Section outline


  • Key unit competence: Evaluate Newton’s law of gravitation and 
    apply Kepler’s laws of planetary motion.
     
    Unit Objectives:
      By the end of this unit I will be able to;
     ◊  Explain the terms, concept and characteristics of waves properly.
     ◊   Explain the properties of waves.
     ◊   Explain the behavior of waves in vibrating strings and applications 

    of waves properly.

    Introductory Activity


    People have always enjoyed viewing stars and planets on clear, dark 
    nights. It is not only the beauty and variety of objects in the sky that is 
    so fascinating, but also the search for answers to questions related to 
    the patterns and motions of those objects.
     
    Until the late 1700s, Jupiter and Saturn were the only outer planets 
    identified in our solar system because they were visible to the naked 
    eye. Combined with the inner planets the solar system was believed 
    to consist of the Sun and six planets, as well as other smaller bodies 
    such as moons. Some of the earliest investigations in physical science 
    started with questions that people asked about the night sky.

     i) Based on the scenario above and the observation from the 
    picture. Briefly summarize what is illustrated in the picture.
    ii) What is the name of belt separating the largest and smallest planets?
    iii) Explain why you think the moon doesn’t fall on the earth.
    iv) Why don’t we fly off into space rather than remaining on the 
        Earth’s surface? Explain your idea.

    v) Explain why  planets move across the sky.

    8.1. INTRODUCTION
    Gravity is the mysterious force that makes everything fall down towards 
    the Earth. But after research it has turned out that all objects have gravity. 
    It’s just that some objects, like the Earth and the Sun, have a stronger 
    gravity than others. How much gravity an object has depends its mass. 
    It also depends on how close you are to the object. The closer you are, the 

    stronger the gravity.

    Gravity is very important to our everyday lives. Without Earth’s gravity 
    we would fly right off it. If you kicked a ball, it would fly off forever. While 
    it might be fun to try for a few minutes, we certainly can’t live without 
    gravity. Gravity also is important on a larger scale. It is the Sun’s gravity 
    that keeps the Earth in orbit around the Sun. Life on Earth needs the Sun’s 
    light and warmth to survive. Gravity helps the Earth to stay at just the 

    right distance from the Sun, so it’s not too hot or too cold.

     8.2. NEWTON’S LAW OF GRAVITATION

    This is also called the universal law of gravitation or inverse square 
    law. It states that “the gravitational force of attraction between two 
    masses m1 and m2 is directly proportional to the product of masses 
    and inversely proportional to the square of their mean distance 
    apart.” Remember two objects exert equal and opposite force of gravitation
    on each other.

    Notes:
     • The value of G in the laboratory was first determined by Cavendish 

    using the torsional balance.


    Properties of Gravitational Force
     • It is always attractive in nature while electric and magnetic force can 
    be attractive or repulsive.
     • It is independent of the medium between the particles while electric 
    and magnetic forces depend on the nature of the medium between the particles.
     • It holds good over a wide range of distances. It is found true for 
    interplanetary to interatomic distances.
     • It is a central force, i.e. it acts along the line joining the centres of two 
    interacting bodies. 
    • It is a two-body interaction, i.e. gravitational force between two particles 
    is independent of the presence or absence of other particles; so, the 
    principle of superposition is valid, i.e. force on a particle due to number 

    of particles is the resultant of forces due to individual particles, i.e. 


     • It is a conservative force, i.e. work done by it is path independent or 
    work done in moving a particle round a closed path under the action 
    of gravitational force is zero. 
    • It is an action reaction pair, i.e. the force with which one body (say, 
    earth) attracts the second body (say, moon) is equal to the force with 
    which moon attracts the earth. This is in accordance with Newton’s 
    third law of motion.
     
    8.3. KEPLER’S LAWS OF PLANETARY MOTION
     Planets are large natural bodies rotating around a star in definite orbits. 
    The planetary system of the star sun, called solar system, consists of eight 
    planets, viz. Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and 
    Neptune . Out of these planets mercury is the smallest, closest to the sun. 
    jupiter is the largest and has the maximum number of moons. Venus is 
    closest to the earth and the brightest planet. Kepler, after a life time study, 
    worked out three empirical laws which govern the motion of these planets 

    and are known as Kepler’s laws of planetary motion. These are stated below.

    1st Law: This law is called the law of orbits and it states that planets move 
    in ellipses with the sun as one of their foci. It can also be stated that planets 

    describe ellipses about the sun as one focus. (Fig. 8.2)


    2nd Law: This is called the law of areas and states that the line joining the 
    sun and the planet sweeps out equal areas in equal periods of time. (Fig. 8.3)


     
    3rd Law: The law of periods states that the square of the period T of 
    revolution of any planet is proportional to the cube of its mean distance R 

    from the sun. (Fig. 8.4)


    8.4. VERIFICATION OF KEPLER’S THIRD LAW OF   
    PLANETARY MOTION
     Assuming that a planet’s orbit is circular (which is not exactly correct but is 
    a good approximation in most cases), then the mean distance from the sun 
    is constant –radius. Suppose, a planet of mass m2 moving around the sun of mass m1
     If the motion of the planet is circular, there are two types of forces: 
    (a)  Gravitational force of attraction F1 between the sun and the planet,


    EXAMPLE 8.1:
     The distance of a planet from the sun is 5 times the distance between the 
    earth and the sun. What is the time period of revolution of the planet?  

    Solution:

     According to Kepler’s law 



    Background information:
     Kepler’s third law (the Harmonic Law), relates the orbital period of a 
    planet (that is, the time it takes a planet to complete one orbit) to 
    its mean distance from the Sun.  This law states that the closest 
    planets travel at the greatest speeds and have the shortest orbital periods. 





     Source of data: lunar and planetary science by National Aeronautics and 
    Space Administration (NASA)
     
    Use the data provided in the tables above and find the orbital period for 
    each orbital radius for each planet.  Enter the data into spreadsheets 
    and plot line graphs for the data, with each planet’s orbital radius on the 

    X-axis and its orbital period on the Y-axis. 


     Describe any general trends you see:  
    a)  Is there a systematic relationship between period and radius for the 
    planets for each case?
     b)  How would you describe this relationship in words?
     c)   Is the relationship you observe consistent with Kepler’s third law?

     d)  How could you improve your test for consistency? 

    Application Activity 8.1
     Using the cross and down clues write the correct words in the numbered 

    grid below.


     Across 
    6. The second largest planet with many rings.
     7. This planet’s blue color is the result of absorption of red light by 
    methane in the upper atmosphere.
     8. A small body that circles the Sun with a highly elliptical orbit.
     9. An object in orbit around a planet.
     10. A large cloud of dust and gas which escapes from the nucleus of an 
    active comet.
     
    DOWN
     1. It is the brightest object in the sky except for the Sun and the Moon.
     2. The largest object in the solar system.
     3. The only planet whose English name does not derive from Greek/
     Roman mythology.
     4. An area seen as a dark spot on the photosphere of the Sun.
     5. This planet is more than twice as massive as all the other planets combined.

    8.5. ACCELERATION DUE TO GRAVITY AT THE 
    SURFACE OF THE EARTH
     The force of attraction exerted by the earth on a body is called gravitational
     pull or gravity. We know that     when force acts on a body, it produces acceleration. 
    Therefore, a body under the effect of gravitational pull must accelerate.
    The acceleration produced     in the motion of a body under the effect of gravity 
    is called acceleration due to gravity (g). Consider a body of mass m lying on the surface
    of earth. Then gravitational force on the body is given by:




    • Acceleration due to gravity is a vector quantity and its direction is 
    always towards the centre of the planet. 
    • Dimensions of [g] = [LT –2]
     • Average value of g is taken as 9.8 m/s2 or 981 cm/s2, on the surface of 
    the earth at mean sea level.
     • In general, the value of acceleration due to gravity vary due to the 
    following factors: (a) Shape of the earth, (b) Height above the earth 
    surface, (c) Depth below the earth surface and (d) Axial rotation of the earth.


    EXAMPLE 8.4:
     The moon’s radius is (1/4)th of that of earth and its mass is 1/80 times that 
    of the earth. If g represents the acceleration due to gravity on the surface 

    of the earth, what is acceleration due to gravity on the surface of the moon?


    8.6. VARIATION OF ACCELERATION DUE TO 
    GRAVITY WITH HEIGHT
     Consider a particle placed at a height h above the surface of the earth where 

    acceleration due to gravity is g′ as shown on the figure below.





    EXAMPLE 8.5:
     The acceleration of a body due to the attraction of the earth (radius R) is 
    g. Find the acceleration due to gravity at a distance 2R from the surface of the earth.

    EXAMPLE 8.6:
     Find the height of the point above the earth’s surface, at which acceleration 
    due to gravity becomes 1% of its value at the surface is (Radius of the earth = R).


    Notes:
    • The value of g decreases on going below the surface of the earth. From 
    equation 8-12, we get g′ ∝ (R – d). So it is clear that if d increases, the 

    value of g decreases.


    Combining the graphs for variation of acceleration due to gravity below and 

    above the surface of the earth will give the graph as shown below:

     EXAMPLE 8.7:
     Weight of a body of mass m decreases by 1% when it is raised to height h 
    above the earth’s surface. If the body is taken to a depth h in a mine, what 

    is the change in its weight?


     8.8. VARIATION IN G DUE TO ROTATION OF EARTH
     As the earth rotates, a body placed on its surface moves along the circular 
    path and hence experiences centrifugal force. Due to it, the apparent weight 
    of the body decreases. 

    Since the magnitude of centrifugal force varies with the latitude of the 
    place, therefore the apparent weight of the body varies with latitude due to 

    variation in the magnitude of centrifugal force on the body.






    EXAMPLE 8.9:

    What is the angular velocity of the earth with which it has to rotate so that 
    acceleration due to gravity on 60° latitude becomes zero? (Radius of earth 

    = 6400 km. At the poles g = 10 ms–2) 


    8.9. VARIATION OF ‘G’ DUE TO SHAPE OF EARTH
     Earth is elliptical in shape. It is flattened at the poles and bulged out at the equator. 

     8.10. ROCKETS
     A rocket is a device that produces thrust by ejecting 
    stored matter. A rocket moves forward when gas expelled from 
    the rear of a rocket pushes it in the opposite direction. From 
    Newton’s laws of motion, for every action, there is an equal 
    and opposite reaction. In a rocket, fuel is burned to make a 
    hot gas and this hot gas is forced out of narrow nozzles in the 
    back of the rocket, propelling the rocket forward.

    Spacecraft Propulsion
     Spacecraft Propulsion is characterized in general by its complete integration 
    within the spacecraft (e.g. satellites). Its function is to provide forces and 
    torques in (empty) space to:
     • transfer the spacecraft: used for interplanetary travel 
    • position the spacecraft: used for orbit control

     • orient the spacecraft: used for altitude control


     The jet propulsion systems for launching rockets are also called primary 
    propulsion systems. Spacecrafts, e.g. satellites, are operated by secondary 
    propulsion systems.
     
    Characteristics of Spacecraft Propulsion Systems 
    In order to fulfill altitude and orbit operational requirements of spacecraft, 
    spacecraft propulsion systems are characterized by: 
    • Very high velocity increment capability (many km/s) 
    • Low thrust levels (1 mN to 500 N) with low acceleration levels 
    • Continuous operation mode for orbit control 
    • Pulsed operation mode for altitude control 
    • Predictable, accurate and repeatable performance (impulse bits) 
    • Reliable, leak-free long time operation (storable propellants) 
    • Minimum and predictable thrust exhaust impingement effects 

    Classification of Propulsion Systems     
    Spacecraft propulsion can be classified according to the source of energy 
    utilized for the ejection of propellant:
    • Chemical propulsion use heat energy produced by a chemical 
    reaction to generate gases at high temperature and pressure in a 
    combustion chamber. These hot gases are accelerated through a 
    nozzle and ejected from the system at a high exit velocity to produce 
    thrust force. 
    • Electric propulsion uses electric or electromagnetic energy to eject 
    matter at high velocity to produce thrust force. 
    • Nuclear propulsion uses energy from a nuclear reactor to heat 
    gases which are then accelerated through a nozzle and ejected from 

    the system at a high exit velocity to produce thrust force. 


    Notes:
     • While chemical and electric systems are used for the propulsion of 
    today’s spacecrafts, nuclear propulsion is still under study. Therefore, 
    only chemical and electric propulsion will be dealt with in this book. 

    8.11. SATELLITES
     A satellite is an artificial or a natural body placed in orbit round the earth 
    or another planet in order to collect information or for communication. 
    Communication satellites are satellites that are used specifically to 
    communicate. Part of that communication will be the usual commands and 
    signals we get from any satellite. The payload of the satellite consists of 
    huge collection of powerful radio transmitters and a big dish or something 
    like that, to enable it to talk to things on the ground. And we’ll use them 
    to transmit TV signals, to transmit radio signals, and in some cases, it 
    might be to be transmit internet signals. So, all of that gets turned into 
    radio somehow and transmitted up into space and then bounced back down 

    somewhere else.

