Question

A solid sphere of uniform density and mass $M$ has radius $4\text{m}$. Its centre is at origin of the coordinate system. Two spheres of radii $1\text{m}$ are taken out so that their centres are at $P(0,-2,0)$ and $Q(0,2,0)$ respectively. This leaves two spherical cavities.

What is the gravitational field at point P?

• A. $\frac{31GM}{1024}$
• B. $\frac{31Gm}{1024}$
• C. $31GM$
• D. $\text{Zero}$

Answer 1

The correct option is A $\frac{31GM}{1024}$

If the cavities are not made then the intensity at the point P(or Q).

$I_R=\frac{GM}{R^3}.x=\frac{GM}{64}\times 2=\frac{GM}{32}$

where$\overrightarrow{I_R}=\overrightarrow{I_P}+\overrightarrow{I_Q}+\overrightarrow{I}$

$\overrightarrow{I_R}-\overrightarrow{I_P}-\overrightarrow{I_Q}=\overrightarrow{I}$

$\therefore$ Mass of big sphere $M=\frac{4\pi \times 4^{3}d}{3}$

mass of small sphere P and Q ,$m=\frac{4\pi 1^{3}d}{3}$

$\therefore m=\frac{M}{64}$

at P $I_P=0,I_Q=\frac{Gm}{r^2}=\frac{G}{4^2}\left(\frac{M}{64}\right)$

$\therefore I=\frac{31GM}{1024}$