Gravitation

From a sphere of mass M and radius R, a smaller sphere of radius $\frac R2$ is carved out such that the cavity made in the original sphere is between its centre and the periphery. (See figure). For the configuration in the figure where the distance between the centre of the original sphere and the removed sphere is 3R, the gravitational force between the two spheres is :

Two hypothetical planets of masses $m_1$ and $m_2$ are at rest when they are infinite distance apart. Because of the gravitational force they move towards each other along the line joining their centres. What is their speed when their separation is 'd' ?

(Speed of $m_1$ is $v_1$ and that of $m_2$ is $v_2$)

The gravitational field in a region is given by $\vec g=5\text{ N/kg}\,\hat i+12\text{ N/kg}\,\hat j$. The change in the gravitational potential energy of a particle of mass 2 kg when it is taken from the origin to a point (7 m, −3 m) is :

Two bodies, each of mass $M$, are kept fixed with a separation $2L$. A particle of mass $m$ is projected from the midpoint of the line joining their centres, perpendicular to the line. The gravitational constant is $G$. The correct statement(s) is (are)

Two spherical planets $P$ and $Q$ have the same uniform density $\rho$, masses $M_P$ and $M_Q$, and surface areas $A$ and $4A$, respectively. A spherical planet $R$ also has uniform density $\rho$ and its mass is $(M_P+M_Q)$. The escape velocities from the planets $P$, $Q$ and $R$, are $V_P$, $V_Q$ and $V_R$, respectively. Then

A satellite is moving with a constant speed '$V$' in a circular orbit about the earth. An object of mass '$m$' is ejected from the satellite such that it just escapes from the gravitational pull of the earth. At the time of its ejection, the kinetic energy of the object is

A thin uniform annular disc (see figure) of mass M has outer radius 4R and inner radius 3R. The work required to take a unit mass from point P on its axis to infinity is

Gravitational acceleration on the surface of a planet is $\dfrac{\sqrt6}{11}g$, where g is the gravitational acceleration on the surface of the earth. The average mass density of the planet is $\dfrac23$ times that of the earth. If the escape speed on the surface of the earth is taken to be 11 km s⁻¹, the escape speed on the surface of the planet in km s⁻¹ will be

Under the influence of the Coulomb field of charge $+Q$, a charge $-q$ is moving around it in an elliptical orbit. Find out the correct statement(s).

A spherically symmetric gravitational system of particles has a mass density
$$\rho=\begin{cases}\rho_0&\text{for }r\le R\\0&\text{for }r>R\end{cases}$$
where $\rho_0$ is a constant. A test mass can undergo circular motion under the influence of the gravitational field of particles. Its speed $V$ as a function of distance $r$ $(0<r<\infty)$ from the centre of the system is represented by

STATEMENT-1: An astronaut in an orbiting space station above the Earth experiences weightlessness.

and

STATEMENT-2: An object moving around the Earth under the influence of Earth's gravitational force is in a state of 'free-fall'.

Column I describes some situations in which a small object moves. Column II describes some characteristics of these motions. Match the situations in Column I with the characteristics in Column II and indicate your answer by darkening appropriate bubbles in the 4 × 4 matrix given in the ORS.