A wheel starting from rest is uniformly accelerate at 4 `rad /s^2` for 10 seconds. It is allowed to rotate uniformly for the next 10 seconds and is finally brought to rest in the next 10 seconds. Find the total angle rotated by the wheel.

Explanation

To solve the problem step by step, we will break down the motion of the wheel into three distinct phases: acceleration, uniform rotation, and deceleration.

### Step 1: Calculate the angular displacement during acceleration
The wheel starts from rest and accelerates uniformly at \( \alpha = 4 \, \text{rad/s}^2 \) for \( t = 10 \, \text{s} \).

Using the formula for angular displacement during uniform acceleration:
\[ \theta_1 = \omega_0 t + \frac{1}{2} \alpha t^2 \]
where \( \omega_0 = 0 \) (initial angular velocity), we can substitute the values:
\[ \theta_1 = 0 \cdot 10 + \frac{1}{2} \cdot 4 \cdot (10)^2 \]
\[ \theta_1 = \frac{1}{2} \cdot 4 \cdot 100 = 200 \, \text{radians} \]

### Step 2: Calculate the final angular velocity after acceleration
To find the final angular velocity after the first 10 seconds, we use:
\[ \omega_f = \omega_0 + \alpha t \]
Substituting the values:
\[ \omega_f = 0 + 4 \cdot 10 = 40 \, \text{rad/s} \]

### Step 3: Calculate the angular displacement during uniform rotation
During the next 10 seconds, the wheel rotates uniformly at \( \omega = 40 \, \text{rad/s} \). The angular displacement during this phase is given by:
\[ \theta_2 = \omega t \]
Substituting the values:
\[ \theta_2 = 40 \cdot 10 = 400 \, \text{radians} \]

### Step 4: Calculate the angular displacement during deceleration
The wheel is brought to rest in the next 10 seconds. The initial angular velocity for this phase is \( \omega_0 = 40 \, \text{rad/s} \) and the final angular velocity \( \omega_f = 0 \).

First, we need to find the angular acceleration during deceleration:
\[ \omega_f = \omega_0 + \alpha t \implies 0 = 40 + \alpha \cdot 10 \]
Solving for \( \alpha \):
\[ \alpha = -\frac{40}{10} = -4 \, \text{rad/s}^2 \]

Now, we can find the angular displacement during this phase:
\[ \theta_3 = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Substituting the values:
\[ \theta_3 = 40 \cdot 10 + \frac{1}{2} \cdot (-4) \cdot (10)^2 \]
\[ \theta_3 = 400 - 200 = 200 \, \text{radians} \]

### Step 5: Calculate the total angular displacement
Now, we can find the total angle rotated by the wheel by summing the angular displacements from all three phases:
\[ \theta_{\text{total}} = \theta_1 + \theta_2 + \theta_3 \]
Substituting the values:
\[ \theta_{\text{total}} = 200 + 400 + 200 = 800 \, \text{radians} \]

Thus, the total angle rotated by the wheel is \( \boxed{800} \, \text{radians} \).