1. Understand the motion

Two identical particles, each of mass $m=1\,\text{kg}$, revolve due to their mutual gravitational attraction.

Since the masses are identical, they will rotate about their common center of mass. For two equal masses, the center of mass lies exactly midway between them.

So if each particle moves in a circle of radius $R$, then the distance between the two particles is:

$d=2R$

2. Gravitational force between the particles

The mutual gravitational force is

$F=\frac{Gm^2}{d^2}$

Here, $m=1$ kg and $d=2R$, so

$F=\frac{G(1)^2}{(2R)^2}=\frac{G}{4R^2}$

3. Centripetal force required for circular motion

Each particle moves in a circle of radius $R$ with angular speed $\omega$.

Hence, required centripetal force for each particle is

$F_c=m\omega^2 R$

Since $m=1$ kg,

$F_c=\omega^2 R$

4. Equate gravitational force and centripetal force

The only force acting on each particle is the gravitational attraction, so

$\omega^2 R=\frac{G}{4R^2}$

Therefore,

$\omega^2=\frac{G}{4R^3}$

Taking square root,

$\omega=\frac{1}{2}\sqrt{\frac{G}{R^3}}$

5. Match with the options

This corresponds to:

$\boxed{\text{B: } \frac{1}{2}\sqrt{\frac{G}{R^3}}}$

6. Comparison with stored correct answer

Stored correct answer: B

Our derived answer: B

So they agree.