Two bodies of mass `10kg` and `5kg` moving in concentric orbits of radii `R` and `r` such that their periods are the same. Then the ratio between their centipetal acceleration is

Options

A. `R//r`
B. `r//R`
C. `R^(2)//r^(2)`
D. `r^(2)//R^(2)`

Explanation

To solve the problem, we need to find the ratio of the centripetal accelerations of two bodies moving in concentric orbits with the same period.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two bodies with masses \( m_1 = 10 \, \text{kg} \) and \( m_2 = 5 \, \text{kg} \) moving in circular orbits of radii \( R \) and \( r \) respectively. Both bodies have the same period \( T \).

2. **Centripetal Acceleration Formula**:
The centripetal acceleration \( a \) of an object moving in a circle is given by the formula:
\[ a = \frac{v^2}{r} \]
where \( v \) is the linear velocity and \( r \) is the radius of the circular path.

3. **Relating Period to Velocity**:
The period \( T \) of an object in circular motion is related to its velocity \( v \) and radius \( r \) by the equation:
\[ T = \frac{2\pi r}{v} \]
Rearranging gives:
\[ v = \frac{2\pi r}{T} \]

4. **Finding Velocities for Both Bodies**:
For the first body (mass \( 10 \, \text{kg} \)):
\[ v_1 = \frac{2\pi R}{T} \]
For the second body (mass \( 5 \, \text{kg} \)):
\[ v_2 = \frac{2\pi r}{T} \]

5. **Calculating Centripetal Accelerations**:
The centripetal acceleration for the first body is:
\[ a_1 = \frac{v_1^2}{R} = \frac{\left(\frac{2\pi R}{T}\right)^2}{R} = \frac{4\pi^2 R}{T^2} \]
The centripetal acceleration for the second body is:
\[ a_2 = \frac{v_2^2}{r} = \frac{\left(\frac{2\pi r}{T}\right)^2}{r} = \frac{4\pi^2 r}{T^2} \]

6. **Finding the Ratio of Centripetal Accelerations**:
Now, we can find the ratio of the centripetal accelerations:
\[ \frac{a_1}{a_2} = \frac{\frac{4\pi^2 R}{T^2}}{\frac{4\pi^2 r}{T^2}} = \frac{R}{r} \]

### Final Answer:
The ratio of the centripetal accelerations of the two bodies is:
\[ \frac{a_1}{a_2} = \frac{R}{r} \]