Three equal masses \(\text{(m)}\) are placed at the three vertices of an equilateral triangle of side \(\text{r}\). Work required to double the separation between masses will be:-

1.	\(Gm^2\over r\)	2.	\(3Gm^2\over r\)
3.	\({3 \over 2}{Gm^2\over r}\)	4.	None

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

An artificial satellite moving in a circular orbit around the earth has a total (kinetic + potential) energy ${\mathrm{E}}_0$ . Its potential energy is?         

1. $-{\mathrm{E}}_0$                       

2. $1.5$ ${\mathrm{E}}_0$

3. $2{\mathrm{E}}_0$                         

4. ${\mathrm{E}}_0$

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

Radii and densities of two planets are ${\mathrm{R}}_1, {\mathrm{R}}_2$ and ${\mathrm{\rho }}_1, {\mathrm{\rho }}_2$ respectively. The ratio of accelerations due to gravity on their surfaces is:

1.  $\frac{{\mathrm{\rho }}_1}{{\mathrm{R}}_1} : \frac{{\mathrm{\rho }}_2}{{\mathrm{R}}_2}$

2.  $\frac{{\mathrm{\rho }}_1}{{\mathrm{R}}_1^2} : \frac{{\mathrm{\rho }}_2}{{\mathrm{R}}_2^2}$

3.  ${\mathrm{\rho }}_1{\mathrm{R}}_1 : {\mathrm{\rho }}_2{\mathrm{R}}_2$

4.  $\frac{1}{{\mathrm{\rho }}_1{\mathrm{R}}_1} : \frac{1}{{\mathrm{\rho }}_2{\mathrm{R}}_2}$

1 kg of sugar has maximum weight:
1. at the pole.
2. at the equator.
3. at a latitude of 45$°$.
4. in India.

A particle is located midway between two point masses each of mass \(\mathrm{M}\) kept at a separation \(2\mathrm{d}.\) The escape speed of the particle is: (neglect the effect of any other gravitational effect)

1.  $\sqrt{\frac{2\mathrm{GM}}{\mathrm{d}}}$

2.  $2\sqrt{\frac{\mathrm{GM}}{\mathrm{d}}}$

3.  $\sqrt{\frac{3\mathrm{GM}}{\mathrm{d}}}$

4.  $\sqrt{\frac{\mathrm{GM}}{2\mathrm{d}}}$

A planet is revolving around a massive star in a circular orbit of radius R. If the gravitational force of attraction between the planet and the star is inversely proportional to ${\mathrm{R}}^3$, then the time period of revolution T is proportional to:

1. ${\mathrm{R}}^5$

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

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

4. R

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 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: