Assuming earth to be a sphere of a uniform density, what is the value of gravitational acceleration in a mine $100\, km$ below the earth’s surface ........ $m/{s^2}$. (Given $R = 6400 \,km$)

At a height of $10 \,km$ above the surface of earth, the value of acceleration due to gravity is the same as that of a particular depth below the surface of earth. Assuming uniform mass density for the earth, the depth is ............. $km$

A particle of mass $10\, g$ is kept of the surface of a uniform sphere of mass $100\, kg$ and a radius of $10\, cm .$ Find the work to be done against the gravitational force between them to take the particle far away from the sphere. (you make take $\left.G=6.67 \times 10^{-11} Nm ^{2} / kg ^{2}\right)$

A hemisspherical shell of mass $2M$ and radius $6R$ and a point mass $M$ are performing circular motion due to their mutual gravitational interaetion Their positions are shown in figure at any moment of time during motion. If $r_1$ and $r_2$ are the radii of circular path of hemispherical shell and point mass respectively and ${\omega _1}$ and  ${\omega _2}$ are the angular speeds of hemi-spherical shell and point mass respectively, then choose the correct option

The mass of the moon is $(1/8)$ of the earth but the gravitational pull is $(1/6)$ of the earth. It is due to the fact that.

At a given place where acceleration due to gravity is $‘g’$ $m/{\sec ^2}$, a sphere of lead of density $‘d’$ $kg/{m^3}$ is gently released in a column of liquid of density $'\rho '\;kg/{m^3}$. If $d > \rho $, the sphere will