The angular speed of a motor wheel is increased from 120 rpm to 3120 rpm in 16 seconds. The angular acceleration of the motor wheel is

Options

A. `2 pi rad s^(-2)`
B. `4pi rad s^(-2)`
C. `6pi rad s^(-2)`
D. `8pi rad s^(-2)`

Explanation

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

### Step 1: Convert the initial and final angular speeds from RPM to radians per second.
1. **Initial angular speed (ω₁)**: 1200 revolutions per minute (rpm).
- Convert to revolutions per second:
\[ \omega_1 = \frac{1200 \text{ revolutions}}{60 \text{ seconds}} = 20 \text{ revolutions/second} \]
- Convert revolutions per second to radians per second:
\[ \omega_1 = 20 \times 2\pi = 40\pi \text{ radians/second} \]

2. **Final angular speed (ω₂)**: 3120 revolutions per minute (rpm).
- Convert to revolutions per second:
\[ \omega_2 = \frac{3120 \text{ revolutions}}{60 \text{ seconds}} = 52 \text{ revolutions/second} \]
- Convert revolutions per second to radians per second:
\[ \omega_2 = 52 \times 2\pi = 104\pi \text{ radians/second} \]

### Step 2: Use the formula for angular acceleration (α).
The formula relating angular acceleration (α), initial angular speed (ω₁), final angular speed (ω₂), and time (t) is:
\[ \omega_2 = \omega_1 + \alpha t \]
Rearranging this gives:
\[ \alpha = \frac{\omega_2 - \omega_1}{t} \]

### Step 3: Substitute the values into the equation.
- We have:
- ω₂ = 104π radians/second
- ω₁ = 40π radians/second
- t = 16 seconds

Substituting these values into the equation:
\[ \alpha = \frac{104\pi - 40\pi}{16} \]
\[ \alpha = \frac{64\pi}{16} \]
\[ \alpha = 4\pi \text{ radians/second}^2 \]

### Final Answer:
The angular acceleration of the motor wheel is \( 4\pi \) radians/second².

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