The angular frequency of a fan increases uniformly from 30 rpm to 60 rpm in `pi` second. A dust particle is present at a distance of 20 cm from axis of rotation. The tangential acceleration of the particle is :

Options

A. `0.2 ms^(-2)`
B. `0.4 ms^(-2)`
C. `0.6 ms^(-2)`
D. `2m s^(-2)`

Explanation

To find the tangential acceleration of the dust particle, we can follow these steps:

### Step 1: Convert angular velocities from rpm to radians per second
The initial angular velocity (ω_initial) is 30 rpm and the final angular velocity (ω_final) is 60 rpm. We need to convert these values into radians per second using the conversion factor \( \frac{2\pi \text{ radians}}{60 \text{ seconds}} \).

\[ \omega_{\text{initial}} = 30 \, \text{rpm} \times \frac{2\pi \, \text{radians}}{60 \, \text{seconds}} = \frac{30 \times 2\pi}{60} = \pi \, \text{radians/second} \]

\[ \omega_{\text{final}} = 60 \, \text{rpm} \times \frac{2\pi \, \text{radians}}{60 \, \text{seconds}} = \frac{60 \times 2\pi}{60} = 2\pi \, \text{radians/second} \]

### Step 2: Calculate the change in angular velocity (Δω)
Now, we can find the change in angular velocity:

\[ \Delta \omega = \omega_{\text{final}} - \omega_{\text{initial}} = 2\pi - \pi = \pi \, \text{radians/second} \]

### Step 3: Calculate angular acceleration (α)
Angular acceleration (α) is defined as the change in angular velocity divided by the time taken. The time taken is given as \( \pi \) seconds.

\[ \alpha = \frac{\Delta \omega}{\Delta t} = \frac{\pi}{\pi} = 1 \, \text{radians/second}^2 \]

### Step 4: Determine the radius (r)
The radius (r) is given as 20 cm. We need to convert this to meters:

\[ r = 20 \, \text{cm} = 20 \times 10^{-2} \, \text{m} = 0.2 \, \text{m} \]

### Step 5: Calculate tangential acceleration (a_t)
Tangential acceleration (a_t) is given by the formula:

\[ a_t = \alpha \times r \]

Substituting the values we found:

\[ a_t = 1 \, \text{radians/second}^2 \times 0.2 \, \text{m} = 0.2 \, \text{m/s}^2 \]

### Final Answer
The tangential acceleration of the particle is \( 0.2 \, \text{m/s}^2 \).

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