If two bodies of mass M and m are revolving around the centre of mass of the system in circular orbits of radii R and r respectively due to mutual interaction. Which of the following formulee is applicable ?

Options

A. `(GMm)/((R+r)^(2))= m omega^(2) r`
B. `(GMm)/R^(2)=m omega^(2) r`
C. `(GMm)/r^(2) = m omega^(2) R`
D. `(GMm)/(R^(2)+r^(2))=m omega^(2) r`

Answer

To solve the problem of two bodies of mass M and m revolving around their center of mass in circular orbits of radii R and r respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the System**:
We have two masses, M and m, revolving around their center of mass. The distances from the center of mass to the masses are R (for mass M) and r (for mass m).

2. **Understand the Center of Mass**:
The center of mass (CM) of the two-body system is given by the formula:
\[ R_{CM} = \frac{M \cdot R + m \cdot r}{M + m} \]
However, for the purpose of this problem, we will focus on the forces acting on the masses.

3. **Centripetal Force Requirement**:
Each mass experiences a centripetal force that keeps it in circular motion. The centripetal force \( F_C \) for mass M is given by:
\[ F_{C_M} = M \cdot \omega^2 \cdot R \]
and for mass m:
\[ F_{C_m} = m \cdot \omega^2 \cdot r \]
where \( \omega \) is the angular velocity of the system.

4. **Gravitational Force**:
The gravitational force \( F_G \) between the two masses is given by Newton's law of gravitation:
\[ F_G = \frac{G \cdot M \cdot m}{(R + r)^2} \]
where G is the gravitational constant.

5. **Equating Forces**:
The centripetal force required for each mass is provided by the gravitational force. Therefore, we can set up the following equations:
For mass M:
\[ \frac{G \cdot M \cdot m}{(R + r)^2} = M \cdot \omega^2 \cdot R \quad \text{(1)} \]
For mass m:
\[ \frac{G \cdot M \cdot m}{(R + r)^2} = m \cdot \omega^2 \cdot r \quad \text{(2)} \]

6. **Final Formulation**:
From equation (1), we can express the gravitational force in terms of centripetal force for mass M:
\[ \frac{G \cdot M \cdot m}{(R + r)^2} = M \cdot \omega^2 \cdot R \]
This can be rearranged to:
\[ \frac{G \cdot M \cdot m}{(R + r)^2} = M \cdot \omega^2 \cdot R \]
Thus, we find that:
\[ \frac{G \cdot M \cdot m}{(R + r)^2} = m \cdot \omega^2 \cdot r \]

7. **Identifying the Correct Formula**:
The correct formula that applies to the system is:
\[ \frac{G \cdot M \cdot m}{(R + r)^2} = M \cdot \omega^2 \cdot R \]
This indicates that the gravitational force equals the centripetal force for both masses.

### Conclusion:
The applicable formula for the system of two bodies revolving around their center of mass is:
\[ \frac{G \cdot M \cdot m}{(R + r)^2} = M \cdot \omega^2 \cdot R \]