Match the following quantities related to a uniform solid sphere of mass M and radius R, and a particle of mass 'm'. List - I List - II (P) Work done to move the particle from the center to

Question

Match the following quantities related to a uniform solid sphere of mass M and radius R, and a particle of mass 'm'.

List - I	List - II
(P) Work done to move the particle from the center to the surface of the sphere.	(1) $\frac{GM m}{2 R}$
(Q) Work done to move the particle from the surface to infinity.	(2) $\frac{11 GM m}{8 R}$
(R) Magnitude of gravitational potential energy of the particle at a distance R/2 from the center of the sphere.	(3) $- \frac{3 GM m}{2 R}$
(S) Work done to move the particle from infinity to the center of the sphere.	(4) $\frac{GM m}{R}$

TestHub · Accepted Answer

Gravitational potential inside a solid sphere is V ( r ) = − 2 R 3 GM ​ ( 3 R 2 − r 2 ) . Potential at surface V ( R ) = − R GM ​ . Potential at center V ( 0 ) = − 2 R 3 GM ​ . Potential at infinity V ( ∞ ) = 0 . P: W C → S ​ = m [ V ( R ) − V ( 0 )] = m [ − R GM ​ − ( − 2 R 3 GM ​ )] = 2 R GM m ​ . Q: W S → ∞ ​ = m [ V ( ∞ ) − V ( R )] = m [ 0 − ( − R GM ​ )] = R GM m ​ . R: PE ( R /2 ) = mV ( R /2 ) = m [ − 2 R 3 GM ​ ( 3 R 2 − ( R /2 ) 2 )] = − 8 R 11 GM m ​ . Magnitude is 8 R 11 GM m ​ . S: W ∞ → C ​ = m [ V ( 0 ) − V ( ∞ )] = m [ − 2 R 3 GM ​ − 0 ] = − 2 R 3 GM m ​ .

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