What is the depth at which the value of acceleration due to gravity becomes \(\dfrac{1}{{n}}\) times it's value at the surface of the earth? (radius of the earth = \(\mathrm{R}\))  

1.	\(\dfrac R {n^2}\)	2.	\(\dfrac {R~(n-1)} n\)
3.	\(\dfrac {Rn} { (n-1)}\)	4.	\(\dfrac R n\)

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 \(m\) is thrown upwards from the surface of the earth, with a velocity \(u.\) The mass and the radius of the earth are, respectively, \(M\) and \(R.\) \(G\) is the gravitational constant and \(g\) is the acceleration due to gravity on the surface of the earth. The minimum value of \(u\) so that the particle does not return back to earth is:

1. \(\sqrt{\dfrac{2 {GM}}{{R}^2}} \)

2. \(\sqrt{\dfrac{2 {GM}}{{R}}} \)

3.\(\sqrt{\dfrac{2 {gM}}{{R}^2}} \)

4. \(\sqrt{ {2gR^2}}\)

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:

(a)		(b)
(c)		(d)

                    
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 \(72\text{ N}\) 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. \(36\text{ N}\)

2. \(32\text{ N}\)

3. \(144\text{ N}\)

4. \(50\text{ N}\)

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. $\frac{1}{2}$