A flywheel rotates with a uniform angular acceleration. Its angular velocity increases from `20pi rad//s` to `40pi rad //s` in 10 seconds. How many rotations did it make in this period ?

Options

A. 80
B. 100
C. 120
D. 150

Explanation

To solve the problem step by step, we will use the equations of motion for rotational dynamics.

### Step 1: Identify the given values
- Initial angular velocity, \( \omega_i = 20\pi \) rad/s
- Final angular velocity, \( \omega_f = 40\pi \) rad/s
- Time, \( \Delta t = 10 \) s

### Step 2: Calculate the angular acceleration (\( \alpha \))
Using the formula for angular acceleration:
\[ \omega_f = \omega_i + \alpha \Delta t \]
We can rearrange this to find \( \alpha \):
\[ \alpha = \frac{\omega_f - \omega_i}{\Delta t} \]
Substituting the values:
\[ \alpha = \frac{40\pi - 20\pi}{10} = \frac{20\pi}{10} = 2\pi \text{ rad/s}^2 \]

### Step 3: Calculate the angular displacement (\( \Delta \theta \))
We can use the following equation for angular displacement:
\[ \omega_f^2 = \omega_i^2 + 2\alpha \Delta \theta \]
Rearranging gives:
\[ \Delta \theta = \frac{\omega_f^2 - \omega_i^2}{2\alpha} \]
Substituting the known values:
\[ \Delta \theta = \frac{(40\pi)^2 - (20\pi)^2}{2 \cdot 2\pi} \]
Calculating the squares:
\[ \Delta \theta = \frac{1600\pi^2 - 400\pi^2}{4\pi} = \frac{1200\pi^2}{4\pi} = 300\pi \text{ radians} \]

### Step 4: Calculate the number of rotations
To find the number of rotations, we divide the total angular displacement by the angular displacement in one complete rotation (which is \( 2\pi \) radians):
\[ \text{Number of rotations} = \frac{\Delta \theta}{2\pi} \]
Substituting the value of \( \Delta \theta \):
\[ \text{Number of rotations} = \frac{300\pi}{2\pi} = 150 \]

### Final Answer
The flywheel made **150 rotations** during the 10 seconds.

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