A mass falls from a height '\(h\)' and its time of fall '\(t\)' is recorded in terms of time period \(T\) of a simple pendulum. On the surface of the earth, it is found that \(t=2T\). The entire setup is taken on the surface of another planet whose mass is half of that of the Earth and whose radius is the same. The same experiment is repeated and corresponding times are noted as '\(t\)' and '\(T\)'. Then we can say:
1. \(t' = \sqrt{2}T\)
2. \(t'>2T'\)
3. \(t'<2T'\)
4. \(t' = 2T'\)

The time period of a geostationary satellite is \(24~\text{h}\) at a height \(6R_E\) \((R_E\) is the radius of the earth) from the surface of the earth. The time period of another satellite whose height is \(2.5R_E\)${}_{}$ from the surface will be:
1. \(6\sqrt{2}~\text{h}\)
2. \(12\sqrt{2}~\text{h}\)
3. \(\frac{24}{2.5}~\text{h}\)
4. \(\frac{12}{2.5}~\text{h}\)

Assuming that the gravitational potential energy of an object at infinity is zero, the change in potential energy (final - initial) of an object of mass \(m\) when taken to a height \(h\) from the surface of the earth (of radius \(R\) and mass \(M\)), is given by:
1. \(-\frac{GMm}{R+h}\)
2. \(\frac{GMmh}{R(R+h)}\)
3. \(mgh\)
4. \(\frac{GMm}{R+h}\)

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

The work done to raise a mass \(m\) from the surface of the earth to a height \(h\), which is equal to the radius of the earth, is:
1. \(\frac{3}{2}mgR\)
2. \(mgR\)
3. \(2mgR\)
4. \(\frac{1}{2}mgR\)

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 \(\mathrm{S}.\) The shaded area \(\mathrm{SCD}\) is twice the shaded area \(\mathrm{SAB}.\) If \(t_1\) is the time for the planet to move from \(\mathrm{C}\) to \(\mathrm{D}\) and \(t_2\) is the time to move from \(\mathrm{A}\) to \(\mathrm{B},\) then:
           

A particle of mass M is situated at the centre of a spherical shell of the same mass 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 circular orbits of two satellites A and B of the earth are \(4R\) and \(R\) respectively. If the speed of satellite A is \(3v,\) then the speed of satellite B will be:
1. \(3v/4\)
2. \(6v\)
3. \(12v\)
4. \(3v/2\)