If a disc starting from rest acquires an angular velocity of `240 "rev min"^(-1)` in 10 s, then its angular acceleration will be

Options

A. `1.52 rads^(-2)`
B. `3.11 rads^(-2)`
C. `2.51 rads^(-2)`
D. `1.13 rads^(-2)`

Explanation

To find the angular acceleration of the disc, we can follow these steps:

### Step 1: Identify the given values
- Initial angular velocity, \( \omega_0 = 0 \) (since the disc starts from rest)
- Final angular velocity, \( \omega = 240 \, \text{rev/min} \)
- Time, \( t = 10 \, \text{s} \)

### Step 2: Convert the final angular velocity to radians per second
To convert revolutions per minute (rev/min) to radians per second (rad/s), we use the conversion factor:
\[ 1 \, \text{rev} = 2\pi \, \text{rad} \quad \text{and} \quad 1 \, \text{min} = 60 \, \text{s} \]
Thus,
\[ \omega = 240 \, \text{rev/min} = 240 \times \frac{2\pi \, \text{rad}}{1 \, \text{rev}} \times \frac{1 \, \text{min}}{60 \, \text{s}} = 240 \times \frac{2\pi}{60} \, \text{rad/s} \]
Calculating this gives:
\[ \omega = 240 \times \frac{2\pi}{60} = 8\pi \, \text{rad/s} \]

### Step 3: Use the angular acceleration formula
The formula relating angular acceleration (\( \alpha \)), initial angular velocity (\( \omega_0 \)), final angular velocity (\( \omega \)), and time (\( t \)) is:
\[ \omega = \omega_0 + \alpha t \]
Since \( \omega_0 = 0 \):
\[ \omega = \alpha t \]
Substituting the known values:
\[ 8\pi = \alpha \times 10 \]

### Step 4: Solve for angular acceleration (\( \alpha \))
Rearranging the equation gives:
\[ \alpha = \frac{8\pi}{10} \]
Calculating this:
\[ \alpha = \frac{8\pi}{10} = \frac{4\pi}{5} \, \text{rad/s}^2 \]

### Step 5: Approximate the value of \( \alpha \)
Using \( \pi \approx \frac{22}{7} \):
\[ \alpha \approx \frac{4 \times \frac{22}{7}}{5} = \frac{88}{35} \approx 2.51 \, \text{rad/s}^2 \]

### Final Answer
Thus, the angular acceleration of the disc is approximately:
\[ \alpha \approx 2.51 \, \text{rad/s}^2 \]