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 made by itin next two seconds is

Options

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

Answer

To solve the problem step by step, we will follow the principles of rotational motion and the equations of motion for constant angular acceleration.

### Step 1: Understanding the Problem
We have a wheel that starts from rest (initial angular velocity, \( \omega_i = 0 \)) and rotates with a constant angular acceleration (\( \alpha \)). It completes 50 rotations in the first 2 seconds.

### Step 2: Convert Rotations to Radians
Since the equations of motion are typically in radians, we need to convert the number of rotations into radians:
\[ \theta = 50 \text{ rotations} \times 2\pi \text{ radians/rotation} = 100\pi \text{ radians} \]

### Step 3: Use the Angular Displacement Equation
We will use the second equation of motion for angular displacement:
\[ \theta = \omega_i t + \frac{1}{2} \alpha t^2 \]
Substituting the known values:
\[ 100\pi = 0 \cdot 2 + \frac{1}{2} \alpha (2^2) \]
This simplifies to:
\[ 100\pi = 2\alpha \]
From this, we can solve for \( \alpha \):
\[ \alpha = \frac{100\pi}{2} = 50\pi \text{ radians/second}^2 \]

### Step 4: Find the Final Angular Velocity After 2 Seconds
Using the first equation of motion for angular velocity:
\[ \omega_f = \omega_i + \alpha t \]
Substituting the known values:
\[ \omega_f = 0 + (50\pi)(2) = 100\pi \text{ radians/second} \]

### Step 5: Calculate the Number of Rotations in the Next 2 Seconds
Now, we need to find out how many rotations the wheel makes in the next 2 seconds (from \( t = 2 \) seconds to \( t = 4 \) seconds). The initial angular velocity for this interval is \( \omega_i = 100\pi \) radians/second.

Using the same angular displacement equation:
\[ \theta = \omega_i t + \frac{1}{2} \alpha t^2 \]
Here, \( t = 2 \) seconds, \( \alpha = 50\pi \):
\[ \theta = (100\pi)(2) + \frac{1}{2}(50\pi)(2^2) \]
Calculating this gives:
\[ \theta = 200\pi + \frac{1}{2}(50\pi)(4) = 200\pi + 100\pi = 300\pi \text{ radians} \]

### Step 6: Convert Radians Back to Rotations
Now, convert the angular displacement back to rotations:
\[ \text{Number of rotations} = \frac{300\pi}{2\pi} = 150 \text{ rotations} \]

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

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