A point P lies on the axis of a ring of mass M and radius 'a' at a distance 'a' from its centre C. A small particle starts from P and reaches C under gravitational attraction. Its speed at C will be :

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

2. $\sqrt{\frac{2\mathrm{GM}}{\mathrm{a}}\left(1-\frac{1}{\sqrt{2}}\right)}$

3. $\sqrt{\frac{2\mathrm{GM}}{\mathrm{a}}\left(\sqrt{2}-1\right)}$

4. zero

If g is the acceleration due to gravity on the earth's surface, the gain in the potential energy of an object of mass m raised from the surface of earth to a height equal to the radius of the earth R, is 

1. $\frac{1}{2}mgR$

2. 2 mgR

3. mgR

4. $\frac{1}{4}mgR$

The change in the potential energy, when a body of mass m is raised to a height nR from the

Earth's surface is: (R = Radius of the Earth)

1. $mgR\left(\frac{n}{n-1}\right)$                     

2. nmgR

3. mgR$\left(\frac{n^2}{n^2+1}\right)$                   

4. $mgR\left(\frac{n}{n+1}\right)$

The masses and radii of the earth and moon are $M_{1, }{R}_1$ and $M_2, R_2$ respectively. Their centres are distance d apart. The minimum velocity with which a particle of mass m should be projected from a point midway between their centres so that it escapes to infinity is:

1. $2\sqrt{\frac{G}{D}\left(M_1+M_2\right)}$                 

2. $2\sqrt{\frac{2G}{D}(M_1+M_2)}$

3. $2\sqrt{\frac{GM}{D}\left(M_1+M_2\right)}$               

4. $2\sqrt{\frac{GM\left(M_1+M_2\right)}{D\left(R_1+R_2\right)}}$

If the mass of the earth is M, the radius is R and the gravitational constant is G, then work done to take

1 kg mass from earth surface to infinity will be:

1.  $\sqrt{\frac{GM}{2R}}$                      

2. $\frac{GM}{R}$

3.   $\sqrt{\frac{2GM}{R}}$                

4.   $\frac{GM}{2R}$

Two identical satellites are at R and 7R away from earth surface, the wrong statement is (R = Radius of 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 the 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. \(2 ~mgR\)
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) ${\mathrm{E}}_0$

A rocket of mass M is launched vertically from the surface of the earth with an initial speed v. Assuming the radius of the earth to be R and negligible air resistance, the maximum height attained by the rocket above the surface of the earth is   

1. $\mathrm{R}/\left(\frac{\mathrm{gR}}{2{\mathrm{v}}^2}-1\right)$                         

2. $\mathrm{R}\left(\frac{\mathrm{gR}}{2{\mathrm{v}}^2}-1\right)$

3. $\mathrm{R}/\left(\frac{2\mathrm{gR}}{{\mathrm{v}}^2}-1\right)$                         

4. $\mathrm{R}\left(\frac{2\mathrm{gR}}{{\mathrm{v}}^2}-1\right)$

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