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 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.  $\frac{\mathrm{R}}{2}$

2.  $\frac{3\mathrm{R}}{2}$

3.  R

4.  $\frac{\mathrm{R}}{4}$

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?

A satellite of mass 1000 kg revolves in a circular orbit around the earth with a constant speed of 100 m/ s. The total mechanical energy of the satellite is: