Chapter 7 Solutions — Gravitation

Exercises

1. 1.
   Answer the following:

   • (a) You can shield a charge from electrical forces by putting it inside a hollow conductor. Can you shield a body from the gravitational influence of nearby matter by putting it inside a hollow sphere or by some other means?
   • (b) An astronaut inside a small space ship orbiting around the earth cannot detect gravity. If the space station orbiting around the earth has a large…
2. 2.
   Choose the correct alternative:

   • (a) Acceleration due to gravity increases/decreases with increasing altitude.
   • (b) Acceleration due to gravity increases/decreases with increasing depth (assume the earth to be a sphere of uniform density).
   • (c) Acceleration due to gravity is independent of mass of the earth/mass of the body.
   • (d) The formula −G M m (1/r₂ − 1/r₁) is more/less accurate than the…
3. 3.
   Suppose there existed a planet that went around the Sun twice as fast as the earth. What would be its orbital size as compared to that of the earth?
4. 4.
   Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 × 10⁸ m. Show that the mass of Jupiter is about one-thousandth that of the sun.
5. 5.
   Let us assume that our galaxy consists of 2.5 × 10¹¹ stars each of one solar mass. How long will a star at a distance of 50,000 ly from the galactic centre take to complete one revolution? Take the diameter of the Milky Way to be 10⁵ ly.
6. 6.
   Choose the correct alternative:

   • (a) If the zero of potential energy is at infinity, the total energy of an orbiting satellite is negative of its kinetic/potential energy.
   • (b) The energy required to launch an orbiting satellite out of earth's gravitational influence is more/less than the energy required to project a stationary object at the same height (as the satellite) out of earth's influence.
7. 7.
   Does the escape speed of a body from the earth depend on

   • (a) the mass of the body,
   • (b) the location from where it is projected,
   • (c) the direction of projection,
   • (d) the height of the location from where the body is launched?
8. 8.
   A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant

   • (a) linear speed,
   • (b) angular speed,
   • (c) angular momentum,
   • (d) kinetic energy,
   • (e) potential energy,
   • (f) total energy throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.
9. 9.
   Which of the following symptoms is likely to afflict an astronaut in space

   • (a) swollen feet,
   • (b) swollen face,
   • (c) headache,
   • (d) orientational problem.
10. 10.
    In the following two exercises, choose the correct answer from among the given ones: The gravitational intensity at the centre of a hemispherical shell of uniform mass density has the direction indicated by the arrow (see Fig 7.11)

    • (i) a,
    • (ii) b,
    • (iii) c,
    • (iv) 0.
11. 11.
    For the above problem, the direction of the gravitational intensity at an arbitrary point P is indicated by the arrow

    • (i) d,
    • (ii) e,
    • (iii) f,
    • (iv) g.
12. 12.
    A rocket is fired from the earth towards the sun. At what distance from the earth's centre is the gravitational force on the rocket zero? Mass of the sun = 2 × 10³⁰ kg, mass of the earth = 6 × 10²⁴ kg. Neglect the effect of other planets etc. (orbital radius = 1.5 × 10¹¹ m).
13. 13.
    How will you 'weigh the sun', that is estimate its mass? The mean orbital radius of the earth around the sun is 1.5 × 10⁸ km.
14. 14.
    A saturn year is 29.5 times the earth year. How far is the saturn from the sun if the earth is 1.50 × 10⁸ km away from the sun?
15. 15.
    A body weighs 63 N on the surface of the earth. What is the gravitational force on it due to the earth at a height equal to half the radius of the earth?
16. 16.
    Assuming the earth to be a sphere of uniform mass density, how much would a body weigh half way down to the centre of the earth if it weighed 250 N on the surface?
17. 17.
    A rocket is fired vertically with a speed of 5 km s⁻¹ from the earth's surface. How far from the earth does the rocket go before returning to the earth? Mass of the earth = 6.0 × 10²⁴ kg; mean radius of the earth = 6.4 × 10⁶ m; G = 6.67 × 10⁻¹¹ N m² kg⁻².
18. 18.
    The escape speed of a projectile on the earth's surface is 11.2 km s⁻¹. A body is projected out with thrice this speed. What is the speed of the body far away from the earth? Ignore the presence of the sun and other planets.
19. 19.
    A satellite orbits the earth at a height of 400 km above the surface. How much energy must be expended to rocket the satellite out of the earth's gravitational influence? Mass of the satellite = 200 kg; mass of the earth = 6.0 × 10²⁴ kg; radius of the earth = 6.4 × 10⁶ m; G = 6.67 × 10⁻¹¹ N m² kg⁻².
20. 20.
    Two stars each of one solar mass (= 2 × 10³⁰ kg) are approaching each other for a head on collision. When they are a distance 10⁹ km, their speeds are negligible. What is the speed with which they collide? The radius of each star is 10⁴ km. Assume the stars to remain undistorted until they collide. (Use the known value of G).
21. 21.
    Two heavy spheres each of mass 100 kg and radius 0.10 m are placed 1.0 m apart on a horizontal table. What is the gravitational force and potential at the mid point of the line joining the centres of the spheres? Is an object placed at that point in equilibrium? If so, is the equilibrium stable or unstable?