B M Sharma Solutions for Chapter: Gravitation, Exercise 2: DPP 6.2

A cavity of radius$\frac{R}{2}$ is made inside a solid sphere of radius $R$. The centre of the cavity is located at a distance $\frac{R}{2}$ from the centre of the sphere. The gravitational force on a particle of mass $m$ at a distance$\frac{R}{2}$ from the centre of the sphere on the line joining both the centres of the sphere and the cavity is (opposite to the centre of cavity). [$\mathrm{Here}, g=\frac{GM}{R^2}$ where $M$ is the mass of the solid sphere.]

A small volume is removed from a uniform spherical shell of mass $M$ and radius $R$. Then the gravitational field intensity near the hollow portion is

If $R$ is the radius of the earth and $g$ the acceleration due to gravity on the earth's surface, the mean density of the earth is?

The depth $d$ at which the value of acceleration due to gravity becomes $\frac{1}{n}$ times the value at the surface is? [$R$= radius of the earth]

At what altitude in metres will the acceleration due to gravity be $25\%$ of that at the earth's surface? (Radius of earth$=R$ $\mathrm{metres}$)

If the radius of the earth shrinks by $1.5\%$(mass remaining the same), then the value of acceleration due to gravity changes by what percentage?

What should be the angular speed of the Earth, so that the body lying on the equator may appear weightless?

$\left(g=10\mathrm{m} {\mathrm{s}}^{-2}, R=6400\mathrm{km}\right)$

If both the mass and the radius of the earth decrease by $1\%,$ the value of the acceleration due to gravity will be?