Match the following quantities related to a uniform ring of mass M and radius R. List - I List - II (P) Gravitational field intensity at a point on the axis at a distance R from the center. (1) 0 (Q)

Question

Match the following quantities related to a uniform ring of mass M and radius R.

List - I	List - II
(P) Gravitational field intensity at a point on the axis at a distance R from the center.	(1) $0$
(Q) Gravitational force on a particle of mass 'm' placed at the center of the ring.	(2) $\frac{GM}{2 2 R ^{2}}$
(R) Gravitational potential at a point on the axis at a distance R from the center.	(3) $- \frac{GM}{2 R}$
(S) Maximum gravitational field intensity on the axis of the ring.	(4) $\frac{2 GM}{3 3 R ^{2}}$

TestHub · Accepted Answer

Gravitational field on the axis of a ring at distance x is E = ( R 2 + x 2 ) 3/2 GM x ​ . Gravitational potential is V = − R 2 + x 2 ​ GM ​ . P: At x = R , E P ​ = ( R 2 + R 2 ) 3/2 GMR ​ = ( 2 R 2 ) 3/2 GMR ​ = 2 2 ​ R 2 GM ​ . Q: At center ( x = 0 ), field is 0, so force is 0. R: At x = R , V R ​ = − R 2 + R 2 ​ GM ​ = − 2 ​ R GM ​ . S: Maximum field occurs at x = R / 2 ​ . E ma x ​ = ( R 2 + ( R / 2 ​ ) 2 ) 3/2 GM ( R / 2 ​ ) ​ = ( 3 R 2 /2 ) 3/2 GM ( R / 2 ​ ) ​ = 3 3 ​ R 2 2 GM ​ .

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