A wheel intially at rest, starts rotating with a uniform angular acceleration. The wheel rotates through an angle `theta_1` , in first second and through a angle `theta_2` , in next two second. The value of `theta_2 / theta_1`

Options

A. 8
B. 4
C. `1/4`
D. `1/8`

Answer

To solve the problem, we need to determine the ratio \( \frac{\theta_2}{\theta_1} \) given that a wheel starts from rest and rotates with a uniform angular acceleration. We will break down the solution step by step.

### Step 1: Understand the given information
- The wheel starts from rest, so the initial angular velocity \( \omega_0 = 0 \).
- The angular acceleration is uniform, denoted as \( \alpha \).
- The angle rotated in the first second is \( \theta_1 \).
- The angle rotated in the next two seconds is \( \theta_2 \).

### Step 2: Calculate \( \theta_1 \)
Using the formula for angular displacement under uniform acceleration:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
For the first second (\( t = 1 \) s):
\[ \theta_1 = 0 \cdot 1 + \frac{1}{2} \alpha (1^2) = \frac{1}{2} \alpha \]

### Step 3: Calculate \( \theta_2 \)
For the total time of 3 seconds (\( t = 3 \) s):
\[ \theta_2 = \text{Total angle in 3 seconds} - \theta_1 \]
The total angle in 3 seconds is:
\[ \theta_{\text{total}} = 0 \cdot 3 + \frac{1}{2} \alpha (3^2) = \frac{1}{2} \alpha \cdot 9 = \frac{9}{2} \alpha \]
Thus, we have:
\[ \theta_2 = \theta_{\text{total}} - \theta_1 = \frac{9}{2} \alpha - \frac{1}{2} \alpha = \frac{8}{2} \alpha = 4 \alpha \]

### Step 4: Find the ratio \( \frac{\theta_2}{\theta_1} \)
Now we can find the ratio:
\[ \frac{\theta_2}{\theta_1} = \frac{4 \alpha}{\frac{1}{2} \alpha} = \frac{4 \alpha}{0.5 \alpha} = \frac{4}{0.5} = 8 \]

### Final Answer
Thus, the value of \( \frac{\theta_2}{\theta_1} \) is \( 8 \).

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