# Numerical Problems , Gravitation

#### LEVEL – I

Q:1. Three identical bodies, each of mass m, are separated by a distance a and are found to start moving towards one another under mutual force of gravitational attraction. If at t = t0, the separation between any two is a/2, find the speed of each body at time t0.

Ans: $\displaystyle \sqrt{\frac{2 G m}{a}}$

Q:2. If the radius and density of a planet are two times and half respectively of those of earth, find the intensity of gravitational field at planet surface and escape velocity from planet.

Ans: 15.84 km/s

Q:3. A mass M is split into two parts ‘ m ‘ and (M – m), which are the separated by a certain distance. What ratio of ( m/M ) maximizes the gravitational force between the parts ?

Q:4. Four massive particles, each of mass m, are kept at the vertices of square of side l. With what speed should the system rotate in its plane about its centre so as to remain stable ?

Ans: $\displaystyle (\frac{G m}{\sqrt{2}l}(\sqrt{2} + \frac{1}{2}))^{1/2}$

Q:5. Two concentric spherical shells have masses M1, M2 and radii R1, R2 (R1 < R2). What is the force exerted by this system on a particle of mass m if it is placed at a distance (R1 + R2)/2 from the centre?

Ans: Force exerted by M2 on m = 0

Q:6. A body stretches a spring by a particular length at the earth’s surface at the equator. At what height above the South Pole will it stretch the same spring to the same length? Assume the earth to be spherical.

Q:7. Find the radius of the circular orbit of a satellite moving with an angular speed equal to the angular speed of earth’s rotation.

Ans: 42297 km

Q:8. What is the true weight of an object, that weighed exactly 10.0 N at the north pole, at the position of a geostationary satellite?

Ans : 0.23 N

Q:9. What should be the period of rotation of the earth so that every object on the equator is weightless ?

Ans: 84 min

Q:10. A particle is fired vertically upward with a speed 15 km/s.

(a) Show that it will escape from the earth and

(b) With what speed will it move in interstellar space? Assume the presence of the earth’s gravitational field only.

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