A planet is revolving around a massive star in a circular orbit of radius \(R\). If the gravitational force of attraction between the planet and the star is inversely proportional to \(R^3,\) then the time period of revolution \(T\) is proportional to:
1. \(R^5\)
2. \(R^3\)
3. \(R^2\)
4. \(R\)

A satellite of mass \(m\) revolving around the earth in a circular orbit of radius \(r\) has its angular momentum equal to \(L\) about the centre of the earth. The potential energy of the satellite is: 
1. \(- \frac{L^{2}}{2 mr}\)
2. \(- \frac{2L^{2}}{mr^2}\)
3. \(- \frac{3L^{2}}{m^2r^2}\)
4. \(- \frac{L^{2}}{mr^2}\)

If \(R\) is the radius of the orbit of a planet and \(T\) is the time period of the planet, then which of the following graphs correctly shows the motion of a planet revolving around the sun?

1.		2.
3.		4.

The figure shows a planet in an elliptical orbit around the sun (\(S\)). The ratio of the momentum of the planet at point \(A\) to that at point \(B\) is:

                              
1. \(\frac{r_1}{r_2}\)
2. \(\frac{r_{1}^{2}}{r_{2}^{2}}\)
3. \(\frac{r_2}{r_1}\)
4. \(\frac{r_{2}^{2}}{r_{1}^{2}}\)

Three identical point masses, each of mass \(1\) kg lie at three points \((0,0)\), \((0,0.2~\text{m})\), \((0.2~\text{m}, 0)\). The net gravitational force on the mass at the origin is:
1. \(6.67\times 10^{-9}(\hat i +\hat j)~\text{N}\)
2. \(1.67\times 10^{-9}(\hat i +\hat j) ~\text{N}\)
3. \(1.67\times 10^{-9}(\hat i -\hat j) ~\text{N}\)
4. \(1.67\times 10^{-9}(-\hat i -\hat j) ~\text{N}\)

Two particles of mass \(m\) and \(4m\) are separated by a distance \(r.\) Their neutral point is at:
1. \(\frac{r}{2}~\text{from}~m\)
2. \(\frac{r}{3}~\text{from}~4m\)
3. \(\frac{r}{3}~\text{from}~m\)
4. \(\frac{r}{4}~\text{from}~4m\)

A particle is located midway between two point masses each of mass \(M\) kept at a separation \(2d.\) The escape speed of the particle is: (neglect the effect of any other gravitational effect)
1. \(\sqrt{\frac{2 GM}{d}}\)
2. \(2 \sqrt{\frac{GM}{d}}\)
3. \(\sqrt{\frac{3 GM}{d}}\)
4. \(\sqrt{\frac{GM}{2 d}}\)

\(1\) kg of sugar has maximum weight:
1. at the pole.
2. at the equator.
3. at a latitude of \(45^{\circ}.\)
4. in India.

Radii and densities of two planets are \(R_1, R_2\) and \(\rho_1, \rho_2\) respectively. The ratio of accelerations due to gravity on their surfaces is:
1. \(\frac{\rho_1}{R_1}:\frac{\rho_2}{R_2}\)
2. \(\frac{\rho_1}{R^2_1}: \frac{\rho_2}{R^2_2}\)
3. \(\rho_1 R_1 : \rho_2R_2\)
4. \(\frac{1}{\rho_1R_1}:\frac{1}{\rho_2R_2}\)