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 denote the masses of the particles as \( M \) and \( m \), their respective radii as \( R \) and \( r \), and their angular velocities as \( \omega_1 \) and \( \omega_2 \).

### Step-by-step Solution:

1. **Understanding the Time Period**:
- The time period \( T \) for an object moving in a circular path is given by the formula:
\[ T = \frac{2\pi r}{v} \]
- Here, \( v \) is the linear velocity of the particle.

2. **Setting Up the Equations**:
- For the first particle (mass \( M \), radius \( R \)):
\[ T_1 = \frac{2\pi R}{v_1} \]
- For the second particle (mass \( m \), radius \( r \)):
\[ T_2 = \frac{2\pi r}{v_2} \]
- Given that the time periods are the same (\( T_1 = T_2 \)), we can equate the two equations:
\[ \frac{2\pi R}{v_1} = \frac{2\pi r}{v_2} \]

3. **Cancelling Common Terms**:
- Cancel \( 2\pi \) from both sides:
\[ \frac{R}{v_1} = \frac{r}{v_2} \]
- Rearranging gives:
\[ \frac{v_2}{v_1} = \frac{r}{R} \]

4. **Relating Linear Velocity to Angular Velocity**:
- The relationship between linear velocity \( v \) and angular velocity \( \omega \) is given by:
\[ v = r \omega \]
- For the first particle:
\[ v_1 = R \omega_1 \]
- For the second particle:
\[ v_2 = r \omega_2 \]

5. **Substituting Linear Velocities**:
- Substitute \( v_1 \) and \( v_2 \) into the ratio:
\[ \frac{v_2}{v_1} = \frac{r \omega_2}{R \omega_1} \]
- From the previous step, we know:
\[ \frac{r}{R} = \frac{r \omega_2}{R \omega_1} \]

6. **Cancelling Out the Common Terms**:
- Cancel \( r \) and \( R \) from both sides:
\[ 1 = \frac{\omega_2}{\omega_1} \]
- Thus, we find:
\[ \frac{\omega_1}{\omega_2} = 1 \]

### Final Answer:
The ratio of angular velocities \( \frac{\omega_1}{\omega_2} \) is \( 1 \).