A uniform ring of mass 'm' is lying at a distance $\sqrt{3}R$ from the center of the sphere of mass 'M' just over the sphere (where 'R' is the radius of the ring as well as of the sphere), then the magnitude of the gravitational force between them is:

Question

Accepted Answer

To determine the gravitational force between a uniform ring and a sphere, we need to consider the gravitational interaction between the two objects. The given problem specifies that the ring is at a distance of $\sqrt{3}R$ from the center of the sphere, where $R$ is the radius of both the ring and the sphere.

### Explanation for Option 2:

The gravitational force between two masses is given by Newton's law of universal gravitation:

$$ F = \frac{G \cdot M \cdot m}{r^2} $$

where $G$ is the gravitational constant, $M$ and $m$ are the masses of the sphere and the ring, respectively, and $r$ is the distance between the centers of mass of the two objects.

In this scenario, the ring can be considered as a collection of point masses, each contributing to the gravitational force on the sphere. Due to symmetry, the net gravitational force can be calculated by integrating the contributions of each infinitesimal mass element of the ring.

Given the distance $\sqrt{3}R$, the effective gravitational force can be derived through integration, considering the geometry and symmetry of the system. The correct expression for the gravitational force, after performing the necessary integration and simplification, is:

$$ F = \frac{\sqrt{3}}{8} \frac{GMm}{R^2} $$

This matches Option 2, which is the correct answer.

### Explanation for Why Other Options are Incorrect:

1. **Option 1: $\frac{\sqrt{3}GMm}{R^{2}}$**

This option suggests a direct application of the gravitational formula without considering the distribution of mass in the ring and the specific distance $\sqrt{3}R$. It does not account for the integration over the ring's mass distribution, leading to an incorrect factor.

2. **Option 3: $\frac{GMm}{8R^{2}}$**

This option lacks the $\sqrt{3}$ factor, which arises from the specific geometry and distance involved in the problem. The absence of this factor indicates an incomplete consideration of the system's geometry.

3. **Option 4: $\frac{G}{2}\frac{Mm}{R^{2}}$**

This option provides a different constant factor, $\frac{1}{2}$, which does not align with the derived expression from integrating the gravitational effects over the ring. It fails to incorporate the necessary geometric considerations that lead to the $\frac{\sqrt{3}}{8}$ factor.

In conclusion, Option 2 correctly accounts for the geometry and distribution of mass in the system, leading to the accurate expression for the gravitational force between the ring and the sphere.