A spaceship moves about the earth in a circular orbit whose radius is $\eta =4.0$ times the radius of earth. It is desired that the spaceship leaves the earth. It is desired that the spaceship leaves the earth's gravitational field by imparting additional velocity $V_r$ to it in the radial direction. Given $g=10 {\mathrm{ms}}^{-2}$, radius of earth $R=6400 \mathrm{km}$, the minimum value of $V_r$ in $\mathrm{k} {\mathrm{m}}^{-1}$ is

Consider a ring where linear mass density $\lambda$ varies with $\theta$ as $\lambda ={\lambda }_0\left|\cos \theta \right|\left({\lambda }_0>0\right)$. The gravitational field at point $O$ will be

Consider the arrangement shown in figure. A thin rod (mass $M$ and length $4R$ ) and a sphere of equal mass $M$ and radius $R$ are located as shown in figure. The centre of the sphere being at a distance $2R$ from the centre of the rod on its perpendicular bisector. Assume that the system of rod and sphere is in gravity free space. Find the force acting on the rod due to sphere.

A planet is revolving around the sun $S$ in an elliptical path with semi-major and semi-minor axes as $a$ and $b$ respectively. If time for one complete revolution is $T$, then time taken by planet to move from point $A$ to $B$ as shown in figure, will be