A planet of mass \(m\) is moving around a star of mass \(M\) and radius \(R\) in a circular orbit of radius \(r.\) The star abruptly shrinks to half its radius without any loss of mass. What change will be there in the orbit of the planet?

1.	The planet will escape from the Star.
2.	The radius of the orbit will increase.
3.	The radius of the orbit will decrease.
4.	The radius of the orbit will not change.

A planet is orbiting the sun in an elliptical orbit. Let \(U\) denote the potential energy and \(K\) denote the kinetic energy of the planet at an arbitrary point in the orbit.
Choose the correct statement from the given ones:

The law of gravitation states that the gravitational force between two bodies of mass \(m_1\) $\mathrm{and}$ \(m_2\) is given by:
\(F=\dfrac{Gm_1m_2}{r^2}\)
$\mathrm{}$\(G\) (gravitational constant) \(=7\times 10^{-11}~\text{N-m}^2\text{kg}^{-2}\)$\mathrm{.}$
\(r\) (distance between the two bodies) in the case of the Earth and Moon $\mathrm{}$\(=4\times 10^8~\text{m}\)
\(m_1~(\text{Earth})=6\times 10^{24}~\text{kg}\)
\(m_2~(\text{Moon})=7\times 10^{22}~\text{kg}\)
${\mathrm{}}_{}$What is the gravitational force between the Earth and the Moon?
1. \(1.8375 \times 10^{19}~\text{N}\)
2. \(1.8375 \times 10^{20}~\text{N}\)
3. \(1.8375 \times 10^{25}~\text{N}\)
4. \(1.8375 \times 10^{26}~\text{N}\)

A planet whose density is double of earth and radius is half of the earth, will produce gravitational field on its surface:
(\(g=\) acceleration due to gravity at the surface of earth)

If the radius of a planet is \(R\) and its density is \(\rho,\) the escape velocity from its surface will be:
1. \(v_e\propto \rho R\)
2. \(v_e\propto \sqrt{\rho} R\)
3. \(v_e\propto \frac{\sqrt{\rho}}{R}\)
4. \(v_e\propto \frac{1}{\sqrt{\rho} R}\)

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:
(neglecting 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. \(\sqrt{\frac{GM}{2 d}}\)

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. \(R\)

The escape velocity of a particle of mass \(m\) varies as:

Three identical point masses, each of mass \(1~\text{kg}\) 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. \(1.67\times 10^{-9}(-\hat i -\hat j) ~\text{N}\)