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: Recall rotational kinematics formula.
For angular motion with constant angular acceleration: \[ \theta = \omega_0 t + \frac{1}{2}\alpha t^2 \]

Step 2: Identify the given values.
Initial angular velocity: \[ \omega_0 = 4 \, \text{rad/s} \] Angular acceleration: \[ \alpha = 2 \, \text{rad/s}^2 \] Time: \[ t = 4 \, s \]

Step 3: Substitute into the formula.
\[ \theta = (4)(4) + \frac{1}{2}(2)(4^2) \]

Step 4: Simplify each term.
First term: \[ 4 \times 4 = 16 \] Second term: \[ \frac{1}{2} \times 2 \times 16 = 16 \]

Step 5: Add both contributions.
\[ \theta = 16 + 16 = 32 \]

Step 6: Interpret physically.
The total angular displacement consists of contribution due to initial velocity and due to acceleration.

Step 7: Final conclusion.
Hence, the angle turned is: \[ \boxed{32 \, \text{radians}} \] Therefore, the correct option is \[ \boxed{(1)\ 32} \]