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(D) The gravitational field $E$ produced by a ring of mass $m$ and radius $R$ at a distance $x$ from its center along its axis is given by:$E = \frac{

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$\frac{\sqrt{8}}{27} \cdot \frac{GmM}{R^2}$

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(D) The gravitational field $E$ produced by a ring of mass $m$ and radius $R$ at a distance $x$ from its center along its axis is given by:$E = \frac{Gmx}{(R^2 + x^2)^{3/2}}$Since the sphere is placed at a distance $x = \sqrt{8}R$ from the center of the ring,the gravitational force $F$ exerted by the ring on the sphere of mass $M$ is:$F = M \cdot E = \frac{GMm(\sqrt{8}R)}{(R^2 + (\sqrt{8}R)^2)^{3/2}}$Substituting the values:$F = \frac{GMm(\sqrt{8}R)}{(R^2 + 8R^2)^{3/2}}$$F = \frac{GMm(\sqrt{8}R)}{(9R^2)^{3/2}}$$F = \frac{GMm(\sqrt{8}R)}{27R^3}$$F = \frac{\sqrt{8}}{27} \cdot \frac{GMm}{R^2}$

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