The additional kinetic energy to be provided to a satellite of mass m revolving around a planet of mass M, to transfer it from a circular orbit of radius ${\mathrm{R}}_1$ to another of radius ${\mathrm{R}}_2({\mathrm{R}}_2>{\mathrm{R}}_1)$ is

(1) $GmM\left(\frac{1}{R_{\mathit{1}}^{\mathit{2}}}-\frac{1}{R_{\mathit{2}}^{\mathit{2}}}\right)$                               

(2) $GmM\left(\frac{1}{R_{\mathit{1}}}-\frac{1}{R_{\mathit{2}}}\right)$

(3) $\mathit{2}GmM\left(\frac{1}{R_{\mathit{1}}}-\frac{1}{R_{\mathit{2}}}\right)$                               

(4) $\frac{\mathit{1}}{\mathit{2}}GmM\left(\frac{1}{R_{\mathit{1}}}-\frac{1}{R_{\mathit{2}}}\right)$

The dependence of acceleration due to gravity g on the distance r from the centre of the earth assumed to be a sphere of radius R of uniform density is as shown in the figure below -

                          (1)                                                           (2)

                             (3)                                                          (4)

The correct figure is

(a)  (1)                                             (b)  (3)

(c)  (2)                                              (d)  (4)

The figure shows elliptical orbit of a planet m about the sun S. The shaded area SCD is twice the shaded area SAB. If $t_1$ is the time for planet to move from C to D and $t_2$ is the time to move from A to B, then

(1) $t_1>t_2$

(2) $t_1=4t_2$

(3) $t_1=2t_2$

(4) $t_1=t_2$

A particle of mass M is situated at the centre of a spherical shell of mass M and radius a.The gravitational potential at a point situated at a/2 distance from the centre will be 

1. $-\frac{3GM}{a}$                                       2. $-\frac{2GM}{a}$

3. $-\frac{GM}{a}$                                         4. $-\frac{4GM}{a}$

The radii of the circular orbits of two satellites \(A\) and \(B\) of the earth are \(4R\) and \(R,\) respectively. If the speed of the satellite \(A\) is \(3v,\) then the speed of the satellite \(B\) will be:

A particle of mass m is thrown upwards from the surface of the earth with a velocity u. The mass and the radius of the earth are, respectively, M and R. G is gravitational constant and g is acceleration due to gravity on the surface of the earth. The minimum value of u so that the particle does not return back to earth is

(1) $\sqrt{\frac{2GM}{R}}$                                       

(2) $\sqrt{\frac{2GM}{R^{\mathit{2}}}}$

(3) $\sqrt{2g{R}^{\mathit{2}}}$                                       

(4) $\sqrt{\frac{2GM}{R^3}}$

A body projected vertically from the earth reaches a height equal to earth's radius before returning to the earth. The power exerted by the gravitational force is greatest:

(1) at the instant just before the body hits the earth

(2) it remains constant all through

(3) at the instant just after the body is projected

(4) at the highest position of the body  

A planet moving along an elliptical orbit is closest to the sun at a distance ${\mathrm{r}}_1$ and farthest away at a distance of ${\mathrm{r}}_2$. If ${\mathrm{v}}_1 \mathrm{and} {\mathrm{v}}_2$ are the linear velocities at these points respectively, then the ratio $\frac{{\mathrm{v}}_1}{{\mathrm{v}}_2}$ is:

(1)  ${\mathrm{r}}_2/{\mathrm{r}}_1$                                   

(2)   $({\mathrm{r}}_2/{\mathrm{r}}_1{)}^2$

(3)  ${\mathrm{r}}_1/{\mathrm{r}}_2$                                     

(4)   $({\mathrm{r}}_1/{\mathrm{r}}_2{)}^2$

The height at which the weight of a body becomes 1/16th, its weight on the surface of the earth (radius R), is:

(1) 5R                                 

(2) 15R

(3) 3R                                 

(4) 4R

Infinite number of bodies, each of mass 2 kg are situated on x-axis at distance 1m, 2m,  4m, 8m, respectively from the origin. The resulting gravitational potential due to this system at the origin will be

(a)-G

(b)-8/3G

(c)-4/3G

(d)-4G