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:

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

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

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. $\frac{1}{2}$

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

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:
           

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:-