What is the depth at which the value of acceleration due to gravity becomes \(\frac{1}{{n^{th}}}\) time it's value at the surface of the earth? (radius of the earth = \(\mathrm{R}\))  
1. \(R \over n^2\)
2. \(R~(n-1) \over n\)
3. \(Rn \over (n-1)\)
4. \(R \over n\)

If $v_e$ is the escape velocity and $v_o$ is the orbital velocity of a satellite for orbit close to the earth's surface, then these  are related by:

Which one of the following plots represents the variation of a gravitational field on a particle with distance \(r\) due to a thin spherical shell of radius \(R?\)
(\(r\) is measured from the centre of the spherical shell)

1.		2.
3.		4.

A particle of mass \(\mathrm{m}\) is thrown upwards from the surface of the earth, with a velocity \(\mathrm{u}\). The mass and the radius of the earth are, respectively, \(\mathrm{M}\) and \(\mathrm{R}\). \(\mathrm{G}\) is the gravitational constant and \(\mathrm{g}\) is the acceleration due to gravity on the surface of the earth. The minimum value of \(\mathrm{u}\) so that the particle does not return back to earth is:
1. \(\sqrt{\frac{2 \mathrm{GM}}{\mathrm{R}^2}} \)
2. \(\sqrt{\frac{2 \mathrm{GM}}{\mathrm{R}}} \)
3.\(\sqrt{\frac{2 \mathrm{gM}}{\mathrm{R}^2}} \)
4. \(\sqrt{ \mathrm{2gR^2}}\)

A particle of mass M is situated at the centre of a spherical shell of the same mass and radius a. The magnitude of the gravitational potential at a point situated at a/2 distance from the centre will be:

1. $-\frac{\mathrm{GM}}{\mathrm{a}}$

2. $-\frac{2\mathrm{GM}}{\mathrm{a}}$

3. $-\frac{3\mathrm{GM}}{\mathrm{a}}$

4. $-\frac{4\mathrm{GM}}{\mathrm{a}}$

The dependence of acceleration due to gravity 'g' on the distance 'r' from the centre of the earth, assumed to be a sphere of radius R of uniform density, is as shown in figure below:
   
The correct figure is:
1. a

2. b

3. c

4. d

The additional kinetic energy to be provided to a satellite of mass \(m\) revolving around a planet of mass \(M,\) to transfer it from a circular orbit of radius \(R_1\) to another of radius \(R_2\) (\(R_2>R_1\)) is:

1. \(GmM\) $\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

2. \(2GmM\) $\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

3. $\frac{1}{2}GmM$ $\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

4. \(GmM\) $\left(\frac{1}{R_1^2}-\frac{1}{R_2^2}\right)$

When a body of weight 72 N moves from the surface of the Earth at a height half of the radius of the earth, then the gravitational force exerted on it will be:

1. 36 N

2. 32 N

3. 144 N

4. 50 N

For a satellite moving in an orbit around the earth, the ratio of kinetic energy to potential energy is:
1. $\frac{1}{\sqrt{2}}$
2. \(2\)
3. $\sqrt{2}$
4. $\frac{1}{2}$

Imagine a new planet having the same density as that of the Earth but 3 times bigger than the Earth in size. If the acceleration due to gravity on the surface of the earth is g and that on the surface of the new planet is g', then: