1. Geometry of the configuration

Since four equal masses are equidistant from each other along a circle, they must be placed at the vertices of a square inscribed in a circle of radius $R$.

• Distance between adjacent particles: $a = \sqrt{2}R$ because the side of an inscribed square is $\sqrt{2}R$.

• Distance between opposite particles: $d = 2R$ which is the diameter.

2. Force on one particle due to the other three

Take one particle at a vertex of the square. The other three particles exert gravitational forces on it.

(i) Force due to each adjacent particle

For adjacent separation $a=\sqrt{2}R$, $F_1 = \frac{GM^2}{a^2} = \frac{GM^2}{2R^2}$

There are two such forces, symmetric about the diagonal toward the center.

The angle between each adjacent-force direction and the diagonal is $45^\circ$. Hence resultant of these two equal forces is: $F_{12} = 2F_1\cos 45^\circ$ $F_{12} = 2\cdot \frac{GM^2}{2R^2}\cdot \frac{1}{\sqrt{2}}$ $F_{12} = \frac{GM^2}{\sqrt{2}R^2}$

This resultant is directed toward the center.

(ii) Force due to the opposite particle

Opposite separation is $2R$, so $F_2 = \frac{GM^2}{(2R)^2} = \frac{GM^2}{4R^2}$ This also acts along the same diagonal, toward the center.

3. Net gravitational force toward the center

Thus total inward force on one particle is $F_{\text{net}} = F_{12} + F_2$ $F_{\text{net}} = \frac{GM^2}{\sqrt{2}R^2} + \frac{GM^2}{4R^2}$

Write $\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$: $F_{\text{net}} = \frac{GM^2}{R^2}\left(\frac{\sqrt{2}}{2} + \frac14\right)$ $F_{\text{net}} = \frac{GM^2}{R^2}\cdot \frac{1+2\sqrt{2}}{4}$

4. Use centripetal force condition

Each particle moves in a circle of radius $R$, so required centripetal force is $\frac{Mv^2}{R}$

Hence, $\frac{Mv^2}{R} = \frac{GM^2}{R^2}\cdot \frac{1+2\sqrt{2}}{4}$

Cancel $M$ and one factor of $R$: $v^2 = \frac{GM}{R}\cdot \frac{1+2\sqrt{2}}{4}$

Therefore, $v = \frac12\sqrt{\frac{GM}{R}(1+2\sqrt{2})}$

5. Match with options

This matches: $\boxed{\text{D: } \frac12\sqrt{\frac{GM}{R}(1+2\sqrt{2})}}$

6. Comparison with stored correct answer

Stored correct answer: D

Our derived answer: D

So the derived answer agrees with the stored answer.