Two bodies of masses ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$  are initially at rest at infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance r between them is

1. ${\left[2\mathrm{G}\frac{({\mathrm{m}}_1-{\mathrm{m}}_2)}{\mathrm{r}}\right]}^{1/2}$                         

2. ${\left[\frac{2\mathrm{G}}{\mathrm{r}}({\mathrm{m}}_1+{\mathrm{m}}_2)\right]}^{1/2}$

3. ${\left[\frac{\mathrm{r}}{2\mathrm{G}({\mathrm{m}}_1{\mathrm{m}}_{2)}}\right]}^{1/2}$                             

4. ${\left[\frac{2\mathrm{G}}{\mathrm{r}}{\mathrm{m}}_1{\mathrm{m}}_2\right]}^{1/2}$

The period of the moon’s rotation around the earth is nearly 29 days. If the moon’s mass were 2 fold of its present value and all other things remained unchanged, the period of moon’s rotation would be nearly:

1. $29\sqrt{2} days$                                     

2. $29/\sqrt{2} days$

3. $29\times 2 days$                                       

4. 29 days

The distance of a planet from the sun is 5 times the distance between the earth and the sun. The time period of the planet is -

1. $5^{3/2}$ years                       

2. $5^{2/3}$ years

3. $5^{1/3}$ years                       

4. $5^{1/2}$ years

The diagram showing the variation of the gravitational potential of the earth with distance r from the centre of the earth is:

In planetary motion the areal velocity of position vector of a planet depends on angular velocity $\left(\omega \right)$ and the distance of the planet from Sun (r). If so the correct relation for areal velocity is -

1. $\frac{dA}{dt}\propto \omega r$                            

2. $\frac{dA}{dt}\propto {\omega }^{2}r$ 

3. $\frac{dA}{dt}\propto \omega r^2$                           

4. $\frac{dA}{dt}\propto \sqrt{\omega r}$ 

A sphere of mass M and radius R₂ has a concentric cavity of radius R₁ as shown in figure. The force F exerted by the sphere on a particle of mass m located at a distance r from the centre of sphere varies as $(0\le \mathrm{r}\le \infty )$. 

Which one of the following graphs represents correctly the variation of the gravitational field (I) with the distance (r) from the centre of a spherical shell of mass M and radius a ?

Which of the following graphs represents the motion of a planet moving about the sun ?

The correct graph representing the variation of total energy (${\mathrm{E}}_{\mathrm{t}}$), kinetic energy (${\mathrm{E}}_{\mathrm{k}}$) and potential energy (U) of a satellite with its distance from the centre of the earth is -

The diameters of two planets are in the ratio 4 : 1 and their mean densities in the ratio 1 : 2. The acceleration due to gravity on the planets will be in ratio

1.  1 : 2                             

2.  2 : 3

3.  2 : 1                           

4.  4 : 1