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 will use the equations of rotational motion under uniform angular acceleration. The wheel is initially at rest, which means its initial angular velocity (ω_initial) is 0.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The wheel rotates through an angle θ₁ in the first second and through an additional angle θ₂ in the next second. We need to find the ratio θ₁/θ₂.

2. **Using the Equation of Motion**:
- The equation for angular displacement under uniform angular acceleration is given by:
\[ \theta = \omega_{\text{initial}} \cdot t + \frac{1}{2} \alpha t^2 \]
- Since the wheel starts from rest, ω_initial = 0. Thus, the equation simplifies to:
\[ \theta = \frac{1}{2} \alpha t^2 \]

3. **Calculating θ₁ (Angle in the First Second)**:
- For the first second (t = 1 s):
\[ \theta_1 = \frac{1}{2} \alpha (1^2) = \frac{\alpha}{2} \]

4. **Calculating θ₁ + θ₂ (Angle in the First Two Seconds)**:
- For the first two seconds (t = 2 s):
\[ \theta_1 + \theta_2 = \frac{1}{2} \alpha (2^2) = \frac{1}{2} \alpha \cdot 4 = 2\alpha \]

5. **Expressing θ₂ in Terms of θ₁**:
- From the previous equations, we have:
\[ \theta_1 + \theta_2 = 2\alpha \]
- Substituting θ₁:
\[ \frac{\alpha}{2} + \theta_2 = 2\alpha \]
- Rearranging gives:
\[ \theta_2 = 2\alpha - \frac{\alpha}{2} = \frac{4\alpha}{2} - \frac{\alpha}{2} = \frac{3\alpha}{2} \]

6. **Finding the Ratio θ₁/θ₂**:
- Now we can find the ratio:
\[ \frac{\theta_1}{\theta_2} = \frac{\frac{\alpha}{2}}{\frac{3\alpha}{2}} = \frac{1}{3} \]

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