done precisely in reverse, the turbulence 
    would be different, and the heat lost to friction would not reverse itself. Thus 
    real processes are irreversible.

    In the Carnot cycle, heat engines work in a cycle, and the cycle for the Carnot 

    engine begins at point a on the PV diagram.

    Note:

    • The gas is first expanded isothermally, with addition of heat QH, 

    along the path ab at temperature TH.

    c

    The equation above gives a Carnot (ideal) efficiency. It expresses the 
    fundamental upper limit to the efficiency. Real engines always have an 
    efficiency lower than this because of losses due to friction. Real engines 
    that are well designed reach 60 to 80% of the Carnot efficiency.

    Otto Cycle and Diesel Cycle

    Otto Cycle 

    An Otto cycle is an idealized thermodynamic cycle which describes the 
    functioning of a typical spark ignition reciprocating piston engine, the 

    thermodynamic cycle most commonly found in automobile engine.

    C

    The Pressure Volume diagram above represents the Otto cycle which has the 
    following strokes; the intake (A) stroke is performed by an isobaric expansion, 
    followed by an adiabatic compression (B) stroke (along 1-2). Through the 
    combustion of fuel, heat is added in an isovolumetric process (2-3), followed 
    by an adiabatic expansion process ( 3-4), characterising the power (C) stroke. 
    The cycle is closed by the exhaust (D) stroke, characterized by isovolumetric 

    cooling and isobaric compression processes.

    The processes are described by:
    Process 1-2 is an isentropic compression of the air as the piston moves from 

    bottom dead centre (BDC) to top dead centre (TDC).

    Process 2-3 is a constant –volume heat transfer to the air from an external 
    source while the piston is at top dead centre. This process is 
    intended to represent the ignition of the fuel –air mixture and 

    the subsequent rapid burning.

    Process 3-4 is an isentropic expansion (power stroke).

    Process 4-1 completes the cycle by a constant-volume process in which heat is 

    rejected from the air while the piston is a bottom dead centre.

    The Otto cycle consists of adiabatic compression, heat addition at constant 
    volume, adiabatic expansion, and rejection of heat at constant volume. In the 
    case of a four-stroke Otto cycle, technically there are two additional processes; 
    one for the exhaust of waste heat and combustion products (by isobaric 
    compression), and one for the intake of cool oxygen –rich) air (by isobaric 
    expansion); however, these are often omitted in a simplified analysis. Even 
    though these two processes are critical to the functioning of a real engine, 
    wherein the details of heat transfer and combustion chemistry are relevant, 
    for the simplified analysis of the thermodynamic cycle, it is simpler and more 
    convenient to assume that all of the waste-heat is removed during a single 

    volume change.

    Diesel Cycle 

    The diesel cycle is the thermodynamic cycle, which approximates the pressure 
    and volume of the combustion chamber of the Diesel engine, invented by 
    Rudolph Diesel in 1897. It is assumed to have constant pressure during the 
    first part of the “combustion” phase V2 to V2
     in the diagram, below). This is an idealised mathematical model;
    real physical diesels do have an increase in 
    pressure during this period, but it is less pronounced than in the Otto cycle. 
    The idealized Otto cycle of a gasoline engine approximates constant volume 

    during that phase, generating more of a spike in a P-V diagram.

    The Idealised Diesel Cycle

    C

    From P-V diagram for the Ideal Diesel cycle, the cycle follows the numbers 

    1-4 in clockwise direction.

    The image on the top shows a P-V diagram for the ideal Diesel cycle; where 
    P is pressure and V is specific volume. The ideal Diesel cycle follows the 
    following four distinct processes (the colour references refers to the colour of 

    the line on the diagram):

    Process 1-2 is isentropic (adiabatic) compression of the fluid (blue colour).

    Process 2-3 is reversible (isobaric constant pressure heating (red).

    Process 3-4 is isentropic (adiabatic) expansion (yellow).

    Process 4-1 is reversible constant volume cooling (green).

    The Diesel is a heat engine; it converts heat into work. The isentropic processes 
    are impermeable to heat; heat flows into the loop through the left expanding 
    isobaric process and some of it flows back out through the right depressurising 

    process, and the heat that remains does the work.

    Work in (Win) is done by the piston compressing the working fluid.

    Heat in (Qin) is done by the combustion of the fuel.

    Work out (Wout) is done by the working fluid expanding on to the piston (this 

    produces usable torque).

    C

    C

    A heat engine is a machine, which changes heat energy, obtained by burning a 
    fuel, to kinetic energy. In an internal combustion engine, e.g petrol, diesel, jet 
    engine, the fuel is burnt in the cylinder of chamber where the energy change 

    occurs. This is not so in other engines e.g steam turbine.

    Petrol engine 

    Activity 16

    (i) How many types of fuels do vehicles use to operate?

    (ii) Have you ever heard of vehicles which use petrol in order to operate?

    (iii) List down four vehicles which use petrol.

    (iv) What type of engine do they have?

    Many vehicles use petrol in order to move. Such vehicles are small cars and 
    motorcycles. The engine they have is called a petrol engine since it uses petrol 
    to operate. It operates by moving the piston. The upward and downward 

    movement of the piston is called a stroke.

    a) Four – stroke engine: On the intake stroke, the piston moves 
    down (due to the starter motor in a car or the kick start in a 
    motor cycle turning the crankshaft) so reducing the pressure 
    inside the cylinder. The inlet value opens and the petrol – air 
    mixture from the carburetor is forced into the cylinder by 

    atmospheric pressure.

    On the compression stroke, both valves are closed and the piston 

    moves up, compressing the mixture.

