Two heavy spheres each of mass \(100\) kg and radius \(0.10\) m are placed \(1.0\) m apart on a horizontal table. What is the gravitational potential at the midpoint of the line joining the centres of the spheres?
1. \(2.67\times10^{-8}\) J kg^-1
2. \(0\)
3.   \(6.67\times10^{-9}\) J kg^-1
4.   \(3.71\times10^{-8}\) J kg^-1

A rocket is fired vertically with a speed of \(5\) km/s from the earth’s surface. How far from the earth does the rocket go before returning to the earth?
1. \(8\times10^6\) m
2. \(1.6\times10^6\) m
3. \(6.4\times10^6\) m
4. \(12\times10^6\) m

Assuming the earth to be a sphere of uniform mass density, how much would a body weigh halfway down to the centre of the earth if it weighed \(250\) N on the surface?
1. \(250\) N
2. \(125\) N
3. \(175\) N
4. \(145\) N

A body weighs \(63\) N on the surface of the earth. What is the gravitational force on it due to the earth at a height equal to half the radius of the earth?
1. \(98\) N
2. \(35\) N
3. \(63\) N
4. \(28\) N

If $v_e$ is the escape velocity and $v_o$ is the orbital velocity of a satellite for orbit close to the earth's surface, then these  are related by:

For a satellite moving in an orbit around the earth, the ratio of kinetic energy to potential energy is:
1. $\frac{1}{\sqrt{2}}$
2. \(2\)
3. $\sqrt{2}$
4. $\frac{1}{2}$

Two spheres of masses \(m\) and \(M\) are situated in air and the gravitational force between them is \(F.\) If the space around the masses is filled with a liquid of specific density \(3,\) the gravitational force will become:
1. \(3F\)
2. \(F\)
3. \(F/3\)
4. \(F/9\)

Let the speed of the planet at the perihelion \(P\) in figure shown below be \(v_{_P}\) and the Sun-planet distance \(\mathrm{SP}\) be \(r_{_P}.\) Relation between \((r_{_P},~v_{_P})\) to the corresponding quantities at the aphelion \((r_{_A},~v_{_A})\) is:

              
1. \(v_{_P} r_{_P} =v_{_A} r_{_A}\)
2. \(v_{_A} r_{_P} =v_{_P} r_{_A}\)
3. \(v_{_A} v_{_P} = r_{_A}r_{_P}\)
4. none of these

Three equal masses of \(m\) kg each are fixed at the vertices of an equilateral triangle \(ABC.\) What is the force acting on a mass \(2m\) placed at the centroid \(G\) of the triangle? 
(Take \(AG=BG=CG=1\) m.)

       

1. \(Gm^2(\hat{i}+\hat{j})\)
2. \(Gm^2(\hat{i}-\hat{j})\)
3. zero
4. \(2Gm^2(\hat{i}+\hat{j})\)

Two uniform solid spheres of equal radii \({R},\) but mass \({M}\) and \(4M\) have a centre to centre separation \(6R,\) as shown in the figure. The two spheres are held fixed. A projectile of mass \(m\) is projected from the surface of the sphere of mass \(M\) directly towards the centre of the second sphere. The expression for the minimum speed \(v\) of the projectile so that it reaches the surface of the second sphere is:

1. \(\left(\frac{3 {GM}}{5 {R}}\right)^{1 / 2}\)
2. \(\left(\frac{2 {GM}}{5 {R}}\right)^{1 / 2}\)
3. \(\left(\frac{3 {GM}}{2 {R}}\right)^{1 / 2}\)
4. \(\left(\frac{5 {GM}}{3 {R}}\right)^{1 / 2}\)