Physics

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 handdles 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 interrest in this unit.

(iii) What if in the process one stops pushing, What would happen?

In daily life, we normally pull the objects from one place to another.

What you should know

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?

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 quantityis 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.

Activity Quick check!

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.)

GROUP WORK

1. ............................ is an example of a scalar quantitya) 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

Moment of a force or torque

Requirements

* Asee-saw

* A knife edge

* A ruler

* Masses

Activity 4

Aim: To find the effect of moment of a force or couple

(i) Pull/push the door by applying the force at its handle.

(ii) Pair yourselves and go out on a see saw balance.

(iii) Sit on one side and your friend to the next.

(iv) What happens?

A door knob is located as far as possible from the hinge line for a good reason

.If you want to open a heavy door, you must certainly apply a force, that done, however, is not enough where you apply that force and in what direction you push are also very important.

Study the diagrams above

What happens when force is applied ;

(i) At the end of the tool used?

(ii) In the middle of the tool used?

Couple of forces

* In daily life, they say that a man and a woman constitute a couple. Why?

* How many of you know how to ride a bicycle? If you know, what do you do when negotiating a corner? * If you have never ridden a bicycle ask your friend?

A couple C consists of two equal and opposite parallel forces whose lines of action do not coincide. It always tends to change rotation. It is clearly indicated in the figure below.

Coplanar forces

Activity 5

(i) As a class, visit a specialist in roofing.

(ii) Ask him/her what they consider when roofing.

(iii) Ask him/her the materials suitable to use while roofing and why do they opt for materials he/she told you ?.

iv) Discuss amongst yourselves what would happen when a kinyarwanda hut (local hut) is roofed using Iron bars. What causes the effects ?

Parallelogram of forces

A force is a vector quantity. So it can be represented in size and direction by a straight line drawn to scale. The sum or resultantof two forces 1and2can be added by one of two vector methods.

Resolved components

Note that in real life, we do a lot of things; name them.

Do we (you) apply physics?

If yes, how?

If no, why?

Activity 6

Aim;To investigate what happens when forces are added together

* Try to lift your seat alone.

* How do you feel?

* Tell your friend to help you and you lift it together

* Are you feeling the same way as before when you were alone?

* What if your friend pulls in opposite direction to that of your force? What would happen?

Therefore, when solving daily problems, it is often helpful to replace one force by a combination of two forces with given directions. Of course, these two forces must be equivalent to the given one.

This means that their vector sum must agree with the given force. If this condition is fulfilled, we say that the force has been resolved into components.

A simple geometrical construction provides the magnitudes of the components:We can draw two lines from the end of the given force vector parallel to the given directions. In this way, we get the so-called parallelogram of forces. The magnitudes of the components now can be read off from the sides of this parallelogram.

Example

When pulling/saving a cow that has fallen into a pit, it is advised to pull it applying forces in one/same direction.Why?

Equilibrium of coplanar forces

Class work

As a class, let us move outside. With the help of a teacher outside the class.

Are you seeing any object outside?

Are they stationary or in motion?

If at rest, what causes them to be at rest?

We say that an object is at rest or in motion under the influence of forces; if a body is at rest,then, there is no net force acting.

Therefore, if the resultant forces acting on a body is zero,the body is said to be in equilibrium.The following pointers will help you to solve problems that involve a body acted on by three co-planar forces:

a) The line of action of three forces must all pass through the same point.

b) The principle of moments: The sum of all clock-wise moments about any point must have the same magnitude as the sum of all anti-clock wise moments about the same point.

Centre of gravity

Activity 7

Requirements

* A stone

* A block of wood

* Retort stand

* Thread

* Pendulum bob or a plumbing bob

* A card board

* A marker

Selectively, choosing some of the apparatus, design a simple experiment to determine centre of gravity of an irregular lamina. State reasons why you chose the apparatus.

Discuss the reasons with your friends.Note down and present it to the whole class.Hint Use the idea of O’level.

From the experiment, what can you say about Centre of gravity?Is it the same as Centre of mass? Defend your answer.

Quiz. Can you locate where your Centre of gravity is?

What about your Centre of mass?Why do big ladies and gentlemen feel complications when standing up?

The term centre of mass is often used interchangeably with centre of gravity, but they are physically different concepts. They happen to coincide in a uniform gravitational field, but where gravity is not uniform, centre of gravity refers to the mean location of the gravitational force acting on a body

By observing the diagrams above.

