Question

A fan is rotating with an angular speed \(300\ \text{rpm}\). The fan is switched off, and it takes \(80\ \text{s}\) to come to rest. Assuming constant angular deceleration, the number of revolutions made by the fan before it comes to rest is:

• A. \(400\)
• B. \(200\)
• C. \(300\)
• D. \(314\)

Hint:

For constant angular deceleration, \[ \omega_{\text{avg}}=\frac{\omega_0+\omega}{2} \] and \[ \text{Number of revolutions}=\omega_{\text{avg}}\times t \] when angular speed is measured in revolutions per second.

Answer 1

The Correct Option is B

Step 1: Record initial and final conditions.
$\omega_0 = 300$ rpm $= 5$ rev/s; $\omega = 0$; $t = 80$ s. Deceleration is uniform.
Step 2: Find the angular deceleration $\alpha$.
Using $\omega = \omega_0 + \alpha t$: \[ 0 = 5 + 80\alpha \Rightarrow \alpha = -\frac{1}{16} \text{ rev/s}^2 \]
Step 3: Use the kinematic equation to find total revolutions.
\[ N = \omega_0 t + \frac{1}{2}\alpha t^2 = 5(80) + \frac{1}{2}\!\left(-\frac{1}{16}\right)\!(80)^2 = 400 - 200 = 200 \text{ rev} \]
Step 4: Cross-check using $\omega^2 = \omega_0^2 + 2\alpha N$.
\[ 0 = 25 + 2\!\left(-\frac{1}{16}\right)\!N \Rightarrow N = 200 \text{ rev} \] Both methods agree.
Step 5: Verify via average angular speed.
$\omega_{\text{avg}} = 2.5$ rev/s. $N = 2.5 \times 80 = 200$ rev. Consistent.
Step 6: State the final answer.
\[ \boxed{N = 200 \text{ revolutions}} \]