A wheel is rotating at 900 rpm about its axis. When 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 the initial angular velocity from RPM to radians per second
The initial angular velocity (ω₀) is given as 900 RPM. To convert this to radians per second, we use the conversion factor:
\[ \omega_0 = 900 \, \text{RPM} \times \frac{2\pi \, \text{radians}}{1 \, \text{revolution}} \times \frac{1 \, \text{minute}}{60 \, \text{seconds}} \]
Calculating this gives:
\[ \omega_0 = 900 \times \frac{2\pi}{60} = 900 \times \frac{\pi}{30} = 30\pi \, \text{rad/s} \]

### Step 2: Identify the final angular velocity
When the power is cut off, the wheel comes to rest, so the final angular velocity (ω) is:
\[ \omega = 0 \, \text{rad/s} \]

### Step 3: Calculate the time taken to come to rest
The time taken (t) for the wheel to come to rest is given as 1 minute, which we convert to seconds:
\[ t = 1 \, \text{minute} = 60 \, \text{seconds} \]

### Step 4: Use the angular motion equation to find angular retardation
We use the equation of motion for angular velocity:
\[ \omega = \omega_0 + \alpha t \]
Since the wheel is coming to rest, we have:
\[ 0 = 30\pi + \alpha \cdot 60 \]
Rearranging this gives:
\[ \alpha \cdot 60 = -30\pi \]
\[ \alpha = -\frac{30\pi}{60} = -\frac{\pi}{2} \, \text{rad/s}^2 \]
The negative sign indicates that this is a retardation.

### Step 5: Conclusion
The angular retardation (magnitude) is:
\[ \alpha = \frac{\pi}{2} \, \text{rad/s}^2 \]

### Final Answer
The angular retardation is \(\frac{\pi}{2} \, \text{rad/s}^2\).

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