Question

A spherical shell has inner radius $R_1$, outer radius $R_2$, and mass $M$, distributed uniformly throughout the shell. The magnitude of the gravitational force exerted on the shell by a point mass particle of mass $m$, located at a distance $d$ from the center, inside the inner radius is :

• A. $0$
• B. $\frac{GMm}{R_1^2}$
• C. $\frac{GMm}{d^2}$
• D. $\frac{GMm}{(R_2^2-d^2)}$

Answer 1

The correct option is C $\frac{GMm}{d^2}$

Mass of the sphere $=M$ point $P$ is at a distance of $d$ from the centre and there is a particle of mass $m$ located there.

The mass of the spherical shell that will cause force will be mass located between the sphere of radius $d$ and radius $R_1$

That is mass of volume

$\frac{4}{3}\pi d^3=\frac{4}{3}\pi R_1^3=\frac{4}{3}\pi (d^3-R_1^3)$

Mass is distributed uniformly in the shell. Thus density of the shell is

$\frac{M}{\left[\frac{4}{3}p\right(R_2^3-R_1^3\left)\right]}$

Thus

$M^{\prime }=\frac{M}{\left[\frac{4}{3}\pi \right(R_2^3-R_1^3]}\times \frac{4}{3}\pi (d^3-R_1^3)=\left[\right(d^3-R_1^3)=\frac{\left[\right(d^3-R_1^3\left)M\right]}{(R_2^3-R_1^3)}$

Thus force ecerted on $m$ is $=\frac{GMm}{d^2}$