Two cars A and B are going around concentric circular paths of `tau_(A)` and `tau_(B)`. If the two cars complete the circular paths in the same time then the ratio of angular speeds of A and B is

Options

A. 1
B. `tau_(A)//tau_(B)`
C. `tau_(B)//tau_(A)`
D. none of the above

Answer

To solve the problem, we need to find the ratio of the angular speeds of two cars A and B that are moving around concentric circular paths. Given that both cars complete their respective circular paths in the same time, we can derive the required ratio step by step.

### Step-by-Step Solution:

1. **Understanding the Circular Motion**:
- Let the radius of the circular path for car A be \( R_A \) and for car B be \( R_B \).
- The distance traveled by car A in one complete revolution is \( L_A = 2\pi R_A \).
- The distance traveled by car B in one complete revolution is \( L_B = 2\pi R_B \).

2. **Time Taken for One Revolution**:
- Let the time taken for both cars to complete one revolution be \( T \). Since both cars complete their circular paths in the same time, we have:
\[ T_A = T_B = T \]

3. **Relating Angular Speed to Time Period**:
- The angular speed \( \omega \) is related to the time period \( T \) by the formula:
\[ \omega = \frac{2\pi}{T} \]
- Therefore, for car A:
\[ \omega_A = \frac{2\pi}{T_A} = \frac{2\pi}{T} \]
- And for car B:
\[ \omega_B = \frac{2\pi}{T_B} = \frac{2\pi}{T} \]

4. **Finding the Ratio of Angular Speeds**:
- Since both angular speeds are equal:
\[ \omega_A = \omega_B \]
- Therefore, the ratio of angular speeds is:
\[ \frac{\omega_A}{\omega_B} = 1 \]

5. **Conclusion**:
- The ratio of the angular speeds of cars A and B is:
\[ \frac{\omega_A}{\omega_B} = 1 \]

### Final Answer:
The ratio of angular speeds of A and B is \( 1:1 \).