* How can you build on and determine the centre of gravity of the body shown in the figure above?

* What if the object was irregular lamina .

Explain an experiment you would perform to determine its Centre of gravity.

* In laboratory, perform the experiment. Types of equilibriumRequirements

* A log of wood

* A bottle

* A table

* A knife edge made of wood or A triangular glass prism

* A rectangular wooden block

Activity 8

Aim;To find out the effect of application of force on to the equilibrium on the stability of a body.

* Displace the desk. What happens when you withdraw the force you applied.

* Place a bottle on a table so that it rests on its horizontal surface.

Displace or roll it. What happens?

* Place a knife edge on a table resting on its tip.

Give it a small displacement.What happens to it?

* From the observations made,how do you conclude?

From what you have done above, before displacing the body, the body was at rest, and we describe this state to be in equilibrium.

Equilibrium has many different meanings, depending on what subject (chemistry or physics) or what topic (energy or forces).

Dealing with energy, there are three types of equilibrium.

From the Activity, a body is in either Stable, Unstable or in neutral equilibrium depending how it behaves when subjected to a small displacement.

Stable is when any sort of movement will raise the object’s centre of gravity. When objects in stable equilibrium are moved, they have a tendency to fall back to their original position. For instance, a skateboarder at the bottom, in the middle, of a ramp. Either way the skateboarder moves, his/her potential energy will increase because he/she will be raising in height. The boarder will also roll back to the bottom of the ramp if he/she doesn’t exert any sort of energy to maintain the new position.

When a body returns to its original position on being slightly disturbed, it is said to be in stable equilibrium.

Unstable is when any sort of movement will lower the object’s centre of gravity. When such objects are moved, they cannot return to their original position without some exertion of energy. For instance, when a coin is placed on its side, it exhibits unstable equilibrium. Any sort of push will cause the coin to fall flat, lowering its centre of mass. The coin will not return to its side unless someone picks it up and resets it.

If the position of a body is disturbed and the body does not return to its original position, it is in unstable equilibrium.

Neutral is when any sort of movement does not affect the object’s centre of gravity. For instance, a ball on a table exhibits neutral equilibrium. If the ball rolls, the centre of mass stays at the same height and thus it maintains the same equilibrium.

A body is said to be in neutral equilibrium if it moves to a new position when it is disturbed. Let us consider a cone to understand these states.

Conditions of equilibrium of objects on a horizontal plane

Activity 9

* Using a cone or a knife edge, or a triangular glass prism.

* Place it on a table resting it on its broad surface.

* Displace it in the direction of arrow shown.

* What happens to the cone?

* State your observations.

* Present them to the whole class.

From whatever we have done, we can conclude and say that a body is stable when:

1. The object’s base is broad.

2. The centre of gravity is as low as possible.

3. The vertical line drawn from the centre of gravity should fall within the base.

Lowering the centre of gravity of an object is important for stability.

Some examples

1. Articles like vases, table lamps, wine glasses, pedestal fans, glass jugs, bunsen burners and lanterns have a broad or heavy base to avoid toppling.

2. Racing cars have a broad base.

3. Motorists do not keep too much luggage in overhead carriers.

4. Cargo ships are loaded at the base.

5. Double-decker buses do not allow standing passengers in the upper deck.

6. Overloaded Lorries are stopped at check posts while going up and down mountain roads.

Figure 3.7: Cone in unstable equilibrium

Group work!

* Place a knife edge on a table so that it rests on its tip

* Displace it as shown in the figure.

* Note what happens to it. Can you say that it was stable,unstable or neutral? Why?

CLASS WORK

* Roll a paper so that you come up with a cone. This should be done individually.

* Place them on a table as shown above.

* Displace it as shown in the figure 3.8.

* What did you observe?

You now know that,

* The centre of gravity of an object is a fixed point through which the entire weight of the object acts in whatever position the object may be placed.

* There are three states of equilibrium namely.

a) stable

b) unstable

c) neutral.

* There are three conditions for an object to be stable:

a) The object should have a broad base.

b) The Centre of gravity should be as low as possible.

c) The vertical line drawn from the centre of gravity should fall within the base.

Other examples

1. In here, we shall look at what happens when a body is rolled on an incline.

Field work

* Outside the class, try to get a place that is sloping.

* Try to get a spherical body; say a used bicycle rim.

* Roll it down the incline.

