1. Use shell theorem for the sphere

Since the given sphere has mass $M$ and radius $R$, and the centre of the ring is at distance $\sqrt{8}R$ from the centre of the sphere, we note that

$\sqrt{8}R > R$

So every point of the ring lies outside the sphere. Hence, by the shell theorem, the gravitational field due to the sphere at any external point is the same as if the entire mass $M$ were concentrated at its centre.

Therefore, we can treat the sphere as a point mass $M$ at its centre.

2. Geometry of the arrangement

• Radius of ring $= R$
• Radius of sphere $= R$
• Distance between centres $= d = \sqrt{8}R$
• Plane of ring is perpendicular to the line joining centres

This means the centre of the sphere lies on the axis of the ring.

For any small mass element $dm$ on the ring, its distance from the sphere’s centre is

$r = \sqrt{d^2 + R^2}$

Substitute $d = \sqrt{8}R$:

$r = \sqrt{8R^2 + R^2} = \sqrt{9R^2} = 3R$

So each element of the ring is at the same distance $3R$ from the sphere’s centre.

3. Force on an element of the ring

A mass element $dm$ experiences gravitational force

$dF = \frac{GM\,dm}{r^2} = \frac{GM\,dm}{(3R)^2} = \frac{GM\,dm}{9R^2}$

This force is directed along the line joining the sphere’s centre to that element.

Because of symmetry, transverse components cancel out, and only the component along the common axis survives.

If $\theta$ is the angle between this line and the axis, then

$\cos\theta = \frac{d}{r} = \frac{\sqrt{8}R}{3R} = \frac{\sqrt{8}}{3}$

So the axial component is

$dF_{\text{axis}} = dF\cos\theta = \frac{GM\,dm}{9R^2}\cdot \frac{\sqrt{8}}{3}$

$dF_{\text{axis}} = \frac{\sqrt{8}}{27}\frac{GM\,dm}{R^2}$

4. Integrate over the whole ring

Now integrate over the full ring. Since $\int dm = m$,

$F = \int dF_{\text{axis}} = \frac{\sqrt{8}}{27}\frac{GM}{R^2}\int dm$

$F = \frac{\sqrt{8}}{27}\frac{GMm}{R^2}$

5. Match with the options

Thus,

$\boxed{F = \frac{\sqrt{8}}{27}\frac{GMm}{R^2}}$

This matches Option C.

6. Comparison with stored correct answer

Stored correct answer: C

Our derived answer: C

So they agree.