A flywheel rotates about a fixed axis and slows down from 300 rpm to 100 rpm in 2 minutes (i) What is the angular acceleration in `"rad min"^(-2)` ? (ii) How many revolutions does the wheel complete during this time ?

Options

A. `100/pi`
B. 100
C. `100pi`
D. `200pi`

Answer

To solve the problem step by step, we will first find the angular acceleration and then calculate the number of revolutions completed during the given time.

### Step 1: Convert RPM to Radians per Minute
The initial angular velocity (ω₀) and final angular velocity (ω) can be calculated from the given RPM values.

- Initial angular velocity (ω₀) = 300 rpm
- Final angular velocity (ω) = 100 rpm

To convert RPM to radians per minute, we use the conversion factor \( 1 \text{ rpm} = \frac{2\pi \text{ radians}}{1 \text{ minute}} \).

\[ \omega_0 = 300 \times 2\pi \text{ rad/min} = 600\pi \text{ rad/min} \]
\[ \omega = 100 \times 2\pi \text{ rad/min} = 200\pi \text{ rad/min} \]

### Step 2: Calculate Angular Acceleration
Angular acceleration (α) can be calculated using the formula:

\[ \alpha = \frac{\Delta \omega}{\Delta t} \]

Where:
- \(\Delta \omega = \omega - \omega_0 = 200\pi - 600\pi = -400\pi \text{ rad/min}\)
- \(\Delta t = 2 \text{ minutes}\)

Now substituting the values:

\[ \alpha = \frac{-400\pi \text{ rad/min}}{2 \text{ min}} = -200\pi \text{ rad/min}^2 \]

### Step 3: Calculate the Number of Revolutions
To find the total number of revolutions (N) during the time interval, we can use the formula:

\[ N = \frac{\omega_0 + \omega}{2} \times t \]

Where:
- \(t = 2 \text{ minutes}\)

Substituting the values:

\[ N = \frac{600\pi + 200\pi}{2} \times 2 \]
\[ N = \frac{800\pi}{2} \times 2 = 400\pi \text{ revolutions} \]

### Final Answers
(i) The angular acceleration is \(-200\pi \text{ rad/min}^2\).

(ii) The number of revolutions completed during this time is \(400\pi\).