Two particles having mass 'M' and 'm' are moving in a circular path having radius R & r respectively. If their time period are same then the ratio of angular velocity will be : -

Options

A. `r/R`
B. `R/r`
C. `1`
D. `sqrt(R/r)`

Answer

To solve the problem, we need to find the ratio of the angular velocities of two particles moving in circular paths with the same time period. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Mass of the first particle: \( M \)
- Mass of the second particle: \( m \)
- Radius of the first particle's circular path: \( R \)
- Radius of the second particle's circular path: \( r \)
- Time period of both particles: \( T_1 = T_2 \)

2. **Write the Formula for Time Period**:
The time period \( T \) of a particle moving in a circular path is given by:
\[ T = \frac{2\pi r}{v} \]
where \( v \) is the linear velocity.

3. **Set Up the Equations for Both Particles**:
For the first particle:
\[ T_1 = \frac{2\pi R}{v_1} \]
For the second particle:
\[ T_2 = \frac{2\pi r}{v_2} \]

4. **Since the Time Periods are Equal**:
We can set \( T_1 = T_2 \):
\[ \frac{2\pi R}{v_1} = \frac{2\pi r}{v_2} \]
The \( 2\pi \) cancels out:
\[ \frac{R}{v_1} = \frac{r}{v_2} \]

5. **Rearranging the Equation**:
From the above equation, we can find the ratio of the velocities:
\[ \frac{v_2}{v_1} = \frac{r}{R} \]

6. **Relate Linear Velocity to Angular Velocity**:
The linear velocity \( v \) is related to angular velocity \( \omega \) by the formula:
\[ v = r\omega \]
Therefore, for both particles:
- For the first particle: \( v_1 = R \omega_1 \)
- For the second particle: \( v_2 = r \omega_2 \)

7. **Substituting the Velocities**:
Substitute \( v_1 \) and \( v_2 \) in the ratio:
\[ \frac{v_2}{v_1} = \frac{r \omega_2}{R \omega_1} \]

8. **Equating the Two Expressions for the Ratio of Velocities**:
From step 5, we have:
\[ \frac{r}{R} = \frac{r \omega_2}{R \omega_1} \]

9. **Canceling Out the Common Terms**:
Since \( r \) and \( R \) are not zero, we can cancel them out:
\[ 1 = \frac{\omega_2}{\omega_1} \]

10. **Final Result**:
Therefore, the ratio of the angular velocities is:
\[ \frac{\omega_2}{\omega_1} = 1 \]

### Conclusion:
The ratio of the angular velocities \( \omega_2 : \omega_1 = 1 : 1 \).