The gravitational field due to a mass distribution is given by $\overrightarrow{I}= \frac{k}{x^2}\hat{i}$, where k is a constant. Assuming the potential to be zero at infinity, find the potential at a point x = a.[This question includes concepts from Gravitation chapter]

1. $\frac{k}{a^2}$

2. $\frac{-k}{a^2}$

3. $\frac{k}{a}$

4. $\frac{-k}{a}$

A body weighs \(200\) N on the surface of the earth. How much will it weigh halfway down the centre of the earth?

The earth is assumed to be a sphere of radius \(R\). A platform is arranged at a height \(R\) from the surface of the earth. The escape velocity of a body from this platform is \(fv_e\), where \(v_e\) is its escape velocity from the surface of the earth. The value of \(f\) is:
1. \(\sqrt{2}\)
2. \(\frac{1}{\sqrt{2}}\)
3. \(\frac{1}{3}\)
4. \(\frac{1}{2}\)

Two satellites of Earth, \(S_1\)_, and \(S_2\), are moving in the same orbit. The mass of \(S_1\) is four times the mass of \(S_2\). Which one of the following statements is true?

The figure shows the 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 the planet to move from \(C\) to \(D\) and \(t_2\) is the time to move from \(A\) to \(B,\) then:
                     

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:

A spherical planet has a mass \(M_p\) and diameter \(D_p\)${\mathrm{}}_{}$. A particle of mass \(m\) falling freely near the surface of this planet will experience acceleration due to gravity equal to:
1. \(\frac{4GM_pm}{D_p^2}\)
2. \(\frac{4GM_p}{D_p^2}\)
3. \(\frac{GM_pm}{D_p^2}\)
4. \(\frac{GM_p}{D_p^2}\)

A body of mass \(m\) is taken from the Earth’s surface to the height equal to twice the radius \((R)\) of the Earth. The change in potential energy of the body will be: