Radii and densities of two planets are \(R_1, R_2\) and \(\rho_1, \rho_2\) respectively. The ratio of accelerations due to gravity on their surfaces is:
1. \(\frac{\rho_1}{R_1}:\frac{\rho_2}{R_2}\)
2. \(\frac{\rho_1}{R^2_1}: \frac{\rho_2}{R^2_2}\)
3. \(\rho_1 R_1 : \rho_2R_2\)
4. \(\frac{1}{\rho_1R_1}:\frac{1}{\rho_2R_2}\)

\(1\) kg of sugar has maximum weight:
1. at the pole.
2. at the equator.
3. at a latitude of \(45^{\circ}.\)
4. in India.

A body is thrown vertically upwards with an initial speed \(\sqrt{gR}\), where \(R\) is the radius of the earth. The maximum height reached by the body from the surface of the earth is:
1. \(\frac{R}{2}\)
2. \(\frac{3R}{2}\)
3. \(R\)
4. \(\frac{R}{4}\)

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

Three identical particles each of mass M are located at the vertices of an equilateral triangle of side a. The escape speed of one particle will be:

1.  $\sqrt{\frac{4\mathrm{GM}}{\mathrm{a}}}$

2.  $\sqrt{\frac{3\mathrm{GM}}{\mathrm{a}}}$

3.  $\sqrt{\frac{2\mathrm{GM}}{\mathrm{a}}}$

4.  $\sqrt{\frac{\mathrm{GM}}{\mathrm{a}}}$

The escape velocities from the surface of two planets of the same mass are in the ratio of $1:\sqrt{2}$. The ratio of their densities is:

Two identical hollow spheres of negligible thickness are placed in contact with each other. The force of gravitation between the spheres will be proportional to (\(R\) = radius of each sphere):
1. \(R\)
2. \(R^2\)
3. \(R^4\)
4. \(R^3\)

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

When a planet revolves around the sun in an elliptical orbit, then which of the following remains constant?