A uniform ring of mass m and radius r is placed directly above a uniform sphere of mass m and of equal radius. The centre of the ring is directly above the centre of the sphere at a distance $r\sqrt{3}$ as shown in the figure. The gravitational force exerted by the sphere on the ring will be

$\sqrt{3} \frac{G Mm}{8 r^2}$

Let the field of the ring at the centre of the sphere be

$E=\frac{G M}{(2 r)^2} \cos \alpha=\frac{G M}{4 r^2} \times \frac{\sqrt{3 r}}{2 r}$

Force on the sphere of mass M

F = ME = $\frac{GMm \sqrt{3}}{8 r^2}$