Kepler's third law states that the square of the period of revolution (\(T\)) of a planet around the sun, is proportional to the third power of average distance \(r\) between the sun and planet i.e. \(T^2 = Kr^3\), here \(K\) is constant. If the masses of the sun and planet are \(M\) and \(m\) respectively, then as per Newton's law of gravitation, the force of attraction between them is \(F = \frac{GMm}{r^2},\) here \(G\) is the gravitational constant. The relation between \(G\) and \(K\) is described as:
1. \(GK = 4\pi^2\)
2. \(GMK = 4\pi^2\)
3. \(K =G\)
4. \(K = \frac{1}{G}\)

A black hole is an object whose gravitational field is so strong that even light cannot escape from it. To what approximate radius would Earth (mass \(= 5.98\times 10^{24}~\text{kg}\)) have to be compressed to be a black hole?
1. \(10^{-9}~\text{m}\)
2. \(10^{-6}~\text{m}\)
3. \(10^{-2}~\text{m}\)
4. \(100​~\text{m}\)

The figure shows the elliptical orbit of a planet \(m\) about the sun \(\mathrm{S}.\) The shaded area \(\mathrm{SCD}\) is twice the shaded area \(\mathrm{SAB}.\) If \(t_1\) is the time for the planet to move from \(\mathrm{C}\) to \(\mathrm{D}\) and \(t_2\) is the time to move from \(\mathrm{A}\) to \(\mathrm{B},\) then:
           

Two satellites of Earth, \(S_1\)_, and \(S_2\), are moving in the same orbit. The mass of \(S_1\) is four times the mass of \(S_2\). Which one of the following statements is true?

A body weighs \(200\) N on the surface of the earth. How much will it weigh halfway down the centre of the earth?

If the mass of the sun were ten times smaller and the universal gravitational constant were ten times larger in magnitude, which of the following statements would not be correct?

A satellite of mass \(m\) is orbiting the earth (of radius \(R\)) at a height \(h\) from its surface. What is the total energy of the satellite in terms of \(g_0?\)
(\(g_0\) is the value of acceleration due to gravity at the earth's surface)
1. \(\frac{mg_0R^2}{2(R+h)}\)
2. \(-\frac{mg_0R^2}{2(R+h)}\)
3. \(\frac{2mg_0R^2}{(R+h)}\)
4. \(-\frac{2mg_0R^2}{(R+h)}\)

Starting from the centre of the earth, having radius \(R,\) the variation of \(g\) (acceleration due to gravity) is shown by:

1.		2.
3.		4.

If the radius of a planet is \(\mathrm{R}\) and its density is $\rho$ , the escape velocity from its surface will be:

1. $$ ${\mathrm{v}}_{\mathrm{e}}\propto \mathrm{\rho R}$

2. $$ ${\mathrm{v}}_{\mathrm{e}}\propto \sqrt{\mathrm{\rho }}\mathrm{R}$

3. $$ ${\mathrm{v}}_{\mathrm{e}}\propto \frac{\sqrt{\mathrm{\rho }}}{\mathrm{R}}$

4. $$ ${\mathrm{v}}_{\mathrm{e}}\propto \frac{1}{\sqrt{\mathrm{\rho }}\mathrm{R}}$

If a particle is dropped from a height h = 3 R from the earth's surface, the speed with which the particle will strike the ground is :

1.  $\sqrt{3 \mathrm{gR}}$

2.  $\sqrt{2 \mathrm{gR}}$

3.  $\sqrt{1.5 \mathrm{gR}}$

4.  $\sqrt{\mathrm{gR}}$