Two racing cars of masses `m_(1)` and `m_(2)` are moving in circles of radii `r_(1)` and `r_2` respectively. Their speeds are such that each makes a complete circle in the same time t. The ratio of the angular speed of the first to the second car is

Options

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

Answer

To solve the problem of finding the ratio of the angular speed of the first car to the second car, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Angular Speed**:
Angular speed (ω) is defined as the angle covered per unit time. For an object moving in a circular path, it can be expressed as:
\[ \omega = \frac{2\pi}{T} \]
where \(T\) is the time period of one complete revolution.

2. **Identifying the Time Periods**:
According to the problem, both cars complete their circular paths in the same time \(t\). Thus, we can denote:
\[ T_1 = T_2 = t \]

3. **Calculating Angular Speeds**:
Now, we can calculate the angular speeds for both cars:
- For the first car (mass \(m_1\)):
\[ \omega_1 = \frac{2\pi}{T_1} = \frac{2\pi}{t} \]
- For the second car (mass \(m_2\)):
\[ \omega_2 = \frac{2\pi}{T_2} = \frac{2\pi}{t} \]

4. **Finding the Ratio of Angular Speeds**:
Now, we can find the ratio of the angular speeds:
\[ \frac{\omega_1}{\omega_2} = \frac{\frac{2\pi}{t}}{\frac{2\pi}{t}} = \frac{2\pi}{t} \times \frac{t}{2\pi} = 1 \]

5. **Conclusion**:
Therefore, the ratio of the angular speed of the first car to the second car is:
\[ \omega_1 : \omega_2 = 1 : 1 \]

### Final Answer:
The ratio of the angular speed of the first car to the second car is \(1 : 1\).

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