Two particles of a rigid body of masses `m_(1)` and `m_(2)` are completing one rotation in paths of radius `r_(1) and r_(2)` respectively in same time. The ratio of their angular velocities is :

Options

A. `m_(1)r_(2)^(2):m_(2)r_(2)^(2)`
B. `r_(1):r_(2)`
C. `r_(2):r_(1)`
D. `1:1`

Answer

To solve the problem, we need to find the ratio of the angular velocities of two particles rotating in circular paths of different radii but completing one rotation in the same time.

### Step-by-Step Solution:

1. **Understanding Angular Velocity**:
Angular velocity (ω) is defined as the rate of change of angular displacement and is given by the formula:
\[ \omega = \frac{2\pi}{T} \]
where \( T \) is the time period for one complete rotation.

2. **Given Information**:
- Two particles with masses \( m_1 \) and \( m_2 \).
- They are rotating in circular paths of radii \( r_1 \) and \( r_2 \) respectively.
- Both complete one rotation in the same time \( T \).

3. **Finding Angular Velocities**:
Since both particles complete one rotation in the same time \( T \), we can express their angular velocities as:
\[ \omega_1 = \frac{2\pi}{T} \quad \text{and} \quad \omega_2 = \frac{2\pi}{T} \]

4. **Calculating the Ratio of Angular Velocities**:
The ratio of their angular velocities is:
\[ \frac{\omega_1}{\omega_2} = \frac{\frac{2\pi}{T}}{\frac{2\pi}{T}} = 1 \]

5. **Conclusion**:
Therefore, the ratio of their angular velocities \( \omega_1 : \omega_2 \) is:
\[ \omega_1 : \omega_2 = 1 : 1 \]

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

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