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 find the ratio of the angular velocities of two particles revolving in circles of different radii but with the same time period, we can follow these steps:

### Step-by-Step Solution

1. **Understand Angular Velocity**:
The angular velocity (ω) of a particle is defined as the rate of change of angular displacement with respect to time. It can also be expressed in terms of the time period (T) as:
\[ \omega = \frac{2\pi}{T} \]

2. **Define Angular Velocities for Both Particles**:
Let the angular velocity of the first particle (mass M, radius R) be \( \omega_M \) and the second particle (mass m, radius r) be \( \omega_m \). Since both particles have the same time period (T), we can write:
\[ \omega_M = \frac{2\pi}{T} \quad \text{and} \quad \omega_m = \frac{2\pi}{T} \]

3. **Set Up the Ratio of Angular Velocities**:
To find the ratio of their angular velocities, we can express it as:
\[ \frac{\omega_M}{\omega_m} = \frac{\frac{2\pi}{T}}{\frac{2\pi}{T}} \]

4. **Simplify the Ratio**:
Since both angular velocities are equal (as they depend on the same time period), the ratio simplifies to:
\[ \frac{\omega_M}{\omega_m} = 1 \]

5. **Conclusion**:
The ratio of the angular velocities of the two particles is:
\[ \frac{\omega_M}{\omega_m} = 1 \]

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

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