Question

Two spherical shells of radii $R$ and $2R$, masses $M$ and $2M$ respectively are arranged concentrically. The net gravitational force acting on a particle of mass $m$ placed at a distance of $23R$ from the common centre of the shells is ($G$ = Universal gravitational constant)

• A. $\frac{4 G M m}{3 R^2}$
• B. $\frac{76 G M m}{9 R^2}$
• C. $\frac{4 G M m}{9 R^2}$
• D. $\frac{68 G M m}{9 R^2}$

Hint:

Use the shell theorem: outside shell → treat as point mass; inside shell → zero force.

Answer 1

The Correct Option is B

1. By the shell theorem, a particle outside a spherical shell experiences a gravitational force as if the shell's entire mass were concentrated at its center.
2. Force due to shell 1: $F_1 = \frac{G M m}{(23R)^2}$
3. Force due to shell 2: $F_2 = \frac{G \cdot 2 M \cdot m}{(23R)^2}$
4. Net force: $F_\text{net} = F_1 + F_2 = \frac{G M m}{(23R)^2} + \frac{2 G M m}{(23R)^2} = \frac{3 G M m}{(23R)^2} = \frac{76 G M m}{9 R^2}$