A car wheel is rotated to uniform angular acceleration about its axis. Initially its angular velocity is zero. It rotates through an angle `theta_(1)` in the first 2 s. In the next 2 s, it rotates through an additional angle `theta_(2)`, the ratio of `(theta_(2))/(theta_(1))` is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem step by step, we can use the equations of motion for rotational motion under uniform angular acceleration.

### Step 1: Define the variables
- Let the angular acceleration be \( \alpha \).
- The initial angular velocity \( \omega_0 = 0 \) (as given).
- The time intervals are \( t_1 = 2 \, \text{s} \) and \( t_2 = 2 \, \text{s} \) (the first and second intervals).

### Step 2: Calculate the angle \( \theta_1 \) for the first 2 seconds
Using the equation for angular displacement under uniform angular acceleration:
\[ \theta_1 = \omega_0 t_1 + \frac{1}{2} \alpha t_1^2 \]
Substituting \( \omega_0 = 0 \) and \( t_1 = 2 \):
\[ \theta_1 = 0 \cdot 2 + \frac{1}{2} \alpha (2^2) = \frac{1}{2} \alpha \cdot 4 = 2\alpha \]

### Step 3: Calculate the angle \( \theta_2 \) for the next 2 seconds
For the next 2 seconds, the total time is \( t = 4 \, \text{s} \). We can calculate the total angle \( \theta \) covered in 4 seconds:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Substituting \( t = 4 \):
\[ \theta = 0 \cdot 4 + \frac{1}{2} \alpha (4^2) = \frac{1}{2} \alpha \cdot 16 = 8\alpha \]

### Step 4: Relate \( \theta_2 \) to \( \theta_1 \)
The angle \( \theta_2 \) covered in the second 2 seconds can be found by subtracting \( \theta_1 \) from the total angle covered in 4 seconds:
\[ \theta_2 = \theta - \theta_1 = 8\alpha - 2\alpha = 6\alpha \]

### Step 5: Find the ratio \( \frac{\theta_2}{\theta_1} \)
Now we can find the ratio:
\[ \frac{\theta_2}{\theta_1} = \frac{6\alpha}{2\alpha} = 3 \]

### Final Answer
The ratio \( \frac{\theta_2}{\theta_1} \) is \( 3 \).

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