A fan rotating with an initial angular velocity of 1000 rev/min is switched off. In 2s, the angular velocity decreases to 200 rev/min. Assuming the angular acceleration is constant, how many revolutions does the blade undergo during this time?

Options

A. 10
B. 100
C. 20
D. 125

Answer

To solve the problem step by step, we will follow the outlined process to find the number of revolutions the fan blades undergo during the time interval given.

### Step 1: Convert Angular Velocities to Radians per Second
The initial angular velocity \( \omega_i \) is given as 1000 revolutions per minute (rev/min). We need to convert this to radians per second (rad/s):
\[ \omega_i = 1000 \, \text{rev/min} \times \frac{2\pi \, \text{rad}}{1 \, \text{rev}} \times \frac{1 \, \text{min}}{60 \, \text{s}} = \frac{1000 \times 2\pi}{60} = \frac{100\pi}{3} \, \text{rad/s} \]

The final angular velocity \( \omega_f \) is given as 200 rev/min. We convert this similarly:
\[ \omega_f = 200 \, \text{rev/min} \times \frac{2\pi \, \text{rad}}{1 \, \text{rev}} \times \frac{1 \, \text{min}}{60 \, \text{s}} = \frac{200 \times 2\pi}{60} = \frac{20\pi}{3} \, \text{rad/s} \]

### Step 2: Calculate Angular Acceleration
We know that the angular acceleration \( \alpha \) is constant. We can use the formula:
\[ \omega_f = \omega_i + \alpha t \]
Rearranging gives:
\[ \alpha = \frac{\omega_f - \omega_i}{t} \]
Substituting the known values:
\[ \alpha = \frac{\frac{20\pi}{3} - \frac{100\pi}{3}}{2} = \frac{-80\pi/3}{2} = -\frac{40\pi}{3} \, \text{rad/s}^2 \]

### Step 3: Calculate Angular Displacement
We can find the angular displacement \( \theta \) using the formula:
\[ \theta = \omega_i t + \frac{1}{2} \alpha t^2 \]
Substituting the values:
\[ \theta = \left(\frac{100\pi}{3}\right)(2) + \frac{1}{2}\left(-\frac{40\pi}{3}\right)(2^2) \]
Calculating each term:
\[ \theta = \frac{200\pi}{3} + \frac{1}{2}\left(-\frac{40\pi}{3}\right)(4) = \frac{200\pi}{3} - \frac{80\pi}{3} = \frac{120\pi}{3} = 40\pi \, \text{radians} \]

### Step 4: Convert Angular Displacement to Revolutions
To find the number of revolutions \( N \), we divide the angular displacement by \( 2\pi \):
\[ N = \frac{\theta}{2\pi} = \frac{40\pi}{2\pi} = 20 \]

### Final Answer
The fan blades undergo **20 revolutions** during the 2 seconds.

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