    There is only one main force acting on a satellite when it is in orbit, and 
    that is the gravitational force exerted on the satellite by the Earth. This 
    force is constantly pulling the satellite towards the centre of the Earth.
     
    A satellite doesn’t fall straight down to the Earth because of its velocity. 
    Throughout a satellite’s orbit there is a perfect balance between the 
    gravitational force due to the Earth, and the centripetal force necessary to 

    maintain the orbit of the satellite.

     8.11.1. Orbital Velocity of Satellite.
     Satellites are natural or artificial bodies describing orbit around a planet under
    its gravitational     attraction. Moon is a natural satellite while INSAT
    1B is an artificial satellite of the earth. Condition for establishment of artificial satellite
     is that the     centre of orbit of satellite must coincide with centre 

    of earth or satellite must move around great circle of earth.


    Orbital velocity of a satellite is the velocity required to put the satellite into 
    its orbit around the earth. For revolution of satellite around the earth, the 

    gravitational pull provides the required centripetal force.


    Notes:

    • Orbital velocity is independent of the mass of the orbiting body and is 
    always along the tangent of the orbit, i.e. satellites of deferent masses 
    have the same orbital velocity, if they are in the same orbit.
     • Orbital velocity depends on the mass of central body and radius of orbit.
    • For a given planet, greater the radius of orbit, lesser will be the orbital 

    velocity of the satellite 

    • Orbital velocity of the satellite when it revolves very close to the 

    surface of the planet:


    EXAMPLE 8.10:
    Two satellites A and B go round a planet P in circular orbits having radii 
    4R and R respectively. If the speed of the satellite A is 3v, what is the speed 

    of the satellite B?


    Orbital velocity increases by 0.5%.
     8.11.2. Time Period of Satellite

     It is the time taken by satellite to go once around the earth.




    EXAMPLE 8.12:
     A satellite is launched into a circular orbit of radius ‘R’ around earth while 
    a second satellite is launched into an orbit of radius 1.02 R. What is the 
    percentage difference in the time periods of the two satellites?
     
    Solution:

     Orbital radius of second satellite is 2% more than the first satellite.



    EXAMPLE 8.13:

    What is the periodic time of a satellite revolving above Earth’s surface at a 

    height equal to R, where R is the radius of Earth?


    By knowing the value of time period we can calculate the height of satellite 
    the surface of the earth.

     EXAMPLE 8.14:
     Given radius of earth ‘R’ and length of a day ‘T’, what is the height of a 

    geostationary satellite


    EXAMPLE 8.15:

    A satellite is revolving round the earth in circular orbit at some height above 
    surface of the earth. It takes 5.26 × 103 seconds to complete a revolution 
    while its centripetal acceleration is 9.32 m/s2. What is the height of satellite 

    above the surface of earth? (Radius of the earth 6.37 × 106 m)


     8.11.4. Geostationary Satellite
     The satellite which appears stationary relative to earth is called 
    geostationary or geosynchronous satellite, e.g. communication satellite.
     A geostationary satellite always stays over the same place above the earth. 
    Such a satellite is never at rest. It appears stationary due to its zero relative 
    velocity with respect to that place on earth. 
    The orbit of a geostationary satellite is known as the parking orbit.
     
    Notes:
     • It should revolve in an orbit concentric and coplanar with the equatorial plane.
     • Its sense of rotation should be same as that of earth about its own 
    axis, i.e. in anti-clockwise direction (from west to east).
     • Its period of revolution around the earth should be the same as that of 
    earth about its own axis.

     T = 24 h = 86400 s


    8.11.5. Energy of Satellite
     When a satellite revolves around a planet in its orbit, it possesses both 
    potential energy (due to its position against gravitational pull of earth) and 

    kinetic energy (due to orbital motion).


    Notes
     • Kinetic energy, potential energy or total energy of a satellite depends 
    on the mass of the satellite and the central body and also on the radius 
    of the orbit.
     • From the above expressions we can say that
     Kinetic energy (K) = – (Total energy)
     Potential energy (U) = 2 (Total energy)

     Potential energy (K) = – 2 (Kinetic energy)


    • If the orbit of a satellite is elliptical, then
     
     2- = constant; where a is semi-major axis.
     (b) Kinetic energy (K) will be maximum when the satellite is closest 
    to the central body (at perigee) and maximum when it is farthest 
    from the central body (at apogee).
     (c) Potential energy (U) will be minimum when kinetic energy is 
    maximum, i.e. when satellite is closest to the central body (at 
    perigee). Potential energy is maximum when kinetic energy is 
    minimum, i.e. the satellite is farthest from the central body (at apogee).
    • Binding Energy: Total energy of a satellite in its orbit is negative. 
    Negative energy means that the satellite is bound to the central body 
    by an attractive force and energy must be supplied to remove it from 
    the orbit to infinity. The energy required to remove the satellite from 

    its orbit to infinity is called Binding Energy of the system, i.e.


    Application Activity 8.2
     1. The distance of Neptune and Saturn from sun are nearly 1013 and 
    1012 metres respectively. Assuming that they move in circular orbits, 
    what will be their periodic times in the ratio?
     2. A spherical planet far out in space has a mass M0
     and diameter D0
     A particle of mass m falling freely near the surface of this planet 
    will experience an acceleration due to gravity which is equal to g. 
    Derive the expression of g in terms of D.
     3. At surface of earth, weight of a person is 72 N. What is his weight at 
    height R/2 from surface of earth (R = radius of earth)?
     4. Assuming earth to be a sphere of a uniform density, what is the 
    value of gravitational acceleration in a mine 100 km below the 
    earth’s surface (Given R = 6400 km)?
     5. If the gravitational force between two objects was proportional to 
    1/R; where R is separation between them, then a particle in circular 
    orbit under such a force would have its orbital speed v proportional 
    to which value? 
    6. An earth satellite S has an orbital radius which is 4 times that of a 

    communication satellite C. What is its period of revolution?

    8.12  TYPES AND APPLICATIONS OF SATELLITE SYSTEMS
     Four different types of satellite orbits have been identified depending on 
    the shape and diameter of each orbit:
     • GEO (Geo-stationary earth orbit)
     • MEO (medium earth orbit)
     • LEO (Low earth orbit) and

     • HEO (Highly elliptical orbit)


    GEO (geostationary orbit)
     A geostationary orbit or geosynchronous equatorial orbit (GEO) has a 
    circular orbit 35,786 kilometres above the Earth’s equator and following the 
    direction of the Earth’s rotation. An object in such an orbit has an orbital 
    period equal to the Earth’s rotational period (one sidereal day) and thus 

    appears motionless, at a fixed position in the sky, to ground observers.


    Most common geostationary satellites are either weather satellites or  
    communication satellites relaying signals between two or more ground 
    stations and satellites that broadcast signals to a large area on the planet. 
    All radio and TV, whether satellite etc. are launched in this orbit.
     Advantages of Geo-Stationary Earth Orbit
     1. It is possible to cover almost all parts of the earth with just 3 geo satellites.
     2. Antennas need not be adjusted every now and then, but can be fixed 
    permanently.
     3. The life-time of a GEO satellite is quite high usually around 15 years.
     
    Disadvantages of Geo-Stationary Earth Orbit
     1. Larger antennas are required for northern/southern regions of the earth.
     2. High buildings in a city limit the transmission quality.
     3. High transmission power is required.
     4. These satellites cannot be used for small mobile phones.
     5. Fixing a satellite at Geo stationary orbit is very expensive.
     
    LEO (Low Earth Orbit)
     Satellites in low Earth orbits are normally military reconnaissance satellites 
    that can locate out tanks from 160 km above the Earth. They orbit the earth 
    very quickly, one complete orbit normally taking 90 minutes. However, 
    these orbits have very short lifetimes in the order of weeks compared with 

    decades for geostationary satellites. Simple launch vehicles can be used to 


     Low Earth Orbit is used for things that we want to visit often with the 
    Space Shuttle, like the Hubble Space Telescope and the International Space 
    Station. This is convenient for installing new instruments, fixing things 
    that are broken, and inspecting damage. It is also about the only way we 
    can have people go up, do experiments, and return in a relatively short time.

     A special type of LEO is the Polar Orbit. This is a 
    LEO with a high inclination angle (close to 
    90 degrees). This means the satellite travels over the poles.


