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

Options

A. (a)4
B. (b)2
C. (c)3
D. (d)1

Explanation

To solve the problem, we need to find the ratio of the angles rotated by a wheel under uniform angular acceleration during two consecutive seconds. Let's denote the angular acceleration as \( \alpha \).

### Step-by-Step Solution:

1. **Define the Variables**:
- Let \( \theta_1 \) be the angle rotated in the first second.
- Let \( \theta_2 \) be the additional angle rotated in the second second.

2. **Use the Angular Displacement Formula**:
The angular displacement \( \theta \) for an object starting from rest with uniform angular acceleration is given by:
\[ \theta = \frac{1}{2} \alpha t^2 \]

3. **Calculate \( \theta_1 \)**:
For the first second (\( t = 1 \) s):
\[ \theta_1 = \frac{1}{2} \alpha (1^2) = \frac{1}{2} \alpha \]

4. **Calculate Total Angle After Two Seconds**:
For the first two seconds (\( t = 2 \) s):
\[ \theta_{total} = \frac{1}{2} \alpha (2^2) = \frac{1}{2} \alpha \cdot 4 = 2\alpha \]

5. **Relate \( \theta_2 \) to \( \theta_1 \)**:
The total angle after two seconds is the sum of the angles rotated in the first and second seconds:
\[ \theta_{total} = \theta_1 + \theta_2 \]
Therefore,
\[ 2\alpha = \theta_1 + \theta_2 \]

6. **Substituting \( \theta_1 \)**:
Substitute \( \theta_1 = \frac{1}{2} \alpha \) into the equation:
\[ 2\alpha = \frac{1}{2} \alpha + \theta_2 \]

7. **Solve for \( \theta_2 \)**:
Rearranging gives:
\[ \theta_2 = 2\alpha - \frac{1}{2} \alpha = \frac{4\alpha}{2} - \frac{1}{2} \alpha = \frac{3}{2} \alpha \]

8. **Find the Ratio \( \frac{\theta_2}{\theta_1} \)**:
Now, we can find the ratio:
\[ \frac{\theta_2}{\theta_1} = \frac{\frac{3}{2} \alpha}{\frac{1}{2} \alpha} \]
The \( \alpha \) cancels out:
\[ \frac{\theta_2}{\theta_1} = \frac{3/2}{1/2} = 3 \]

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