A wheel initially at rest, is rotated with a uniform angular acceleration. The wheel rotates through an angle `theta_(1)` in the first one second and through an additional angle `theta_(2)` in the next one second. The ratio `theta_(1)//theta_(2)` is:

Options

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

Explanation

To solve the problem, we need to find the angles rotated by the wheel in the first and second seconds, denoted as \( \theta_1 \) and \( \theta_2 \), and then calculate the ratio \( \frac{\theta_1}{\theta_2} \).

### Step-by-step Solution:

1. **Understanding the Motion**:
The wheel starts from rest and rotates with a uniform angular acceleration \( \alpha \). The angular displacement in any time interval can be calculated using the formula:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
where \( \omega_0 \) is the initial angular velocity. Since the wheel starts from rest, \( \omega_0 = 0 \).

2. **Calculating \( \theta_1 \)**:
For the first second (from \( t = 0 \) to \( t = 1 \)):
\[ \theta_1 = 0 + \frac{1}{2} \alpha (1)^2 = \frac{1}{2} \alpha \]

3. **Calculating \( \theta_2 \)**:
For the second second (from \( t = 1 \) to \( t = 2 \)), we first need to find the total angle rotated from \( t = 0 \) to \( t = 2 \):
\[ \theta_{\text{total}} = \frac{1}{2} \alpha (2)^2 = 2\alpha \]
We already calculated \( \theta_1 \) for the first second, so now we can find \( \theta_2 \):
\[ \theta_2 = \theta_{\text{total}} - \theta_1 = 2\alpha - \frac{1}{2} \alpha = 2\alpha - 0.5\alpha = 1.5\alpha = \frac{3}{2} \alpha \]

4. **Finding the Ratio \( \frac{\theta_1}{\theta_2} \)**:
Now we can find the ratio of the angles:
\[ \frac{\theta_1}{\theta_2} = \frac{\frac{1}{2} \alpha}{\frac{3}{2} \alpha} = \frac{1/2}{3/2} = \frac{1}{3} \]

### Final Answer:
The ratio \( \frac{\theta_1}{\theta_2} \) is \( \frac{1}{3} \).