A flywheel is revolving at 150 revolutions per minutes .If deccelerates at a constant rate of `2 pi rad //s^(2)`, then time requred to stop it is

Options

A. `10 s`
B. `5 s`
C. `2.5 s`
D. `1.25 s`

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Convert revolutions per minute to radians per second
The flywheel is revolving at 150 revolutions per minute (rpm). To convert this to radians per second, we use the conversion factor that 1 revolution = 2π radians and there are 60 seconds in a minute.

\[ \text{Angular velocity} (\omega) = 150 \text{ revolutions/minute} \times \frac{2\pi \text{ radians}}{1 \text{ revolution}} \times \frac{1 \text{ minute}}{60 \text{ seconds}} \]

Calculating this gives:

\[ \omega = 150 \times \frac{2\pi}{60} = 5\pi \text{ radians/second} \]

### Step 2: Identify the angular deceleration
The problem states that the flywheel decelerates at a constant rate of \(2\pi \text{ radians/second}^2\). Since it is deceleration, we will take this as negative:

\[ \alpha = -2\pi \text{ radians/second}^2 \]

### Step 3: Use the angular motion equation
We need to find the time required to stop the flywheel. We can use the angular motion equation:

\[ \omega_f = \omega_0 + \alpha t \]

Where:
- \(\omega_f\) is the final angular velocity (which is 0 when the flywheel stops),
- \(\omega_0\) is the initial angular velocity (which we calculated as \(5\pi\)),
- \(\alpha\) is the angular deceleration (\(-2\pi\)),
- \(t\) is the time we want to find.

Substituting the known values into the equation:

\[ 0 = 5\pi - 2\pi t \]

### Step 4: Solve for time \(t\)
Rearranging the equation to solve for \(t\):

\[ 2\pi t = 5\pi \]

Dividing both sides by \(2\pi\):

\[ t = \frac{5\pi}{2\pi} = 2.5 \text{ seconds} \]

### Conclusion
The time required to stop the flywheel is **2.5 seconds**.

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