    Advantages of Low Earth Orbit
      1. The antennas can have low transmission power of about 1 watt.
     2. The delay of packets is relatively low.
     3. Useful for smaller foot prints

         Disadvantages of Low Earth Orbit
     1. If global coverage is required, it requires at least 50-200 satellites in this orbit.
     2. Special handover mechanisms are required.
     3. These satellites involve complex design.
     4. Very short life: Time of 5-8 years. Assuming 48 satellites with a life-time 
    of 8 years each, a new satellite is needed every 2 months.
    5. Data packets should be routed from satellite to satellite.
     MEO (Medium Earth Orbit) or ICO (Intermediate Circular Orbit)
     Medium Earth Orbit satellites move around the earth at a height of 6000
    20000 km above earth’s surface. Their signal takes 50 to 150 milliseconds 
    to make the round trip. MEO satellites cover more earth area than LEOs 
    but have a higher latency. MEOS are often used in conjunction with GEO 

    satellite systems.

    Advantages of Medium Earth Orbit
     1. Compared to LEO system, MEO requires only a dozen satellites.
     2. Simple in design.

     3. Requires very few handovers.

     Disadvantages of Medium Earth Orbit
     1. Satellites require higher transmission power.
     2. Special antennas are required.
     HEO (Highly Elliptical Orbit)
     A satellite in elliptical orbit follows an oval-shaped path. One part of the orbit 
    is closest to the centre of Earth (perigee) and another part is farthest away 
    (apogee). A satellite in this type of orbit generally has an inclination angle of 
    64 degrees and takes about 12 hours to circle the planet. This type of orbit 

    covers regions of high latitude for a large fraction of its orbital period


    8.13. COSMIC VELOCITY FIRST, SECOND AND THIRD
     The cosmic velocity is the initial velocity which a body must have to be able 
    to overcome the gravity of another object.

     We have:

     1. The first cosmic velocity
     2. Second cosmic velocity

     3. The third cosmic velocity

     8.13.1. The first cosmic velocity
     As you know the satellites which were sent by a human are orbiting around 
    the Earth. They had to be launched with a very high velocity, namely, with 

    the first cosmic velocity. 


    This velocity can be calculated using the gravitational force and the 

     centripetal force of the satellite:


    Satellites must have extremely high velocity to orbit around the Earth. In 
    fact, satellites go around the Earth at the height h = 160 km in order not to 

    break into the atmosphere.

    8.13.2. Second cosmic velocity (escape velocity)
     In the previous section we calculated the velocity which a body has to have 
    to go around the Earth, which means that we calculated the value of the first 
    cosmic velocity. Now it is time to give attention to calculating the second cosmic 
    velocity -it is the speed needed to “break free” from the gravitational 
    attraction of the Earth or celestial body to which it is attract.
     In order to understand this issue we should know something about kinetic 

    and potential energy.


     This value is calculated using the fact that as the body moves away from 
    the Earth, the kinetic energy decreases and the potential energy increases. 
    At infinity, both the energies are equal to zero, because, when the distance 
    between the body and the Earth increases, the kinetic energy decreases and 

    at infinity, it has the value of 0.

     The potential energy at infinity has got the highest value but if we put 
    infinity in the previous formula, we will obtain zero (or an extremely small 
    fraction).

     The value of the second cosmic velocity is calculated as follows;


     We can also obtain the value of the second cosmic velocity by multiply the 

    value of the first cosmic velocity by the square root of two.


     8.13.3. The 3rd cosmic velocity
     The third cosmic velocity is the initial velocity which a body has to have to 
    leave the Solar System and its value is:
    v3= 16.7 km/s at solar system
     At the surface of the Earth, this velocity is about 42 km/s. But due to its 
    revolution, it is enough to launch the body with velocity 16.7 km/s in the 
    direction of this movement.
     
    8.13.4. The fourth cosmic velocity

     It is the initial velocity which a body should have to leave the Milky Way.


     This velocity is about 350 km/s but since Sun is going around the galaxy 
    centre, so it is enough to launch the body with the velocity of 130 km/s in 

    the direction of the Sun’s movement.

    Application Activity 8.3
     The grid shown below contains terms used in this unit. Highlight at 
    least 25 terms. Construct 10 sentences in context of motion in orbits 

    using those words found in the grid.


    Application Activity 8.4

    Using the Across and Down clues, write the correct words in the 

    numbered grid below.


    ACROSS
     1. The only natural satellite of Earth.
     5. An object in orbit around a planet.
     6. The smallest planet and farthest from the Sun.
     7. This planet probably got this name due to its red color and is 
    sometimes referred to as the Red Planet.
     9. This planet’s blue color is the result of absorption of red light by 
    methane in the upper atmosphere.
     10. It is the brightest object in the sky except for the Sun and the moon.

         DOWN
     2. Named after the Roman god of the sea.
     3. The closest planet to the Sun and the eighth largest.
     4. A large cloud of dust and gas which escapes from the nucleus of an 
    active comet.

     8. The largest object in the solar system.

    END OF UNIT ASSESSMENT

    1.  A satellite A of mass m is at a distance of r from the centre of the earth. 
    Another satellite B of mass 2m is at distance of 2r from the earth’s 
    centre. What is the ratio of their time periods?
     2.  Mass of moon is 7.34 × 1022 kg. If the acceleration due to gravity 
    on the moon is 1.4 m/s2, find the radius of moon. Use (G = 6.67 × 
    10–11 Nm2/kg2).
     
    3.  A planet has mass 1/10 of that of earth, while radius is 1/3 that of 
    earth. If a person can throw a stone on earth surface to a height of 90 m, 

    to what height will he be able to throw the stone on that planet?

    4. If the distance between centres of earth and moon is D and the mass of 
    earth is 81 times the mass of moon, then at what distance from centre of 

    earth the gravitational force will be zero?


    5. What is the depth d at which the value of acceleration due to gravity 
    becomes n/1
     times the value at the surface? [R = radius of the earth] 
    6.  The distance between centre of the earth and moon is 384000 km. If the 
    mass of the earth is 6 × 1024 kg and G = 6.67 × 10–11 Nm2/kg2, what is the 
    speed of the moon?
     7.  One project after deviation from its path, starts moving round the earth 
    in a circular path at radius equal to nine times the radius at earth R, 
    what is its time?
     8.  A satellite A of mass m is revolving round the earth at a height ‘r’ from 
    the centre. Another satellite B of mass 2m is revolving at a height 2r. 

    What is the ratio of their time periods?

    UNIT SUMMARY
     Newton’s law of gravitation
     This is also called the universal law of gravitation or inverse square law. 
    And sates that “the gravitational force of attraction between two masses 
    m1 and m2 is directly proportional to the product of masses and inversely 

    proportional to the square of their mean distance apart.”


    Kepler’s laws of planetary motion
     1st Law: This law is called the law of orbits and states that planets move in 
    ellipses with the sun as one of their foci. It can also be stated that planets 
    describe ellipses about the sun as one focus.
     2nd Law: This is called the law of areas and states that the line joining the 
    sun and the planet sweeps out equal areas in equal periods of time.
     3rd Law: The law of periods states that the square of the periods T of 
    revolution of planets are proportional to the cubes of their mean distances 

    R from the sun.



     The depth d is measured from the surface of the earth. The value of 
    acceleration due to gravity increases  as we move towards the surface. At 
    centre of earth g = 0.
     
    Variation in g Due to Rotation of Earth
     As the earth rotates, a body placed on its surface moves along the circular 
    path and hence experiences centrifugal force, due to which the apparent 
    weight of the body decreases. 

    By solving, the acceleration due to gravity is given by;


    Rockets and spacecraft

    A rocket is a device that produces thrust by ejecting stored matter. Spacecraft 
    Propulsion is characterized in general by its complete integration within 

    the spacecraft (e.g. satellites). 

    Satellites 
    A satellite is an artificial body placed in orbit round the earth or another 

    planet in order to collect information or for communication. 


    It is seen that angular momentum of satellite depends on both the mass of 
    orbiting and central body as well as the radius of orbit.
     
    Energy of Satellite
     When a satellite revolves around a planet in its orbit, it possesses both 
    potential energy (due to its position against gravitational pull of earth) and 

    kinetic energy (due to orbital motion).


    Types and applications of Satellite Systems
     • GEO (Geo-stationary earth orbit)
     • MEO (medium earth orbit)
     • LEO (Low earth orbit) and
     • HEO (Highly elliptical orbit)
     
    Cosmic velocity
     The first cosmic velocity
     v1= 7900 m/s
     Second cosmic velocity 
    This is also called the escape velocity, v2 = 11200 m/s
     
    Third cosmic velocity 
    The third cosmic velocity is the initial velocity which a body has to have to  
    escape the Solar System and its value is given by;
     v3 = 16.7 km/s