The energy required to move a satellite of mass m from an orbit of radius 2R to 3R around the Earth having mass M is:

1.	\(\frac{\mathrm{GMm}}{\mathrm{12R}} \)	2.	\(\frac{\mathrm{GMm}}{\mathrm{R}} \)
3.	\(\frac{\mathrm{GMm}}{8 \mathrm{R}} \)	4.	\(\frac{\mathrm{GMm}}{2 \mathrm{R}}\)

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:-

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

The gravitational force between two point masses $m_1$  and $m_2$  at separation r is given by $F=k\frac{m_1{m}_2}{r^2}$

The constant k:

The centripetal force acting on a satellite orbiting around the earth and the gravitational force of the earth acting on the satellite, both are equal to F. The net force on the satellite is:

1. Zero                                  

2. F

3. $F\sqrt{2}$                                 

4. 2 F

Two identical solid copper spheres of radius R are placed in contact with each other. The gravitational attraction between them is proportional to

1. $R^2$                                               

2. $R^{‐2}$

3. $R^4$                                               

4. $R^{‐4}$

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. ${\mathrm{E}}_0$

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

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:

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