Question

The angular speed of a rigid body rotating about a fixed axis is \( (8-2t) \) rad s\(^{-1}\). The angle through which the body rotates before it comes to rest is:

• A. 8 rad
• B. 12 rad
• C. 16 rad
• D. 20 rad

Hint:

When dealing with angular motion, remember that the total angle rotated is the integral of the angular velocity over time.

Answer 1

The Correct Option is C

We are given that the angular speed \( \omega(t) = 8 - 2t \) rad/s and that the body eventually comes to rest. When the body comes to rest, \( \omega = 0 \). \[ \omega(t) = 0 \quad \Rightarrow \quad 8 - 2t = 0 \quad \Rightarrow \quad t = 4 \text{ seconds} \] Next, we need to calculate the total angular displacement (the angle through which the body rotates) before it comes to rest. The angle \( \theta \) is given by the integral of the angular velocity: \[ \theta = \int_0^4 \omega(t) \, dt = \int_0^4 (8 - 2t) \, dt \] Evaluating this integral: \[ \theta = \left[ 8t - t^2 \right]_0^4 = (8 \times 4 - 4^2) - (8 \times 0 - 0^2) \] \[ \theta = (32 - 16) - 0 = 16 \text{ rad} \] Thus, the total angle the body rotates before coming to rest is \( 16 \) radians.

Answer 2

Step 1: Write down the given data
Angular speed, \( \omega = 8 - 2t \) rad/s
The body comes to rest when \( \omega = 0 \)

Step 2: Find the time when the body stops
Set \( \omega = 0 \):
\[ 8 - 2t = 0 \implies t = 4 \, \text{s} \]

Step 3: Find the angular displacement \( \theta \)
Angular displacement is the integral of angular velocity over time:
\[ \theta = \int_0^{4} (8 - 2t) dt \]
Calculate the integral:
\[ \theta = \left[ 8t - t^2 \right]_0^{4} = (8 \times 4) - (4)^2 - 0 = 32 - 16 = 16 \, \text{rad} \]

Final answer: The angle through which the body rotates before coming to rest is 16 radians.