A particle of mass `M` is revolving along a circule of radius `R` and nother particle of mass `m` is recolving in a circle of radius `r`. If time periods of both particles are same, then the ratio of their angular velocities is

Options

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

Answer

To solve the problem, we need to find the ratio of the angular velocities of two particles revolving in circles of different radii but with the same time period. Let's denote the angular velocities of the two particles as \( \omega_M \) for mass \( M \) and \( \omega_m \) for mass \( m \).

### Step-by-Step Solution:

1. **Understanding the Relationship Between Angular Velocity and Time Period**:
- The angular velocity \( \omega \) is related to the time period \( T \) by the formula:
\[ \omega = \frac{2\pi}{T} \]
- This means that angular velocity is inversely proportional to the time period.

2. **Setting Up the Given Information**:
- We are given that the time periods of both particles are the same:
\[ T_M = T_m \]

3. **Calculating the Angular Velocities**:
- For the particle of mass \( M \):
\[ \omega_M = \frac{2\pi}{T_M} \]
- For the particle of mass \( m \):
\[ \omega_m = \frac{2\pi}{T_m} \]

4. **Using the Equality of Time Periods**:
- Since \( T_M = T_m \), we can substitute \( T_m \) in the equation for \( \omega_m \):
\[ \omega_m = \frac{2\pi}{T_M} \]

5. **Finding the Ratio of Angular Velocities**:
- Now, we can find the ratio of the angular velocities:
\[ \frac{\omega_M}{\omega_m} = \frac{\frac{2\pi}{T_M}}{\frac{2\pi}{T_m}} = \frac{T_m}{T_M} \]
- Since \( T_M = T_m \), we have:
\[ \frac{\omega_M}{\omega_m} = \frac{T_m}{T_m} = 1 \]

6. **Conclusion**:
- Therefore, the ratio of their angular velocities is:
\[ \frac{\omega_M}{\omega_m} = 1 \]

### Final Answer:
The ratio of their angular velocities is \( 1:1 \).

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