Question

A rotating fly wheel with an initial angular speed of \( 4 \, \text{rad s}^{-1} \) has an angular acceleration of \( 2 \, \text{rad s}^{-2} \). The angle (in radian) it will turn in a time of \( 4 \, s \) from the start is

• A. \( 32 \)
• B. \( 16 \)
• C. \( 8 \)
• D. \( 64 \)
• E. \( 24 \)

Hint:

For rotational motion, use \( \theta = \omega_0 t + \frac{1}{2}\alpha t^2 \), which is analogous to linear motion \( s = ut + \frac{1}{2}at^2 \).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a problem of rotational kinematics with constant angular acceleration. The equations of rotational motion are analogous to the linear equations of motion. We need to find the angular displacement given the initial angular velocity, angular acceleration, and time.
Step 2: Key Formula or Approach:
The rotational equation of motion that relates angular displacement (\(\theta\)), initial angular velocity (\(\omega_0\)), time (\(t\)), and constant angular acceleration (\(\alpha\)) is:
\[ \theta = \omega_0 t + \frac{1}{2}\alpha t^2 \] This is analogous to the linear equation \(s = ut + \frac{1}{2}at^2\).
Step 3: Detailed Explanation:
We are given the following values:
- Initial angular speed, \( \omega_0 = 4 \) rad s\(^{-1}\)
- Angular acceleration, \( \alpha = 2 \) rad s\(^{-2}\)
- Time, \( t = 4 \) s
We need to find the angular displacement, \( \theta \).
Substitute the given values into the equation of motion:
\[ \theta = (4 \text{ rad s}^{-1})(4 \text{ s}) + \frac{1}{2}(2 \text{ rad s}^{-2})(4 \text{ s})^2 \] \[ \theta = 16 \text{ rad} + \frac{1}{2}(2 \text{ rad s}^{-2})(16 \text{ s}^2) \] \[ \theta = 16 \text{ rad} + (1 \text{ rad s}^{-2})(16 \text{ s}^2) \] \[ \theta = 16 \text{ rad} + 16 \text{ rad} \] \[ \theta = 32 \text{ rad} \] Step 4: Final Answer:
The angle the flywheel will turn in 4 seconds is 32 radians. This corresponds to option (A).