An electric fan has blades of length 30 cm neasured from the axis of rotation. If the fan rotating at 120 rev/min. the acceleration of a point on the tip of the blade is
(a) `1600ms^(-2)` (b) `47.4ms^(-2)` (c) `23.7ms^(-2)` (d) `50.55ms^(-2)`

Options

A. `1600ms^(-2)`
B. `47.4ms^(-2)`
C. `23.7ms^(-2)`
D. `50.55ms^(-2)`

Answer

To solve the problem of finding the acceleration of a point on the tip of the blade of an electric fan, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Length of the blade (radius, \( r \)) = 30 cm = 0.3 m
- Rotational speed = 120 revolutions per minute (rev/min)

2. **Convert Revolutions per Minute to Revolutions per Second:**
\[ \text{Frequency} (f) = \frac{120 \text{ rev/min}}{60 \text{ s/min}} = 2 \text{ rev/s} \]

3. **Calculate Angular Velocity (\( \omega \)):**
The angular velocity in radians per second can be calculated using the formula:
\[ \omega = 2\pi f \]
Substituting the value of \( f \):
\[ \omega = 2\pi \times 2 = 4\pi \text{ rad/s} \]

4. **Calculate Centripetal Acceleration (\( a_c \)):**
The formula for centripetal acceleration is given by:
\[ a_c = \omega^2 r \]
Substituting the values of \( \omega \) and \( r \):
\[ a_c = (4\pi)^2 \times 0.3 \]

5. **Calculate \( (4\pi)^2 \):**
\[ (4\pi)^2 = 16\pi^2 \]
Using \( \pi \approx \frac{22}{7} \):
\[ \pi^2 \approx \left(\frac{22}{7}\right)^2 = \frac{484}{49} \]
Thus,
\[ 16\pi^2 \approx 16 \times \frac{484}{49} = \frac{7744}{49} \]

6. **Calculate the Centripetal Acceleration:**
Now substituting back into the centripetal acceleration formula:
\[ a_c = \frac{7744}{49} \times 0.3 = \frac{2323.2}{49} \approx 47.4 \text{ m/s}^2 \]

7. **Final Result:**
The acceleration of a point on the tip of the blade is approximately \( 47.4 \text{ m/s}^2 \).

### Conclusion:
The correct answer is (b) \( 47.4 \text{ m/s}^2 \).

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