Question

A wheel initially at rest is subjected to a uniform angular acceleration about its axis. In the first \(2\) sec it rotates through an angle \(\theta_1\) and in the next \(2\) sec it rotates an angle \(\theta_2\). Find the ratio \(\theta_2/\theta_1\).

• A. \(5\)
• B. \(2\)
• C. \(4\)
• D. \(3\)

Hint:

For motion starting from rest under constant angular acceleration: \[ \theta \propto t^2 \] So angular displacement in successive time intervals increases rapidly with time.

Answer 1

The Correct Option is D

Concept: For rotational motion with constant angular acceleration, \[ \theta = \omega_0 t + \frac{1}{2}\alpha t^2 \] Since the wheel starts from rest, \[ \omega_0 = 0 \] \[ \theta = \frac{1}{2}\alpha t^2 \]
Step 1: Angular displacement in the first \(2\) seconds. \[ \theta_1 = \frac{1}{2}\alpha (2)^2 \] \[ \theta_1 = 2\alpha \qquad ...(1) \]
Step 2: Angular displacement in the first \(4\) seconds. \[ \theta' = \frac{1}{2}\alpha (4)^2 \] \[ \theta' = 8\alpha \qquad ...(2) \]
Step 3: Angular displacement in the next \(2\) seconds. \[ \theta_2 = \theta' - \theta_1 \] \[ \theta_2 = 8\alpha - 2\alpha \] \[ \theta_2 = 6\alpha \]
Step 4: Find the required ratio. \[ \frac{\theta_2}{\theta_1} = \frac{6\alpha}{2\alpha} \] \[ \frac{\theta_2}{\theta_1} = 3 \] Thus, \[ \boxed{\frac{\theta_2}{\theta_1} = 3} \]