Find the gravitational force of attraction between the ring and sphere as shown in the diagram, where the plane of the ring is perpendicular to the line joining the centres. If $\sqrt{8}R$ is the distance between the centres of a ring (of mass $m$) and a sphere (mass $M$) where both have equal radius $R$

In the reported figure of earth, the value of acceleration due to gravity is same at point $A$ and $C$ but it is smaller than that of its value at point $B$ (surface of the earth). The value of $OA:AB$ will be $x:5$. The value of $x$ is

The height $h$ at which the weight of a body will be the same as that at the same depth $h$ from the surface of the earth is,

(Radius of the earth is $R$ and effect of the rotation of the earth is neglected) 

A solid sphere of radius $R$ carries a charge $\left(Q+q\right)$ distributed uniformly over its volume. A very small point-like piece of it of mass $m$ gets detached from the bottom of the sphere and falls down vertically under gravity. This piece carries charge $q$. If it acquires a speed $v$ when it has fallen through a vertical height $y$ (see figure), then (assume the remaining portion to be spherical)