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 along concentric circular paths of different radii, denoted as \( \tau_A \) and \( \tau_B \). Given that both cars complete their respective circular paths in the same time, we can derive the ratio of their angular speeds.

### Step-by-Step Solution:

1. **Understanding Angular Speed**:
Angular speed (\( \omega \)) is defined as the angle covered per unit time. The formula for angular speed is given by:
\[ \omega = \frac{\theta}{t} \]
where \( \theta \) is the angle in radians and \( t \) is the time taken.

2. **Time Taken to Complete Circular Path**:
For a complete circular path, the angle covered is \( 2\pi \) radians. Therefore, the time taken to complete one full circular path can be expressed as:
\[ t_A = \frac{2\pi}{\omega_A} \quad \text{(for car A)} \]
\[ t_B = \frac{2\pi}{\omega_B} \quad \text{(for car B)} \]

3. **Equating the Times**:
Since both cars complete their paths in the same time, we can set the two time equations equal to each other:
\[ t_A = t_B \]
This leads to:
\[ \frac{2\pi}{\omega_A} = \frac{2\pi}{\omega_B} \]

4. **Simplifying the Equation**:
We can cancel \( 2\pi \) from both sides of the equation:
\[ \frac{1}{\omega_A} = \frac{1}{\omega_B} \]

5. **Finding the Ratio of Angular Speeds**:
Taking the reciprocal of both sides gives us:
\[ \frac{\omega_A}{\omega_B} = 1 \]

6. **Conclusion**:
Therefore, 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 \( 1:1 \).

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