* In groups of 5, discuss what happens.

* Present your finds to the class.

Considering the diagram below, we can see that the body will ascend downwards when put on the incline.

Have you ever done this alone?

On a plane without friction, if l is its length, h its height, the object is in equilibrium if there is a force acting on the object upward on the plane having a magnitude:

Note that there is a frictional force.

Equilibrium of a suspended object

Let us consider a ruler such that its centre of gravity is at a point G and can be suspended at a given point O

The centre of gravity is below the point of suspension o The immobility being realized, if we move the ruler from its equilibrium position, it has the tendency to come back. We have the case of a stable equilibrium.

The centre of gravity is above the point of suspension oThis equilibrium is difficult to realize. When you move lightly the ruler, it moves away from the equilibrium position. We have an unstable equilibrium

The centre of gravity is at the point of suspension oIn this case, the action and the reaction being at the same point, when you move lightly the ruler from its equilibrium position, it takes a new equilibrium position. We have a neutral equilibrium.

Equilibrium of a solid submitted to several forces

For a body to be at rest, the sum of the forces acting on it must add up to zero. Since force is a vector, the components of the net force must each be zero. Hence, if the forces on the object act in a plane, a condition for equilibrium is that: ΣFx = 0, ΣFy = 0.

We must remember that, if a particular force component points along the negative axis, it must have a negative sign. Equations above are called the first condition for equilibrium.

Although these equations must be true if an object is to be in equilibrium, it is not a sufficient condition. If the body is to remain at rest, the net torque applied to it must be zero. Thus we have the second condition for equilibrium: that the sum of the torques acting on a body must be zero: Στy = 0.

This will assure that the angular acceleration about any axis will be zero; if the body is not rotating initially it will not start rotating. The first and second conditions are the only requirements for a body to be in equilibrium.

This has several applications among them; we have the balancing of a seesaw, beam balance, etc.

Examples

QUIZ

1. Why do you think that when a person with huge bums after sitting down feels it hard to stand up? Do you think that it is easy for the lady below to stand up? Why?

2. A bottle resting on its top falls to the next position when slightly displaced. Explain this observation.

Stevinus proof

Stevinus (some times called Stevin) proof of the law of equilibrium on an inclined plane known as the “Epitaph of Stevinus”.

He derived the condition for the balance of forces on inclined planes using a diagram with a “wreath” containing evenly spaced round masses resting on the planes of a triangular prism (see the illustration on the figure 3.14). He concluded that the weights required were proportional to the lengths of the sides on which they rested assuming the third side was horizontal and that the effect of a weight was reduced in a similar manner.

It’s implicit that the reduction factor is the height of the triangle divided by the side (the sine of the angle of the side with respect to the horizontal). The proof diagram of this concept is known as the “Epitaph of Stevinus”.

Archimedes and the principles of the lever

Activity 10

1. In pairs, discuss levers using knowledge of O’level Machines.

2. What is your friend telling you. Is it what you knew before?

Building up from the earliest remaining writings regarding levers date from the 3rd century BC and were provided by Archimedes. “Give me a place to stand, and I shall move the Earth with it”is a remark of Archimedes who formally stated the correct mathematical principle of lever.

Force and levers, law of the lever

A lever is a beam connected to ground by a hinge or pivot called a fulcrum. \

In other words we say that the lever is a movable bar that pivots on a fulcrum attached to a fixed point. The lever operates by applying forces at different distances from the fulcrum, or a pivot.

The ideal lever does not dissipate or store energy, which means there is no friction in the hinge or bending in the beam. In this case, the power into the lever equals the power out, and the ratio of output to input force is given by the ratio of the distances from the fulcrum to the points of application of these forces. This is known as the law of the lever.

Note: You can draw a diagram masses replaced by learners.The mechanical advantage of a lever can be determined by considering the balance of moments or torque, τ, about the fulcrum, and we write:

where F1 is the input force to the lever and F2 is the output force. The distances a and b are the perpendicular distances between the forces and the fulcrum.

For the case of the figure 3.16, knowing that the forces considered are weights of objects and W = Mg, we can write the mechanical advantage of the lever as a ratio,

Activity 11

In groups determine the mass of the object using the principle of moments.

