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:
           

1.	\(t_1>t_2\)	2.	\(t_1=4t_2\)
3.	\(t_1=2t_2\)	4.	\(t_1=t_2\)

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?

The earth is assumed to be a sphere of radius R. A platform is arranged at a height R from the surface of the earth. The escape velocity of a body from this platform is fv_e, where v_e is its escape velocity from the surface of the earth. The value of f is:

1. $\sqrt{2}$

2. $\frac{1}{\sqrt{2}}$

3. $\frac{1}{3}$

4. $\frac{1}{2}$

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

Two particles of equal masses go around 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. $\mathrm{v}=\sqrt{\frac{\mathrm{Gm}}{2\mathrm{R}}}$

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

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

Mass \(M\) is divided into two parts \(xM\) and \((1-x)M.\) For a given separation, the value of \(x\) for which the gravitational attraction between the two pieces becomes maximum is:

 An object weighs 72 N on earth. Its weight at a height R/2 from the surface of the earth will be:

A body projected vertically from the earth reaches a height equal to earth’s radius before returning to the earth. The power exerted by the gravitational force:

A body of mass \(m\) is taken from the Earth’s surface to the height equal to twice the radius \((R)\) of the Earth. The change in potential energy of the body will be: 

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. \(100​~\text{m}\)