Umakant Kondapure, Collin Fernandes, Nipun Bhatia, Vikram Bathula and, Ketki Deshpande Solutions for Chapter: Gravitation, Exercise 3: Competitive Thinking

A spherically symmetric gravitational system of particles has a mass density $\rho =\left\{\begin{array}{r}{\rho }_0\text{ for}r\le R\\ 0\text{ for }r>R\end{array}\right.$ where ${\rho }_0$ is a constant. A test mass can undergo circular motion under the influence of the gravitational field or particles. Its speed $v$ as a function of distance $r(0<r<\infty )$ from the centre of the system is represented by 

Assertion: An astronaut in an orbiting space station above the Earth experiences weightlessness.

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

Infinite number of spheres, each of mass $m$ are placed on the $X$-axis at distances $1,2,4,8,16$ meters from origin. The magnitude of the gravitational field at the origin is

Infinite number of bodies, each of mass $2 \mathrm{kg}$ are situated on x -axis at distance $1\mathrm{m}, 2\mathrm{m}, 4\mathrm{m}, 8\mathrm{m},\dots ..$, respectively, from the origin. The resulting gravitational potential due to this system at the origin will be

Two heavy spheres of mass $m$ Are kept separated by a distance $2r$. The gravitational field and potential at the midpoint of the line joining the centres of the spheres are

At what height from the surface of earth the gravitation potential and the value of g are $-5.4\times {10}^7\mathrm{J} {\mathrm{kg}}^{-2}$ and $6.0 \mathrm{m} {\mathrm{s}}^{-2}$ Respectively?
Take the radius of earth as $6400 \mathrm{km}$.

A satellite of mass '$m$' is revolving in circular orbit of radius '$r$' around the earth. Its angular momentum with respect to the centre of its orbit is $\mathit{(}M=$ mass of earth, $G=$ universal gravitational constant) 
 

The radii of a planet and its satellite are $2r$ and $r$ and their densities are $\rho$ and $2\rho$ respectively. Their centres are separated by a distance $d$. The minimum speed with which a body should be projected from the mid point of the line joining their centres, so that the body escapes to infinity is ($G$-universal gravitational constant)