Two cars having masses `m_1 and m_2` move in circles of radii `r_1 and r_2` respectively. If they complete the circle is equal time the ratio of their angular speeds is `omega_1/omega_2` is

Options

A. `m_1/m_2`
B. `r_1/r_2`
C. `m_1r_1/m_2r_2`
D. 1

Answer

To solve the problem, we need to determine the ratio of the angular speeds of two cars moving in circular paths of different radii but completing their circles in equal time.

### Step-by-Step Solution:

1. **Understand Angular Speed**:
Angular speed (ω) is defined as the angle covered per unit time. It can be calculated using the formula:
\[ \omega = \frac{2\pi}{T} \]
where \(T\) is the time period, or the time taken to complete one full circle.

2. **Identify Given Information**:
We have two cars with masses \(m_1\) and \(m_2\) moving in circles of radii \(r_1\) and \(r_2\) respectively. The problem states that they complete their circles in equal time.

3. **Set Up the Equations**:
Since both cars complete their circles in equal time, we can denote the time taken for both cars to complete one circle as \(T\). Therefore:
\[ T_1 = T_2 = T \]

4. **Calculate Angular Speeds**:
Using the formula for angular speed, we can express the angular speeds for both cars:
\[ \omega_1 = \frac{2\pi}{T_1} = \frac{2\pi}{T} \]
\[ \omega_2 = \frac{2\pi}{T_2} = \frac{2\pi}{T} \]

5. **Find the Ratio of Angular Speeds**:
Now, we can find the ratio of their angular speeds:
\[ \frac{\omega_1}{\omega_2} = \frac{\frac{2\pi}{T}}{\frac{2\pi}{T}} = 1 \]

6. **Conclusion**:
The ratio of their angular speeds is:
\[ \frac{\omega_1}{\omega_2} = 1 \]

### Final Answer:
The ratio of their angular speeds \(\frac{\omega_1}{\omega_2}\) is \(1\).

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