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A projectile is fired upwards from the surface of the earth with a velocity ${\mathrm{kv}}_{\mathrm{e}}$ where ${\mathrm{v}}_{\mathrm{e}}$ is the escape velocity and k < 1. If r is the maximum distance from the center of the earth to which it rises and R is the radius of the earth, then r equals:
1. \(\frac{R}{k^2}\)
2. \(\frac{R}{1-k^2}\)
3. \(\frac{2R}{1-k^2}\)
4. \(\frac{2R}{1+k^2}\)

Two bodies of masses m and 4m are placed at a distance r. The gravitational potential at a point on the line joining them where the gravitational field is zero is
1.  $-\frac{5\mathrm{Gm}}{\mathrm{r}}$
2.  $-\frac{6\mathrm{Gm}}{\mathrm{r}}$
3.  $-\frac{9\mathrm{Gm}}{\mathrm{r}}$
4.  0

In planetary motion, the areal velocity of the position vector of a planet depends on the angular velocity ($\omega$) and the distance of the planet from the sun (r). The correct relation for areal velocity is:
1. $\frac{dA}{dt}$ $\alpha$ $\omega r$
2. $\frac{dA}{dt}$ $\alpha$ ${\omega }^{2}r$
3. $\frac{dA}{dt}$ $\alpha$ $\omega r^2$
4. $\frac{dA}{dt}$ $\alpha$ $\sqrt{\omega r}$

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                                                                
1. Depends on system of units only
2. Depends on medium between masses only
3. Depends on both 1 and 2
4. Is independent of both 1 and 2

A thin rod of length L is bent to form a semicircle. The mass of the rod is M. The gravitational potential at the centre of the circle is :
1. $-\frac{\mathrm{GM}}{\mathrm{L}}$
2. $-\frac{\mathrm{GM}}{2\mathrm{\pi L}}$
3. $-\frac{\mathrm{\pi GM}}{2\mathrm{L}}$
4. $-\frac{\mathrm{\pi GM}}{\mathrm{L}}$

For the moon to cease as the earth's satellite, its orbital velocity has to be increased by a factor of -

Two spherical bodies of masses M and 5M and radii R and 2R are released in free space with initial separation between their centres equal to 12R. If they attract each other due to gravitational force only, then the distance covered by the smaller body before collision is
(1)2.5 R

(2)4.5 R

(3)7.5 R

(4)1.5 R

Change in acceleration due to gravity is same at height h above the Earth surface and  depth \(x ,\) then relation between \(x\) and \(h\) is: (\(h\) and \(x<<R_e\))
1. \(x=h\)
2. \(x=2h\)
3. \(x=\frac h2\)
4. \(x=h^2\)

At what height from the surface of earth the gravitation potential and the value of g are $-5.4\times {10}^{7}J$ $k{g}^{-2}$ and $6.0$ $m{s}^{-2}$ respectively? (Take, the radius of earth as 6400 km.)
(a) 1600 km             (b) 1400 km 
(c) 2000 km             (d) 2600 km

Two particles of equal mass go round a circle of radius R under the action of their mutual
gravitational attraction. The speed of each particle is
1. $\nu =\frac{1}{2R}\sqrt{\frac{1}{Gm}}$                             
2. $\nu =\sqrt{\frac{Gm}{2R}}$
3. $\nu =\frac{1}{2}\sqrt{\frac{Gm}{R}}$                               
4. $\nu =\sqrt{\frac{4Gm}{R}}$   

The time period of a simple pendulum on a freely moving artificial satellite is
(1) Zero                           
(2) 2 sec
(3)  3 sec                          
(4)  Infinite

Two identical solid copper spheres of radius R were placed in contact with each other. The gravitational attraction between them is proportional to:
1. $R^2$                                               
2. $R$
3. $R^4$                                               
4. $R^5$

If there were a smaller gravitational effect, which of the following forces do you think would alter in some respect?
(1)  Viscous forces                   
(2)   Archimedes uplift
(3)   Electrostatic force              
(4)   None of the above

A planet is moving in an elliptical orbit. If T, V, E, and L stand, respectively, for its kinetic energy, gravitational potential energy, total energy and angular momentum about the center of the orbit, then:

Suppose the law of gravitational attraction suddenly changes and becomes an inverse cube law i.e., $F\propto 1/r^3$ but the force still remains a central force, then: