A wheel of a vehicle is rotated to a uniform angular acceleration about its axis. Initially its angular velocity is zero. It rotates through an angle `theta_1` in the first 2 s and in the next 3 s, it rotates through an additional angle `theta_2`. The ratio of `theta_2/theta_1` is

Options

A. `4/21`
B. `21/4`
C. `4/25`
D. `25/4`

Answer

To solve the problem, we need to find the ratio of the angles rotated by a wheel of a vehicle under uniform angular acceleration. Let's break down the solution step by step.

### Step 1: Understand the Problem
We have a wheel that starts from rest (initial angular velocity, \( \omega_0 = 0 \)) and rotates with a constant angular acceleration \( \alpha \). The wheel rotates through an angle \( \theta_1 \) in the first 2 seconds and through an additional angle \( \theta_2 \) in the next 3 seconds.

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

Substituting in the formula:
\[ \theta_1 = 0 \cdot 2 + \frac{1}{2} \alpha (2^2) = \frac{1}{2} \alpha \cdot 4 = 2\alpha \]

### Step 3: Calculate \( \theta_2 \)
Now, we need to calculate the total angular displacement after 5 seconds (which includes both \( \theta_1 \) and \( \theta_2 \)):
- Total time \( t = 5 \) seconds

Using the same formula:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Substituting:
\[ \theta = 0 \cdot 5 + \frac{1}{2} \alpha (5^2) = \frac{1}{2} \alpha \cdot 25 = \frac{25}{2} \alpha \]

### Step 4: Relate \( \theta_1 \) and \( \theta_2 \)
We know that:
\[ \theta = \theta_1 + \theta_2 \]
Substituting the expressions we found:
\[ \frac{25}{2} \alpha = \theta_1 + \theta_2 \]
Substituting \( \theta_1 = 2\alpha \):
\[ \frac{25}{2} \alpha = 2\alpha + \theta_2 \]

### Step 5: Solve for \( \theta_2 \)
Rearranging the equation:
\[ \theta_2 = \frac{25}{2} \alpha - 2\alpha \]
Expressing \( 2\alpha \) in terms of a common denominator:
\[ \theta_2 = \frac{25}{2} \alpha - \frac{4}{2} \alpha = \frac{21}{2} \alpha \]

### Step 6: Find the Ratio \( \frac{\theta_2}{\theta_1} \)
Now we can find the ratio:
\[ \frac{\theta_2}{\theta_1} = \frac{\frac{21}{2} \alpha}{2\alpha} \]
The \( \alpha \) cancels out:
\[ \frac{\theta_2}{\theta_1} = \frac{21/2}{2} = \frac{21}{4} \]

### Conclusion
Thus, the ratio of \( \frac{\theta_2}{\theta_1} \) is:
\[ \frac{\theta_2}{\theta_1} = \frac{21}{4} \]