A stationary wheel starts rotating about its own axis at constant angular acceleration. If the wheel completes `50` rotations in first `2` seconds, then the number of rotations mades by it in next two seconds is

Options

A. `75`
B. `100`
C. `125`
D. `150`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
The wheel starts from rest and rotates with a constant angular acceleration. We know that it completes 50 rotations in the first 2 seconds. We need to find out how many rotations it makes in the next 2 seconds.

### Step 2: Convert rotations to radians
The total angular displacement (θ) in radians for 50 rotations can be calculated as:
\[ \theta = \text{number of rotations} \times 2\pi = 50 \times 2\pi = 100\pi \text{ radians} \]

### Step 3: Use the angular displacement formula
For an object starting from rest with constant angular acceleration, the angular displacement can be expressed as:
\[ \theta = \omega_i t + \frac{1}{2} \alpha t^2 \]
Since the initial angular velocity (ω_i) is 0 (the wheel is stationary), the equation simplifies to:
\[ \theta = \frac{1}{2} \alpha t^2 \]
Substituting θ = 100π and t = 2 seconds:
\[ 100\pi = \frac{1}{2} \alpha (2^2) \]
\[ 100\pi = \frac{1}{2} \alpha \cdot 4 \]
\[ 100\pi = 2\alpha \]
From this, we can solve for α:
\[ \alpha = \frac{100\pi}{2} = 50\pi \text{ radians/second}^2 \]

### Step 4: Find angular displacement in 4 seconds
Now, we need to find the total angular displacement (θ) after 4 seconds using the same formula:
\[ \theta = \frac{1}{2} \alpha t^2 \]
Substituting α = 50π and t = 4 seconds:
\[ \theta = \frac{1}{2} (50\pi) (4^2) \]
\[ \theta = \frac{1}{2} (50\pi) (16) = 400\pi \text{ radians} \]

### Step 5: Calculate angular displacement for the last 2 seconds
To find the angular displacement during the last 2 seconds (from t = 2s to t = 4s), we subtract the angular displacement in the first 2 seconds from the total angular displacement in 4 seconds:
\[ \theta_{\text{last 2s}} = \theta_{\text{4s}} - \theta_{\text{2s}} = 400\pi - 100\pi = 300\pi \text{ radians} \]

### Step 6: Convert angular displacement to rotations
To find the number of rotations made in the last 2 seconds, we convert radians back to rotations:
\[ \text{Number of rotations} = \frac{\theta_{\text{last 2s}}}{2\pi} = \frac{300\pi}{2\pi} = 150 \]

### Final Answer
The number of rotations made by the wheel in the next 2 seconds is **150 rotations**.

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