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 initial angular velocity (ω₀) in revolutions per minute (rpm) needs to be converted to radians per second (rad/s).

Given:
- Initial speed = 900 rpm

Using the conversion factor:
\[ \text{Angular velocity in rad/s} = \text{RPM} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} \]
\[ \omega_0 = 900 \times \frac{2\pi}{60} \]
\[ \omega_0 = 900 \times \frac{\pi}{30} = 30\pi \text{ rad/s} \]

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

### Step 3: Determine Time Taken to Come to Rest
The time taken to come to rest (t) is given as:
\[ t = 1 \text{ minute} = 60 \text{ seconds} \]

### Step 4: Use the Angular Motion Equation
We can use the equation of motion for angular motion:
\[ \omega = \omega_0 + \alpha t \]
Where:
- ω is the final angular velocity
- ω₀ is the initial angular velocity
- α is the angular retardation (which we need to find)
- t is the time

Substituting the known values:
\[ 0 = 30\pi + \alpha \times 60 \]

### Step 5: Solve for Angular Retardation
Rearranging the equation to solve for α:
\[ \alpha \times 60 = -30\pi \]
\[ \alpha = \frac{-30\pi}{60} = -\frac{\pi}{2} \text{ rad/s}^2 \]

### Step 6: Conclusion
The angular retardation (α) 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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