On applying a constant torque, a wheel at rest, turns through `400` radian in `10 s`. Find angular acceleration. If same torque continues to act, what will be angular veclocity of the wheel after `20 s` from stars ?

Options

A. `"160 rad s"^(-1)`
B. `"150 rad s"^(-1)`
C. `"120 rad s"^(-1)`
D. `"130 rad s"^(-1)`

Answer

To find the angular acceleration and the angular velocity of the wheel, we can use the following steps:

### Step 1: Find Angular Acceleration
We know that the angular displacement (θ) is given as 400 radians and the time (t) is 10 seconds. The formula for angular displacement under constant angular acceleration is:

\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]

Where:
- \(\theta\) = angular displacement (400 rad)
- \(\omega_0\) = initial angular velocity (0 rad/s, since the wheel is at rest)
- \(\alpha\) = angular acceleration (unknown)
- \(t\) = time (10 s)

Since the wheel starts from rest, \(\omega_0 = 0\). The equation simplifies to:

\[ \theta = \frac{1}{2} \alpha t^2 \]

Substituting the known values:

\[ 400 = \frac{1}{2} \alpha (10)^2 \]

This simplifies to:

\[ 400 = 5\alpha \]

Now, solving for \(\alpha\):

\[ \alpha = \frac{400}{5} = 80 \, \text{rad/s}^2 \]

### Step 2: Find Angular Velocity after 20 seconds
To find the angular velocity (\(\omega\)) after 20 seconds, we can use the formula:

\[ \omega = \omega_0 + \alpha t \]

Substituting the known values:

\[ \omega = 0 + 80 \times 20 \]

Calculating this gives:

\[ \omega = 1600 \, \text{rad/s} \]

### Final Answers
- Angular acceleration (\(\alpha\)) = 80 rad/s²
- Angular velocity after 20 seconds (\(\omega\)) = 1600 rad/s

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