Question

A uniform ring of mass m is lying at a distance 1.73 a from the centre of a sphere of mass M just over the sphere where a is the small radius of the ring as well as that of the sphere. Then gravitational force exerted is

• A. $\frac { G M m } { 8 a ^ { 2 } }$
• B. $\frac { G M m } { ( 1.73 a ) ^ { 2 } }$
• C. $\sqrt { 3 } \frac { G M m } { a ^ { 2 } }$
• D. $1.73 \frac { G M m } { 8 a ^ { 2 } }$

Answer 1

Answer: (D)

$1.73 \frac { G M m } { 8 a ^ { 2 } }$

Answer 2

Intensity due to uniform circular ring at a point on its axis $I = \frac { G m r } { \left( a ^ { 2 } + r ^ { 2 } \right) ^ { 3 / 2 } }$

$\therefore$ Force on sphere $F = \frac { G M m r } { \left( a ^ { 2 } + r ^ { 2 } \right) ^ { 3 / 2 } }$ $= \frac { G M m \sqrt { 3 } a } { \left( a ^ { 2 } + ( \sqrt { 3 } a ) ^ { 2 } \right) ^ { 3 / 2 } }$ $= \frac { G M m \sqrt { 3 } a } { \left( 4 a ^ { 2 } \right) ^ { 3 / 2 } } = \frac { \sqrt { 3 } G M m } { 8 a ^ { 2 } }$ [As $r = \sqrt { 3 } a$]