A wheel rotating at `12 rev s^(-1)` is brought to rest in 6 s . The average angular deceleration in rad `s^(-2)` of the wheel during this process is

Options

A. `4pi`
B. 4
C. 72
D. `pi`

Answer

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

### Step 1: Convert the initial angular velocity from revolutions per second to radians per second.
The initial angular velocity \( \omega_i \) is given as \( 12 \, \text{rev/s} \).

To convert revolutions to radians, we use the fact that one revolution is equal to \( 2\pi \) radians.

\[ \omega_i = 12 \, \text{rev/s} \times 2\pi \, \text{rad/rev} = 24\pi \, \text{rad/s} \]

### Step 2: Identify the final angular velocity.
The wheel is brought to rest, so the final angular velocity \( \omega_f \) is:

\[ \omega_f = 0 \, \text{rad/s} \]

### Step 3: Use the formula for angular acceleration.
The formula relating initial angular velocity, final angular velocity, angular acceleration \( \alpha \), and time \( t \) is:

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

### Step 4: Rearrange the formula to solve for angular deceleration \( \alpha \).
Substituting the known values into the equation:

\[ 0 = 24\pi + \alpha \cdot 6 \]

Rearranging gives:

\[ \alpha \cdot 6 = -24\pi \]

\[ \alpha = \frac{-24\pi}{6} = -4\pi \, \text{rad/s}^2 \]

### Step 5: State the average angular deceleration.
The average angular deceleration is:

\[ \alpha = -4\pi \, \text{rad/s}^2 \]

### Final Answer:
The average angular deceleration of the wheel is \( 4\pi \, \text{rad/s}^2 \).

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