Calculate the angular acceleration if a flywheel gains a speed of 540 r.p.m. in 6 seconds.

Options

A. 3`pi`rad`s^-2`
B. 6`pi`rad`s^-2`
C. 9`pi`rad`s^-2`
D. 12`pi`rad`s^-2`

Answer

To solve the problem of calculating the angular acceleration of a flywheel that gains a speed of 540 r.p.m. in 6 seconds, we will follow these steps:

### Step 1: Convert the final angular speed from r.p.m. to rad/s
- The final speed given is 540 revolutions per minute (r.p.m.).
- To convert this to radians per second (rad/s), we use the conversion factor:
\[ \text{Angular speed} (\omega) = \text{Revolutions per minute} \times \frac{2\pi \text{ rad}}{1 \text{ revolution}} \times \frac{1 \text{ minute}}{60 \text{ seconds}} \]
- Substituting the values:
\[ \omega = 540 \times \frac{2\pi}{60} = 540 \times \frac{2\pi}{60} = 18\pi \text{ rad/s} \]

### Step 2: Determine the change in angular speed
- The initial angular speed (\(\omega_0\)) is 0 rad/s (since it starts from rest).
- The final angular speed (\(\omega_f\)) is \(18\pi\) rad/s.
- The change in angular speed (\(\Delta \omega\)) is:
\[ \Delta \omega = \omega_f - \omega_0 = 18\pi - 0 = 18\pi \text{ rad/s} \]

### Step 3: Calculate the angular acceleration
- Angular acceleration (\(\alpha\)) is defined as the rate of change of angular speed:
\[ \alpha = \frac{\Delta \omega}{\Delta t} \]
- Here, \(\Delta t\) is the time taken, which is 6 seconds.
- Substituting the values:
\[ \alpha = \frac{18\pi}{6} = 3\pi \text{ rad/s}^2 \]

### Final Answer
The angular acceleration of the flywheel is \(3\pi \text{ rad/s}^2\).

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