Imagine a light planet revolving around a very massive star in a circular orbit of radius R with a period of revolution T. If the gravitational force of attraction between planet and star is proportional to ${\mathrm{R}}^{\frac{-5}{2}}$, then ${\mathrm{T}}^2$ is proportional to:

1. ${\mathrm{R}}^3$                               

2. ${\mathrm{R}}^{7/2_{}^{}}$

3. ${\mathrm{R}}^{5/2}$                           

4. ${\mathrm{R}}^{3/2}$

A mass M is split into two parts, m and (M–m), which are then separated by a certain distance. What ratio of m/M maximizes the gravitational force between the two parts

1. 1/3                           

2. 1/2

3. 1/4                           

4. 1/5

Two identical satellites are at R and 7R away from the earth surface, the wrong statement is (R = Radius of the earth)

1. Ratio of total energy will be 4

2. Ratio of kinetic energies will be 4

3. Ratio of potential energies will be 4

4. Ratio of total energy will be 4 but ratio of potential and kinetic energies will be 2

A body of mass m is taken from earth's surface to the height h equal to the radius of the earth, the increase in potential energy will be:

1. mgR                                

2. $\frac{1}{2}$mgR

3. 2mgR                             

4. $\frac{1}{4}$ mgR

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. ${\mathrm{E}}_0$

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