A wheel starting from rest is uniformly accelerated at `2 red/s^(2)` for 20 seconds. It is allowed to rotate uniformly for the next 10 seconds and is finally brought to rest in next 20 seconds. The total angle rotated by the wheel (in radian) is :-

Options

A. 600
B. 1200
C. 1800
D. 300

Answer

To solve the problem, we will break it down into three segments based on the motion of the wheel:

1. **From rest with uniform acceleration for 20 seconds.**
2. **Rotating uniformly for the next 10 seconds.**
3. **Coming to rest in the final 20 seconds.**

### Step 1: Calculate the angle rotated during the acceleration phase (0 to 20 seconds)

- **Given:**
- Initial angular velocity, \( \omega_0 = 0 \) rad/s (starting from rest)
- Angular acceleration, \( \alpha = 2 \) rad/s²
- Time, \( t = 20 \) s

- **Final angular velocity after 20 seconds:**
\[ \omega_f = \omega_0 + \alpha t = 0 + (2 \times 20) = 40 \text{ rad/s} \]

- **Angle rotated during this phase using the equation:**
\[ \theta_1 = \omega_0 t + \frac{1}{2} \alpha t^2 \]
\[ \theta_1 = 0 \times 20 + \frac{1}{2} \times 2 \times (20)^2 = 0 + \frac{1}{2} \times 2 \times 400 = 400 \text{ radians} \]

### Step 2: Calculate the angle rotated during the uniform motion phase (20 to 30 seconds)

- **Given:**
- Angular velocity during this phase, \( \omega = 40 \) rad/s (constant)
- Time, \( t = 10 \) s

- **Angle rotated during this phase:**
\[ \theta_2 = \omega \times t = 40 \times 10 = 400 \text{ radians} \]

### Step 3: Calculate the angle rotated during the deceleration phase (30 to 50 seconds)

- **Given:**
- Initial angular velocity, \( \omega_0 = 40 \) rad/s
- Final angular velocity, \( \omega_f = 0 \) rad/s (comes to rest)
- Time, \( t = 20 \) s

- **Using the equation for angular deceleration:**
\[ \omega_f = \omega_0 - \alpha t \]
Rearranging gives:
\[ 0 = 40 - \alpha \times 20 \implies \alpha = \frac{40}{20} = 2 \text{ rad/s}^2 \]

- **Using the equation for angle rotated during deceleration:**
\[ \theta_3 = \omega_0 t - \frac{1}{2} \alpha t^2 \]
\[ \theta_3 = 40 \times 20 - \frac{1}{2} \times 2 \times (20)^2 = 800 - \frac{1}{2} \times 2 \times 400 = 800 - 400 = 400 \text{ radians} \]

### Step 4: Total angle rotated

- **Total angle rotated:**
\[ \theta_{total} = \theta_1 + \theta_2 + \theta_3 = 400 + 400 + 400 = 1200 \text{ radians} \]

### Final Answer:
The total angle rotated by the wheel is **1200 radians**.

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