A wheel is rotating at 900 rpm about its axis. When the power is cut off, it comes to rest in 1 min. The angular retardation (in rad `s^(-2)`) is

Options

A. `(pi)/(2)`
B. `(pi)/(4)`
C. `(pi)/(6)`
D. `(pi)/(8)`

Explanation

To find the angular retardation of the wheel, we can follow these steps:

### Step 1: Convert RPM to Radians per Second
The wheel is rotating at 900 revolutions per minute (RPM). To convert this to radians per second, we can use the following conversion factor:

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

Substituting the values:

\[ \omega = 900 \times \frac{2\pi}{60} \]

Calculating this gives:

\[ \omega = 900 \times \frac{2\pi}{60} = 900 \times \frac{\pi}{30} = 30\pi \text{ rad/s} \]

### Step 2: Determine the Time for Deceleration
The wheel comes to rest in 1 minute. We need to convert this time into seconds:

\[ t = 1 \text{ minute} = 60 \text{ seconds} \]

### Step 3: Use the Equation of Motion for Angular Motion
We can use the equation of motion for angular motion, which relates initial angular velocity, final angular velocity, angular acceleration (retardation in this case), and time:

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

Where:
- \(\omega_f\) = final angular velocity = 0 (since it comes to rest)
- \(\omega_i\) = initial angular velocity = \(30\pi \text{ rad/s}\)
- \(\alpha\) = angular retardation (which we need to find)
- \(t\) = time = 60 seconds

Substituting the known values into the equation:

\[ 0 = 30\pi + \alpha \times 60 \]

### Step 4: Solve for Angular Retardation
Rearranging the equation to solve for \(\alpha\):

\[ \alpha \times 60 = -30\pi \]

\[ \alpha = \frac{-30\pi}{60} = -\frac{\pi}{2} \text{ rad/s}^2 \]

### Final Answer
The angular retardation is:

\[ \alpha = -\frac{\pi}{2} \text{ rad/s}^2 \]