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. ${\mathrm{E}}_0$

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

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:

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: