1. A body is taken from the centre of the earth to the moon. What will be the change in the weight of the body?

The weight of the body at the centre of earth will be zero (g = 0). As the body is moved from the centre to the earth surface, its weight will increase due to the increase in the value of g. At the surface of the earth, the weight of the body will be maximum. As the body is moved above the surface of the earth its weight will decrease due to decrease in the value of g. At one place where the gravitational force of earth and moon are equal and opposite, the weight will become zero. Beyond this upto moon, gravitational force of the moon and hence weight of the body will go on increasing.

2. An artificial satellite revolving coplanar with the equator around the earth, appears stationary to an observer on the earth. Calculate the height of the satellite above earth's surface. Given that g = 9.8ms^-2and radius of the earth R = 6.37×10⁶ m.

Let h be the height of the satellite above the surface of the earth. The period of revolution of a satellite is given by:

Since the satellite appears stationary to an observer on the earth, its period of revolution is equal to the period of axial rotation (24 hours) of the earth (also the satellite is revolving from west to east) that is

3. Prove that the angular momentum of a satellite of mass M_s revolving round the earth of mass M_e in an orbit of radius r is equal to

The gravitational force between the earth and satellite provides the necessary centripetal force to move the satellite in the circular orbit. If v is the orbital velocity of the satellite, then,

4. Show that moon would depart forever if its speed were increased by 42%.

Mass of earth = M_E; Mass of moon = M_M

If the distance between the earth and moon is r, then necessary centripetal force( = M_Mv²/r)is provided by the gravitational attraction ( = GM_EM_M/r²) between earth and moon.

5. Satellites remain in orbit instead of falling to the earth because they are beyond the main pull of earth's gravity. Is this statement correct?

No, not at all. If any object were beyond the pull of gravity, it would move in a straight line and would not curve around the earth. Satellites remain in the orbit because they are being pulled by gravity, not because they are beyond it. For the altitudes of most earth satellites, the earth's gravitational field is only a few percent weaker than at the earth's surface.

6. A satellite orbiting close to earth's surface in a circular path falls about 4.9 m each second. Why does not this distance accumulate and send satellite crashing to earth?

In each second, the satellite falls about 4.9 m below the straight-line tangent it would have taken if there were no gravity. The earth's surface also curves 4.9 m beneath a straight line 8 km tangent. The process of falling with the curvature of the earth continues from tangent line to tangent line, so the curved path of the satellite and the curve of earth's surface "match" all the way around the earth.

7. Why are space rockets usually launched from west to east?

We know that earth rotates about its axis from west to east. Therefore, any point on the earth's surface has linear velocity from west to east. When a rocket is launched from west to east, the linear velocity of earth is added to the launching velocity of the rocket.

8. Why does earth not retain hydrogen and helium molecules in its atmosphere?

The average velocity of a gas molecule depends upon its mass and temperature. Lighter molecules such as hydrogen and helium have a high average velocity than the heavier molecules. Therefore, lighter molecules have a greater chance to escape from the planet. This explains why the earth does not retain hydrogen and helium molecules in its atmosphere while much heavier molecules, such as oxygen and nitrogen, do not escape.

9. The mass of earth is M = 6×10²⁴ kg and its radius R = 6.4×10⁶ m. How much work will have to be done in taking a 10 kg body from the surface of the earth to infinity? What is the gravitational potential energy of the body on the earth's surface? If this body falls from infinity to earth, what will be its velocity when striking the earth?

The work done in taking a body of mass m from the surface of the earth to infinity is

10. A satellite of mass 1000 kg moves in a circular orbit of radius 7000 km around the earth. Calculate the total energy required to place the satellite in the orbit from earth's surface.

Take g =10ms^-2 and earth's radius R = 6400 km.

Suppose a satellite mass m is launched from the surface of earth (radius R, mass M) into an orbit of radius r(= R + h = 7000 km). If v is the orbital speed of the satellite in the orbit, then,

Total energy required, W = Gain in K.E. + Gain in P.E. 

11. A particle is projected vertically upward from the surface of the earth (radius R) with a kinetic energy equal to half of the minimum value needed for it to escape. Calculate the height through which it rises above the surface of earth.