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 of how many rotations a flywheel makes while it rotates with uniform angular acceleration, we can follow these steps:

### Step 1: Calculate Angular Acceleration (α)
We know the initial angular velocity (ω_i) and the final angular velocity (ω_f) along with the time (t) during which this change occurs.

Given:
- ω_i = 20π rad/s
- ω_f = 40π rad/s
- t = 10 s

Using the formula for angular acceleration:
\[ \alpha = \frac{\omega_f - \omega_i}{t} \]
Substituting the values:
\[ \alpha = \frac{40\pi - 20\pi}{10} = \frac{20\pi}{10} = 2\pi \text{ rad/s}^2 \]

### Step 2: Calculate Angular Displacement (Δθ)
Now, we will use the angular displacement formula that relates initial and final angular velocities, angular acceleration, and displacement:
\[ \omega_f^2 = \omega_i^2 + 2\alpha \Delta\theta \]
Rearranging gives us:
\[ \Delta\theta = \frac{\omega_f^2 - \omega_i^2}{2\alpha} \]
Substituting the known values:
\[ \Delta\theta = \frac{(40\pi)^2 - (20\pi)^2}{2(2\pi)} \]
Calculating the squares:
\[ \Delta\theta = \frac{1600\pi^2 - 400\pi^2}{4\pi} = \frac{1200\pi^2}{4\pi} \]
Simplifying:
\[ \Delta\theta = 300\pi \text{ radians} \]

### Step 3: Convert Angular Displacement to Rotations
To find the number of rotations (N), we convert the angular displacement from radians to rotations. Since one complete rotation is \(2\pi\) radians:
\[ N = \frac{\Delta\theta}{2\pi} \]
Substituting the value of Δθ:
\[ N = \frac{300\pi}{2\pi} = 150 \]

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

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