A wheel rotating wilth unifrom angular acceleration covers 50 revolutions in the first five seconds after the start. Find the angular acceleration and the angular velocity at the end of five seconds.

Explanation

To solve the problem step by step, we will use the equations of motion for rotational motion under uniform angular acceleration.

### Step 1: Convert Revolutions to Radians
The first step is to convert the number of revolutions into radians. We know that:
1 revolution = \(2\pi\) radians.

Given that the wheel covers 50 revolutions, we can calculate the total angle in radians (\(\theta\)):
\[ \theta = 50 \text{ revolutions} \times 2\pi \text{ radians/revolution} = 100\pi \text{ radians} \]

### Step 2: Identify Initial Conditions
The problem states that the wheel starts from rest, which means the initial angular velocity (\(\omega_0\)) is:
\[ \omega_0 = 0 \text{ rad/s} \]

### Step 3: Use the Angular Displacement Formula
We will use the angular displacement formula for uniformly accelerated motion:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Substituting the known values:
- \(\theta = 100\pi\) radians
- \(\omega_0 = 0\) rad/s
- \(t = 5\) seconds

The equation simplifies to:
\[ 100\pi = 0 \cdot 5 + \frac{1}{2} \alpha (5^2) \]
This simplifies to:
\[ 100\pi = \frac{1}{2} \alpha (25) \]
\[ 100\pi = \frac{25}{2} \alpha \]

### Step 4: Solve for Angular Acceleration (\(\alpha\))
To find \(\alpha\), we rearrange the equation:
\[ \alpha = \frac{100\pi \times 2}{25} = \frac{200\pi}{25} = 8\pi \text{ rad/s}^2 \]

### Step 5: Calculate Angular Velocity at the End of 5 Seconds
Now, we will calculate the angular velocity (\(\omega\)) at the end of 5 seconds using the formula:
\[ \omega = \omega_0 + \alpha t \]
Substituting the known values:
\[ \omega = 0 + (8\pi)(5) \]
\[ \omega = 40\pi \text{ rad/s} \]

### Final Answers
- The angular acceleration (\(\alpha\)) is \(8\pi \text{ rad/s}^2\).
- The angular velocity (\(\omega\)) at the end of 5 seconds is \(40\pi \text{ rad/s}\).