    On the power stroke, a spark jumps across the points of the 

    sparking plug and explodes the mixture, forcing the piston down.

    On the exhaust stroke, the outlet valve opens and the piston rises, 

    pushing the exhaust gases out of the cylinder.

    The crankshaft turns a flywheel (a heavy wheel) whose 

    momentum keeps the piston moving between power strokes.

    Most cars have atleast four cylinders on the same crankshaft. 
    Each cylinder fires in turn in the order 1-3-4-2, giving a power 

    stroke every half revolution of the crankshaft. Smoother running results.

    b) Two-stroke engine: This is used in mopeds, lawnmovers and 
    small boats. Valves are replaced by ports on the side of the 

    cylinder which are opened and closed by the piston as it moves.

    Diesel engine

    Activity 17

    (i) Have you ever heard of vehicles which use diesel in order to move?

    (ii) What kind of vehicles are they?

    (iii) What is the name of the engine in such vehicles? 

    The engine which uses diesel is called a diesel engine. A diesel engine can 

    operate by making two or more strokes.

    The operation of two and four stroke diesel engines is similar to that of the 
    petrol varieties. However, fuel oils is used instead of petrol, there is no 

    sparking plug and the carburetor is replaced by a fuel injector.

    Air is drawn into the cylinder on the down stroke of the piston and on the 
    upstroke it is compressed to about one-sixteenth of its original volume (which 
    is twice the compression in a petrol engine). This very high compression 
    increases the temperature of the air considerably and when, at the end of the 
    compression stroke, fuel is pumped into the cylinder by the fuel injector, it 
    ignites automatically. The resulting explosion drives the piston down on its 
    power stroke. (You may have noticed that the air in a bicycle pump gets hot 

    when it is squeezed. The same applies here.)

    Activity: 18

    State the advantages of a diesel engine over a petrol engine.

    Diesel engines, sometimes called compression ignition (C.I) engines, though 
    heavier than petrol engines, are reliable and economical. Their efficiency of 
    about 40% is higher than that of any other heat engine. A disadvantage of the 
    diesel engine is that its higher compression ratio means that it needs to be 

    more robust, and is therefore more massive.

    The Refrigerator 

    Activity: 19

    * How many of you have seen a 
    refrigerator?
    * With the help of a teacher visit any 
    place where there is a refrigeration 
    and observe it carefully.
    * How useful is it to our daily lives?
    * Who can describe how it works?
    * Write your suggestions in the 

    notebook.

    C

    A refrigerator is a device used to cool substances. It cools things by evaporation 
    of a volatile liquid called Freon. The coiled pipe around the freezer at the top 
    contains Freon which evaporates and takes latent heat from the surroundings 
    so causing cooling. The electrically driven pump removes the vapor and 
    forces it into the heat exchanger (pipes with cooling fins outside the rear of 

    the refrigerator).

    Here the vapor is compressed and liquefies (condenses) giving out latent heat 
    of vaporization to the surrounding air. The liquid returns to the coils around 
    the freezer and the cycle is repeated. An adjustable thermostat switches the 
    pump on and off, controlling the rate of evaporation and so the temperature of 

    the refrigerator.

    The operating principle of refrigerators is just the reverse of a heat engine. Each 

    operates to transfer heat out of a cool environment into warm environment. 

    C

    By doing work W, heat is taken from a low-temperature region, QL (such 
    as inside a refrigerator), and a greater amount of heat is exhausted at a high 
    temperature, QH (the room). You can often feel this heat blowing out beneath 

    a refrigerator. 

    A perfect refrigerator is the one in which no work is required to take heat from 
    the low-temperature region to the high temperature region is not possible. 
    This is Clausius statement of the second law of thermodynamics, already 

    mentioned can be stated formally as:

    “No device is possible whose sole effect is to transfer heat from one system 

    at a temperature TL into a second system at a higher temperature TH”.

    To make heat flow from a low-temperature object (or system) to one at a higher 

    temperature, work must be done. Thus, there can be no perfect refrigerator.

    The coefficient of performance (COP) of a refrigerator is defined as the heat 
    QL removed from the low-temperature area (inside the generator) divided by 

    the work W done to remove the heat:

    C

    This makes sense since the more heat, QL, that can be removed from inside 
    the refrigerator for a given amount of work, the better (more efficient) the 
    refrigerator is. Energy is conserved, so from the first law of thermodynamics 

    we can write

    C

    Example

    An ideal refrigerator-freezer operates with a COP = 7.0 in a 24 oC 

    room. What is the temperature inside the freezer?

    Solution

    C


    END UNIT ASSESSMENT

    1. What is the heat capacity at constant volume considered to be more 

    important than at constant pressure?

    2. Define: isothermal; isobaric; isovolumic and adiabatic processes

    3. What is the relationship between the specific heat (or het capacities) 

    at constant pressure and at constant volume?

    4. One mole of helium gas, initially at STP ( p1 latm kpa =  1.03
     T1 =00c  = 273.15 k) undergoes an isovolumetric process in 

    which its pressure falls to half its initial value. 

    a) What is the work done by the gas?

    b) What is the final temperature of the gas?

    c) The helium gas then expands isobaric ally to twice its volume, 

    what is the work done by the gas?

    5. Find out the internal energy of a system which has constant volume 

    and the heat around the system is increased by 50 J?

    6. In a certain process 8.0kcal of heat is furnished to the system while 
    the system does 6.00 KJ of work. By how much does the internal 

    energy of the system change during the process?

    7. The specific heat of water is 4184J/kg.k. By how many joules does 
    the internal energy of 50g of water changes as it is heated from 210 c 
    to 37    0 c.
    C