A thin rod of length L is bent to form a semicircle. The mass of the rod is M. The gravitational potential at the centre of the circle is :

1. $-\frac{\mathrm{GM}}{\mathrm{L}}$

2. $-\frac{\mathrm{GM}}{2\mathrm{\pi L}}$

3. $-\frac{\mathrm{\pi GM}}{2\mathrm{L}}$

4. $-\frac{\mathrm{\pi GM}}{\mathrm{L}}$

Two particles of equal mass go round a circle of radius R under the action of their mutual gravitational attraction. The speed of each particle is:

1.  $\mathrm{v}=\frac{1}{2\mathrm{R}}\sqrt{\frac{1}{\mathrm{Gm}}}$                             

2. v$=\sqrt{\frac{\mathrm{Gm}}{2\mathrm{R}}}$

3.  v$=\frac{1}{2}\sqrt{\frac{\mathrm{Gm}}{\mathrm{R}}}$                               

4. v$=\sqrt{\frac{4\mathrm{Gm}}{\mathrm{R}}}$   

The distance between the centres of moon and earth is D. The mass of earth is 81 times the mass of the moon. At what distance from the centre of the earth the gravitational force will be zero ?

1. $\frac{D}{2}$                                2. $\frac{2D}{2}$                                

3. $\frac{4D}{3}$                              4. $\frac{9D}{10}$

3 particles each of mass m are kept at vertices of an equilateral triangle of side L. The gravitational field at centre due to these particles is

1. zero                                                                       

2. $\frac{3GM}{L^2}$

3. $\frac{9GM}{L^2}$                                                                     

4. $\frac{12}{\sqrt{3}}\frac{GM}{L^2}$

The centripetal force acting on a satellite orbiting around the earth and the gravitational force of the earth acting on the satellite, both are equal to \(F\). The net force on the satellite is:
1. zero
2. \(F\)
3. \(F\sqrt{2}\)
4. \(2F\)

Two sphere of mass m and M are situated in air and the gravitational force between them is F. The space around the masses is now filled with a liquid of specific gravity 3. The gravitational force will now be

1. F                                        

2. $\frac{F}{3}$  

3. $\frac{F}{9}$                                      

4. 3 F

Two identical solid copper spheres of radius \(R\) are placed in contact with each other. The gravitational attraction between them is proportional to:
1. \(R^2\)
2. \(R^{-2}\)
3. \(R^4\)
4. \(R^{-4}\)

An astronaut orbiting the earth in a circular orbit 120 km above the surface of the earth, gently drops a spoon out of the spaceship. The spoon will:

1. Fall vertically down to the earth

2. Move towards the moon

3. Will move along with space-ship

4. Will move in an irregular way then fall down

Imagine a light planet revolving around a very massive star in a circular orbit of radius R with a period of revolution T. If the gravitational force of attraction between planet and star is proportional to ${\mathrm{R}}^{\frac{-5}{2}}$, then ${\mathrm{T}}^2$ is proportional to:

1. ${\mathrm{R}}^3$                               

2. ${\mathrm{R}}^{7/2_{}^{}}$

3. ${\mathrm{R}}^{5/2}$                           

4. ${\mathrm{R}}^{3/2}$