JEE PYQ: Gravitation Question 16
Question 16 - 2021 (26 Feb Shift 1)
Find the gravitational force of attraction between the ring and sphere as shown in the diagram, where the plane of the ring is perpendicular to the line joining the centres. If $\sqrt{8}R$ is the distance between the centres of a ring (of mass $m$) and a sphere (mass $M$) where both have equal radius $R$.
(1) $\frac{\sqrt{8}}{9} \cdot \frac{GmM}{R^2}$
(2) $\frac{\sqrt{8}}{27} \cdot \frac{GmM}{R^2}$
(3) $\frac{2\sqrt{2}}{3} \cdot \frac{GMm}{R^2}$
(4) $\frac{1}{3\sqrt{8}} \cdot \frac{GMm}{R^2}$
Show Answer
Answer: (2)
Solution
The gravitational field of the ring at distance $d = \sqrt{8}R$ along its axis: $E = \frac{Gmd}{(R^2+d^2)^{3/2}} = \frac{Gm\sqrt{8}R}{(R^2+8R^2)^{3/2}} = \frac{Gm\sqrt{8}R}{(9R^2)^{3/2}} = \frac{Gm\sqrt{8}R}{27R^3} = \frac{\sqrt{8}Gm}{27R^2}$. Force $= ME = \frac{\sqrt{8}GmM}{27R^2}$.
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