Gravitation

(Assume g = 10 m/s² or 10 N/kg unless otherwise stated)

1. A satellite is to be put into orbit 500 km above the earth's surface. If its vertical velocity after launching is 2000 m/s at this height, calculate the magnitude and direction of the impulse required to put the satellite directly into orbit, if its mass is 50 kg. Assume radius of earth, r_E = 6400 km. [Answer: 4.0 x 10⁵ kg m/s, 14.6°]

2. A  satellite of mass 1000 kg moves in a circular orbit of radius 7000 km round the earth, assumed to be a sphere of radius 6400 km. Calculate the total energy needed to place the satellite in orbit from the earth. [Answer: 3.5 x 10¹⁰ J]

3. A satellite in a stable orbit contains two closed vessels: one of these is filled with water while the other is filled with hot steam. Explain why the water exerts very little pressure on its container but the steam exerts almost the same pressure as it would on earth when at the same temperature.

4. The gravitational force on a mass of 1 kg at the earth's surface is 10 N. Assuming the earth is a sphere of radius R, calculate the gravitational force on a satellite of mass 100 kg in a circular orbit of radius 2R from the centre of the earth. [Answer: 250 N]

5. Assuming the earth is a uniform sphere of mass M and radius R, show that the acceleration of free-fall at the earth's surface is given by g = GM / R2. What is the acceleration of a satellite moving in a circular orbit round the earth of radius 2R? [Answer: 0.25g]

6. A planet of mass m moves round the sun of mass M in a circular orbit of radius r with an angular speed w. Show (i) that w is independent of the mass of the planet, (ii) that in a circular orbit of radius 4r round the sun, the angular speed decreases to w/8.

7. A satellite X moves round the earth in a circular orbit of radius R. Another satellite Y of the same mass moves round the earth in a circular orbit of radius 4R. Show that (i) the speed of X is twice that of Y, (ii) the kinetic energy of X is greater than that of Y, (iii) the potential energy of X is less than that of Y. Has X or Y the greater total energy (kinetic plus potential energy)? [Answer: Y]

8. Find the period of revolution of a satellite moving in a circular orbit round the earth at a height of 3.6 x 10⁶ m above the earth's surface. Assuming the earth is a uniform sphere of radius 6.4 x 10⁶ m, the earth's mass is 6 x10²⁴ kg and G is 6.7 x 10^-11 Nm²kg^-2. [Answer: 9910 s]

9. If the acceleration of the free fall at the earth's surface is 9.8 m/s2, and the radius of the earth is 6400 km, calculate a value of for the mass of the earth (G = 6.7 x 10^-11 Nm²kg^-2). Give the theory. [Answer: 6 x 10²⁴ kg]

10. Two stars, masses 10²⁰ kg and 2 x 10²⁰ kg respectively, rotate about their common centre of mass with an angular speed w. Assuming that the only force on a star is the mutual gravitational force between them, calculate w. Assume that the distance between the stars is 106 km and that G is 6.7 x 10^-11 Nm²kg^-2. [Answer: 4.5 x 10-9 rad/s]

11. A preliminary stage of spacecraft Apollo 11's journey to the moon was to place it in an earth parking orbit. This orbit was circular, maintaining an almost constant distance 189 km from the earth's surface. Assuming the gravitational field strength in this orbit is 9.4 N kg^-1, calculate (a) the speed of the spacecraft in this orbit and (b) the time to complete one orbit. (Radius of the earth = 6370 km) [Answer: (a) 7852 m/s (b) 5250 s]

12. Explorer 38, a radio-astronomy research satellite of mass 200 kg, circles the earth in an orbit of average radius 3R/2, where R is the radius of the earth. Assuming the gravitational pull on a mass of 1 kg at the earth's surface to be 10 N, calculate the pull on the satellite. [Answer: 889 N]

13. A satellite of mass 66 kg is in orbit round the earth at a distance of 5.7 R above its surface, where R is the value of the mean radius of the earth. If the gravitational field strength at the earth's surface is 9.8 N kg-1, calculate the centripetal force acting on the satellite. Assuming the earth's mean radius to be 6400 km, calculate the period of the satellite in orbit in hours. [Answer: 14.4 N, 24.5 h]

14. (a) Explain what is meant by gravitational field strength. In what units is it measured? Starting with Newton's law of gravitation, derive an expression for g, the acceleration of free-fall on the earth's surface, stating clearly the meaning of each symbol used. ( Assume that the earth may be considered as a point mass located at its centre.)
(b) g may be found by measuring the acceleration of a free-falling body. Outline how you would measure g in this way, indicating the measurements needed and how you would calculate a value of g from them.
(c) At one point on the line between the earth and the moon, the gravitational field caused by the two bodies is zero. Briefly explain why is it so. If this point is 4 x 10⁴ km from the moon, calculate the ration of mass of the moon to the mass of earth. (Distance from earth to moon = 4.0 x 10⁵ km) [Answer: (c) 0.012]