A uniform solid sphere of mass M 1 and radius R 1 is surrounded by a concentric thin uniform spherical shell of mass M 2 and radius R 2 ( R 2 R 1 ). A particle of mass 'm' is c

Question

A uniform solid sphere of mass M 1 ​ and radius R 1 ​ is surrounded by a concentric thin uniform spherical shell of mass M 2 ​ and radius R 2 ​ ( R 2 ​ > R 1 ​ ). A particle of mass 'm' is considered. Match the following quantities related to this system. List - I List - II (P) Magnitude of gravitational potential at the center of the solid sphere ( r = 0 ). (1) R 1 ​ G M 1 ​ ​ + R 2 ​ G M 2 ​ ​ (Q) Magnitude of gravitational potential at the surface of the solid sphere ( r = R 1 ​ ). (2) 2 R 1 ​ 3 G M 1 ​ ​ + R 2 ​ G M 2 ​ ​ (R) Work done by an external agent to move the particle 'm' from infinity to the center of the solid sphere ( r = 0 ). (3) m ( R 1 ​ G M 1 ​ ​ + R 2 ​ G M 2 ​ ​ ) (S) Minimum speed required for the particle 'm' to escape from the surface of the solid sphere ( r = R 1 ​ ) to infinity, assuming the shell is removed. (4) m ( 2 R 1 ​ 3 G M 1 ​ ​ + R 2 ​ G M 2 ​ ​ ) (5) R 1 ​ 2 G M 1 ​ ​ ​ (6) R 1 ​ G M 1 ​ ​ ​

TestHub · Accepted Answer

P: Gravitational potential at the center ( $r = 0$ ) due to solid sphere is $V_{s p h ere} (0) = - \frac{3 G M _{1}}{2 R _{1}}$ and due to shell is $V_{s h e ll} (0) = - \frac{G M _{2}}{R _{2}}$ . Total potential $V (0) = - (\frac{3 G M _{1}}{2 R _{1}} + \frac{G M _{2}}{R _{2}})$ . Magnitude matches (2).

Q: Gravitational potential at the surface of the solid sphere ( $r = R_{1}$ ) due to solid sphere is $V_{s p h ere} (R_{1}) = - \frac{G M _{1}}{R _{1}}$ and due to shell is $V_{s h e ll} (R_{1}) = - \frac{G M _{2}}{R _{2}}$ (since $R_{1} < R_{2}$ ). Total potential $V (R_{1}) = - (\frac{G M _{1}}{R _{1}} + \frac{G M _{2}}{R _{2}})$ . Magnitude matches (1). R: Work done by external agent $W_{e x t} = m (V_{f} - V_{i}) = m (V (0) - V (\infty)) = mV (0)$ . Magnitude matches (4). S: Escape velocity from the surface of a solid sphere $M_{1}, R_{1}$ is $v_{esc} = \frac{2 G M _{1}}{R _{1}}$ . Matches (5).

List - I	List - II
(P) Magnitude of gravitational potential at the center of the solid sphere ( $r = 0$ ).	(1) $\frac{G M _{1}}{R _{1}} + \frac{G M _{2}}{R _{2}}$
(Q) Magnitude of gravitational potential at the surface of the solid sphere ( $r = R_{1}$ ).	(2) $\frac{3 G M _{1}}{2 R _{1}} + \frac{G M _{2}}{R _{2}}$
(R) Work done by an external agent to move the particle 'm' from infinity to the center of the solid sphere ( $r = 0$ ).	(3) $m (\frac{G M _{1}}{R _{1}} + \frac{G M _{2}}{R _{2}})$
(S) Minimum speed required for the particle 'm' to escape from the surface of the solid sphere ( $r = R_{1}$ ) to infinity, assuming the shell is removed.	(4) $m (\frac{3 G M _{1}}{2 R _{1}} + \frac{G M _{2}}{R _{2}})$
	(5) $\frac{2 G M _{1}}{R _{1}}$
	(6) $\frac{G M _{1}}{R _{1}}$

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JEE Advanced 2027 (Class 12+)