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

Step 1: Understanding the Concept:
The problem involves rotational kinematics under constant angular acceleration.
We can use the rotational equivalent of the second equation of motion to find the angular displacement.
Step 2: Key Formula or Approach:
For an object starting from rest ($\omega_0 = 0$), the angular displacement is $\theta(t) = \frac{1}{2}\alpha t^2$.
We will evaluate this for different time intervals to find the required ratio.
Step 3: Detailed Explanation:
The angle rotated in the first 2 seconds ($t = 2 \text{ s}$) is:
\[ \theta_1 = \frac{1}{2}\alpha (2)^2 = 2\alpha \quad \dots \text{(1)} \]
The angle rotated from $t = 0$ to $t = 4 \text{ s}$ is:
\[ \theta_{\text{total}} = \frac{1}{2}\alpha (4)^2 = 8\alpha \quad \dots \text{(2)} \]
The angle rotated in the next 2 seconds (from $t=2$ to $t=4$) is $\theta_2$:
\[ \theta_2 = \theta_{\text{total}} - \theta_1 = 8\alpha - 2\alpha = 6\alpha \]
Taking the ratio of $\theta_2$ to $\theta_1$:
\[ \frac{\theta_2}{\theta_1} = \frac{6\alpha}{2\alpha} = 3 \]
Step 4: Final Answer:
The ratio of $\theta_2/\theta_1$ is 3.