A wheel having a diameter of 3 m starts from rest and accelerates uniformly to an angular velocity of 210 r.p.m. in 5 seconds. Angular acceleration of the wheel is

Options

A. `4.4rads^-2`
B. `3.3rads^-2`
C. `2.2rads^-2`
D. `1.1rads^-2`

Answer

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

### Step 1: Convert the final angular velocity from RPM to radians per second
The final angular velocity is given as 210 revolutions per minute (RPM). We need to convert this to radians per second.

1. **Convert RPM to revolutions per second (RPS)**:
\[ \text{Final angular velocity in RPS} = \frac{210 \text{ RPM}}{60} = 3.5 \text{ RPS} \]

2. **Convert revolutions per second to radians per second**:
Since one revolution is \(2\pi\) radians,
\[ \omega_f = 3.5 \text{ RPS} \times 2\pi \text{ radians/revolution} = 7\pi \text{ radians/second} \]

### Step 2: Identify the initial angular velocity
The wheel starts from rest, so the initial angular velocity (\(\omega_i\)) is:
\[ \omega_i = 0 \text{ radians/second} \]

### Step 3: Calculate the angular acceleration
Angular acceleration (\(\alpha\)) can be calculated using the formula:
\[ \alpha = \frac{\Delta \omega}{\Delta t} \]
where \(\Delta \omega = \omega_f - \omega_i\) and \(\Delta t\) is the time taken.

1. **Calculate the change in angular velocity**:
\[ \Delta \omega = \omega_f - \omega_i = 7\pi - 0 = 7\pi \text{ radians/second} \]

2. **Use the time taken**:
Given that the time taken (\(\Delta t\)) is 5 seconds.

3. **Calculate angular acceleration**:
\[ \alpha = \frac{7\pi}{5} \text{ radians/second}^2 \]

### Step 4: Calculate the numerical value of angular acceleration
Using \(\pi \approx 3.14\):
\[ \alpha \approx \frac{7 \times 3.14}{5} \approx \frac{21.98}{5} \approx 4.396 \text{ radians/second}^2 \]

### Final Answer
The angular acceleration of the wheel is approximately:
\[ \alpha \approx 4.4 \text{ radians/second}^2 \]

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