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 it down into three parts corresponding to the three phases of motion described in the question.

### Step 1: Calculate the angle rotated during the acceleration phase

1. **Identify the initial conditions**:
- Initial angular velocity (\( \omega_0 \)) = 0 rad/s (starting from rest)
- Angular acceleration (\( \alpha \)) = 4 rad/s²
- Time (\( t_1 \)) = 10 s

2. **Calculate the final angular velocity after 10 seconds**:
\[ \omega = \omega_0 + \alpha t_1 = 0 + 4 \times 10 = 40 \text{ rad/s} \]

3. **Calculate the angle rotated during this phase (\( \theta_1 \))** using the formula:
\[ \theta_1 = \omega_0 t_1 + \frac{1}{2} \alpha t_1^2 \]
Substituting the values:
\[ \theta_1 = 0 \times 10 + \frac{1}{2} \times 4 \times (10)^2 = 0 + 200 = 200 \text{ radians} \]

### Step 2: Calculate the angle rotated during the uniform motion phase

1. **Identify the conditions for this phase**:
- Angular velocity (\( \omega \)) = 40 rad/s (constant)
- Time (\( t_2 \)) = 10 s

2. **Calculate the angle rotated during this phase (\( \theta_2 \))**:
\[ \theta_2 = \omega \times t_2 = 40 \times 10 = 400 \text{ radians} \]

### Step 3: Calculate the angle rotated during the deceleration phase

1. **Identify the conditions for this phase**:
- Initial angular velocity (\( \omega_0 \)) = 40 rad/s
- Final angular velocity (\( \omega_f \)) = 0 rad/s
- Time (\( t_3 \)) = 10 s

2. **Calculate the angular acceleration (\( \alpha \))**:
Using the formula:
\[ \omega_f = \omega_0 + \alpha t_3 \]
Substituting the values:
\[ 0 = 40 + \alpha \times 10 \implies \alpha = -4 \text{ rad/s}^2 \]

3. **Calculate the angle rotated during this phase (\( \theta_3 \))** using the formula:
\[ \omega_f^2 = \omega_0^2 + 2\alpha\theta_3 \]
Rearranging gives:
\[ 0 = 40^2 + 2(-4)\theta_3 \implies 1600 = 8\theta_3 \implies \theta_3 = \frac{1600}{8} = 200 \text{ radians} \]

### Step 4: Calculate the total angle rotated

1. **Sum the angles from all three phases**:
\[ \text{Total angle} = \theta_1 + \theta_2 + \theta_3 = 200 + 400 + 200 = 800 \text{ radians} \]

### Final Answer:
The total angle rotated by the wheel is **800 radians**.

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