A wheel is making revolutions about its axis with uniform angular acceleration. Starting from rest, till it reaches 100 rev/sec in 4 seconds. Find the angular acceleration. Find the angle rotated during these four seconds.

Explanation

To solve the problem step by step, we will follow the given information and apply the relevant formulas for rotational motion.

### Step 1: Convert revolutions per second to radians per second
We are given that the wheel reaches a final angular velocity of 100 revolutions per second. To convert this to radians per second, we use the conversion factor \(2\pi\) radians per revolution.

\[ \omega = N \times 2\pi = 100 \, \text{rev/sec} \times 2\pi \, \text{rad/rev} = 200\pi \, \text{rad/sec} \]

### Step 2: Identify initial angular velocity
The wheel starts from rest, so the initial angular velocity (\(\omega_0\)) is:

\[ \omega_0 = 0 \, \text{rad/sec} \]

### Step 3: Use the angular acceleration formula
We can use the formula for angular velocity with uniform angular acceleration:

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

Substituting the known values:

\[ 200\pi = 0 + \alpha \times 4 \]

### Step 4: Solve for angular acceleration (\(\alpha\))
Rearranging the equation to solve for \(\alpha\):

\[ \alpha = \frac{200\pi}{4} = 50\pi \, \text{rad/sec}^2 \]

### Step 5: Calculate the angle rotated (\(\theta\))
To find the angle rotated during these 4 seconds, we can use the formula:

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

Since \(\omega_0 = 0\), the equation simplifies to:

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

Substituting the values of \(\alpha\) and \(t\):

\[ \theta = \frac{1}{2} \times 50\pi \times (4)^2 \]

Calculating this gives:

\[ \theta = \frac{1}{2} \times 50\pi \times 16 = 400\pi \, \text{radians} \]

### Final Answers
- Angular acceleration (\(\alpha\)) = \(50\pi \, \text{rad/sec}^2\)
- Angle rotated (\(\theta\)) = \(400\pi \, \text{radians}\)

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