If the radius of a planet is \(R\) and its density is \(\rho,\) the escape velocity from its surface will be:
1. \(v_e\propto \rho R\)
2. \(v_e\propto \sqrt{\rho} R\)
3. \(v_e\propto \frac{\sqrt{\rho}}{R}\)
4. \(v_e\propto \frac{1}{\sqrt{\rho} R}\)

Three equal masses \(m\) are placed at the three vertices of an equilateral triangle of sides \(r.\) The work required to double the separation between masses 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. \(v = \frac{1}{2 R} \sqrt{\frac{1}{Gm}}\)
2. \(v = \sqrt{\frac{Gm}{2 R}}\)
3. \(v = \frac{1}{2} \sqrt{\frac{G m}{R}}\)
4. \(v = \sqrt{\frac{4 Gm}{R}}\)

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