The wheel of a toy car rotates about a fixed axes. It slows down from 400 rps to 200 rps in 2 s. Then, its angular retardation in `"rad s"^(-2)` is (rps = revolutions per second)

Options

A. `200pi`
B. `100pi`
C. `400pi`
D. None of these

Explanation

To find the angular retardation of the toy car's wheel, we can follow these steps:

### Step 1: Identify the initial and final angular velocities
- The initial angular velocity (\( \omega_i \)) is 400 revolutions per second (rps).
- The final angular velocity (\( \omega_f \)) is 200 revolutions per second (rps).

### Step 2: Convert revolutions per second to radians per second
- We know that 1 revolution = \( 2\pi \) radians.
- Therefore, we can convert the angular velocities:
- \( \omega_i = 400 \, \text{rps} \times 2\pi \, \text{rad/rev} = 800\pi \, \text{rad/s} \)
- \( \omega_f = 200 \, \text{rps} \times 2\pi \, \text{rad/rev} = 400\pi \, \text{rad/s} \)

### Step 3: Calculate the change in angular velocity
- The change in angular velocity (\( \Delta \omega \)) is given by:
\[ \Delta \omega = \omega_f - \omega_i = 400\pi \, \text{rad/s} - 800\pi \, \text{rad/s} = -400\pi \, \text{rad/s} \]

### Step 4: Determine the time interval
- The time interval (\( \Delta t \)) is given as 2 seconds.

### Step 5: Calculate the angular retardation
- Angular retardation (\( \alpha \)) can be calculated using the formula:
\[ \alpha = \frac{\Delta \omega}{\Delta t} \]
- Substituting the values:
\[ \alpha = \frac{-400\pi \, \text{rad/s}}{2 \, \text{s}} = -200\pi \, \text{rad/s}^2 \]

### Step 6: Interpret the result
- Since we are looking for the magnitude of angular retardation, we take the absolute value:
\[ \text{Angular retardation} = 200\pi \, \text{rad/s}^2 \]

### Final Answer
The angular retardation of the wheel is \( 200\pi \, \text{rad/s}^2 \).

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