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. \(F/9\)

For moon, its mass is 1/81 of Earth's mass and its diameter is 1/3.7 of Earth's diameter. If acceleration due to gravity at Earth's surface is 9.8 m/$s^2$, then at the moon, its value is: 

Rohini satellite is at a height of 500 km and Insat-B is at a height of 3600 km from the surface of the earth. The relation between their orbital velocity (\(v_R,~v_i\)) is:

1. \(v_R>v_i\)

2. \(v_R<v_i\)

3. \(v_R=v_i\)

4. No relation 

For a planet having mass equal to the mass of the earth but radius equal to one-fourth of the radius of the earth, its escape velocity will be:

Imagine a new planet having the same density as that of the Earth but 3 times bigger than the Earth in size. If the acceleration due to gravity on the surface of the earth is g and that on the surface of the new planet is g', then:

A planet moves around the sun. At a point P, it is closest to the sun at a distance \(d_1\) and has speed \(v_1.\) At another point Q, when it is farthest from the sun at distance \(d_2,\) its speed will be:

The gravitational potential energy of an isolated system of three particles, each of mass \(\mathrm{m}\) placed at three corners of an equilateral triangle of side \(\mathrm{l}\) is: 

Two satellites S₁ and S₂ are revolving around a planet in coplanar and concentric circular orbits of radii R₁ and R₂ in the same direction respectively. Their respective periods of revolution are 1 hr and 8 hr. The radius of the orbit of satellite S₁ is equal to 10⁴ km. Find the relative speed when they are closest to each other. 

1. $2\mathrm{\pi }\times {10}^4$ $\mathrm{kmph}$

2. $\mathrm{\pi }\times {10}^4$ $\mathrm{kmph}$

3. $\frac{\mathrm{\pi }}{2}\times {10}^4$ $\mathrm{kmph}$

4. $\frac{\mathrm{\pi }}{3}\times {10}^4$ $\mathrm{kmph}$ 

A body of mass m is situated at a distance 4${\mathrm{R}}_{\mathrm{e}}$ above the Earth's surface, where ${\mathrm{R}}_{\mathrm{e}}$ is the radius of the Earth. What minimum energy should be given to the body so that it may escape?