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. 4
B. 2
C. 3
D. 1

Answer

To solve the problem, we will use the equations of rotational motion under uniform angular acceleration.

1. **Understanding the problem**: We have a wheel that starts from rest and rotates with a uniform angular acceleration (α). We need to find the angles θ₁ and θ₂, which are the angles rotated in the first and second seconds, respectively.

2. **Finding θ₁**: The angle rotated in the first second (θ₁) can be calculated using the formula for angular displacement under uniform acceleration:
\[ \theta_1 = \frac{1}{2} \alpha t^2 \]
For the first second (t = 1 s):
\[ \theta_1 = \frac{1}{2} \alpha (1^2) = \frac{1}{2} \alpha \]

3. **Finding θ₂**: The angle rotated in the second second (θ₂) can be calculated by finding the total angle rotated in the first two seconds and then subtracting θ₁ from it. The total angle after 2 seconds (t = 2 s) is:
\[ \theta_{total} = \frac{1}{2} \alpha (2^2) = \frac{1}{2} \alpha \cdot 4 = 2\alpha \]
Thus, the angle rotated in the second second (θ₂) is:
\[ \theta_2 = \theta_{total} - \theta_1 = 2\alpha - \frac{1}{2} \alpha = \frac{4}{2} \alpha - \frac{1}{2} \alpha = \frac{3}{2} \alpha \]

4. **Finding the ratio θ₂/θ₁**: Now, we can find the ratio of θ₂ to θ₁:
\[ \frac{\theta_2}{\theta_1} = \frac{\frac{3}{2} \alpha}{\frac{1}{2} \alpha} = \frac{3/2}{1/2} = 3 \]

Thus, the ratio \( \frac{\theta_2}{\theta_1} \) is **3**.