Apparatus required

* 1 uniform meter rule

* A knife edge

* Unknown mass

Instructions

In this experiment, the learner determines the mass M of the provided body (meter ruler).

a) Weigh the meter ruler provided to obtain its mass, M.Balance the meter ruler on a knife edge. Read Q the balance point. Find Z.

b) Balance the meter ruler with its graduated face upwards on a knife edge. With the mass M provided at P=10 cm from A as shown,

c) Measure and record distance X and Y.

d) Repeat procedure (b) and (c) for values of P=15,20,25,30 and 35cm.

e) Tabulate your results including values of (Y-Z) and (X-P).

f) Plot a graph of (Y-Z) against (X-P).

g) Find the slope “S’’of your graph.

h) Calculate the mass M of the body from Sm=M.

EXAMPLE

1. 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. The support (called the fulcrum) is under the centre of gravity of the board. The father is a distance d from the centre, and the daughter is at distance l/2 from the centre.

(a) Determine the magnitude of the upward force (reaction) n exerted by the support on the board.

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

Student’s trials

1. A person holds a 50.0N sphere in his hand. The forearm is horizontal, as shown in Figure. The biceps muscle is attached 3.0cm from the joint, and the sphere is 35.0cm from the joint. Find the upward force exerted by the biceps on the forearm and the downward force exerted by the upper arm on the forearm and acting at the joint. Neglect the weight of the forearm.

2. A uniform horizontal beam with a length of 8.00m and a weight of 200N is attached to a wall by a pin connection. Its far end is supported by a cable that makes an angle of 53.0° with the beam. If a 600N person stands 2.00m from the wall, find the tension in the cable as well as the magnitude and direction of the force exerted by the wall on the beam.

3. Calculate the magnitudes F_A and F_B of the tensions in the two cords that are connected to the vertical cord supporting the 200kg chandelier in the figure.

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.

5. The bar in the figure is being used as a lever to pry up a large rock. The small rock acts as a fulcrum (pivot point). The force FP required at the long end of the bar can be quite a bit smaller than the rock’s weight mg, since it is torques that balance in the rotation about the fulcrum. If, however, the leverage isn’t sufficient, and the large rock isn’t budged, what are the two ways to increase the leverage?

6. The bridge on a river of a country of Central Africa is supposed to have a length of 100m and mass 10⁵kg. It leans on two pillars to its extremities.

What are forces exerted on the pillars when three cars (one Mercedes of 1500kg, a Renault of 1200kg and a Fiat 1000kg) are respectively to 30, 80 and 60m from the extremity leaning on the left bank.

7. A horizontal rod AB of negligible weight, 51cm long is submitted in A and in B to two forces1and 2 of magnitudes respectively 14N and 7N. The force 1makes an angle of 45° with the vertical and the force 2is perpendicular to the rod. Their direction is oriented downward. Determine the characteristics of the force which will make the rod in equilibrium.

8. 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

9. A sphere of 50N stands against two inclined planes making respectively angles of 30° and 45°. Calculate the forces of reaction of the two planes on the sphere.

10. A block of mass 330kg is suspended by three unstretchable ropes as shown on the figure below. If the system is in equilibrium,

a) determine T₁,

b) If O₁ = 15°, θ₂ = 30°, find the tensions in the ropes.

11. A ladder AB weighing 160N rests against a smooth vertical wall and makes an angle of 60° with the ground as shown In the figure below. The ladder has small wheels at the point A such that the friction with the vertical wall is negligible. Find the forces acting on the ladder at point A and point B.

The coefficient of static friction between the ladder and the ground is 0.53. How far up the ladder can the firefighter go before the ladder starts to slip?

12. A homogeneous beam of length 2.20m and of massm=25.0kg is fixed on a wall by a hinge and is held in horizontal position by a metallic string making angle of θ = 30.0° as shown in the figure below. It holds a mass M = 280kg suspended at its extremity. Determine the components of the force exerted by the wall on the beam at the hinge and components of the tension in the metallic string.

Extension exercise

1. a) State the conditions under which a rigid body is in equilibrium under the action of coplanar forces.

b) Forces of 2.83N, 4.00N and 6.00N act on an object O as shown the figure below.

2. Find the resultant force on the object.When three concurrent forces act on a body which is in equilibrium, the resultant of the two forces should be equal and opposite to the third force. Prove this statement.

3. A uniform ladder of massm and length L leans against a smooth vertical wall making an angle φ with a horizontal floor. The coefficient of static friction between the ladder and the floor is μ

Find (in terms of μ) the minimum angle θ_m at which the ladder does not slip