What is the gravitational force exerted by th | vedclass

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Question

(A) The gravitational field $I$ at a distance $r$ along the axis of a ring of mass $m$ and radius $a$ is given by $I = \frac{Gmr}{(a^2 + r^2)^{3/2}}$.

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$\frac{GMm}{8a^2}$

Vedclass · Answer

(A) The gravitational field $I$ at a distance $r$ along the axis of a ring of mass $m$ and radius $a$ is given by $I = \frac{Gmr}{(a^2 + r^2)^{3/2}}$.The gravitational force $F$ exerted by the sphere of mass $M$ on the ring is $F = M 	imes I = \frac{GMmr}{(a^2 + r^2)^{3/2}}$.Given $r = \sqrt{3}a$,substitute this into the formula:$F = \frac{GMm(\sqrt{3}a)}{(a^2 + (\sqrt{3}a)^2)^{3/2}}$$F = \frac{GMm\sqrt{3}a}{(a^2 + 3a^2)^{3/2}}$$F = \frac{GMm\sqrt{3}a}{(4a^2)^{3/2}}$$F = \frac{GMm\sqrt{3}a}{8a^3}$$F = \frac{\sqrt{3}GMm}{8a^2}$

What is the gravitational force exerted by the sphere of mass $M$ and radius $a$ on a ring of mass $m$ and radius $a$,where the distance between the center of the ring and the center of the sphere is $r = 1.73a = \sqrt{3}a$?

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