Ratio of Angles Rotated in Successive Time Intervals

To find the ratio $\frac{\theta_2}{\theta_1}$ of the angles rotated by the wheel, we use the kinematic equations for rotational motion with constant angular acceleration.

Given:

• Initial angular velocity of the wheel, $\omega_0 = 0$ (since it starts from rest)
• Uniform angular acceleration, $\alpha$
• First time interval, $t_1 = 2\text{ s}$
• Next time interval, $t_2 = 2\text{ s}$ (total time elapsed from start $t = 4\text{ s}$)

1. Angle rotated in the first $2\text{ s}$ ($\theta_1$): Using the equation of rotational motion $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$: $\theta_1 = 0 \cdot (2) + \frac{1}{2} \alpha (2)^2$ $\theta_1 = 2\alpha$

2. Total angle rotated in the first $4\text{ s}$ ($\theta_{\text{total}}$): $\theta_{\text{total}} = 0 \cdot (4) + \frac{1}{2} \alpha (4)^2$ $\theta_{\text{total}} = 8\alpha$

3. Angle rotated in the next $2\text{ s}$ ($\theta_2$): The angle $\theta_2$ rotated during the interval from $t = 2\text{ s}$ to $t = 4\text{ s}$ is: $\theta_2 = \theta_{\text{total}} - \theta_1$ $\theta_2 = 8\alpha - 2\alpha = 6\alpha$

4. Ratio $\frac{\theta_2}{\theta_1}$: $\frac{\theta_2}{\theta_1} = \frac{6\alpha}{2\alpha} = 3$

Thus, the ratio $\frac{\theta_2}{\theta_1}$ is $3$, which corresponds to Option B.