An electric fan has blades of length 30 cm neasured from the axis of rotation. If the fanrotating at 120 rev/min. the acceleration of a point on the tip if the blade is

Options

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

Answer

To find the acceleration of a point on the tip of the blade of the electric fan, we need to calculate the centripetal acceleration. The formula for centripetal acceleration (a_c) is given by:

\[ a_c = \frac{v^2}{r} \]

where:
- \( v \) is the linear velocity of the tip of the blade,
- \( r \) is the radius (length of the blade).

### Step 1: Convert the length of the blade to meters
The length of the blades is given as 30 cm. We need to convert this to meters.

\[ r = 30 \, \text{cm} = 0.3 \, \text{m} \]

### Step 2: Convert the rotational speed from revolutions per minute (rev/min) to radians per second
The fan rotates at 120 rev/min. To convert this to radians per second, we use the conversion factor \( 2\pi \) radians per revolution and divide by 60 seconds per minute.

\[ \omega = 120 \, \text{rev/min} \times \frac{2\pi \, \text{radians}}{1 \, \text{rev}} \times \frac{1 \, \text{min}}{60 \, \text{s}} \]

Calculating this:

\[ \omega = 120 \times \frac{2\pi}{60} = 4\pi \, \text{radians/s} \]

### Step 3: Calculate the linear velocity (v) of the tip of the blade
The linear velocity can be calculated using the formula:

\[ v = r \cdot \omega \]

Substituting the values we have:

\[ v = 0.3 \, \text{m} \cdot 4\pi \, \text{radians/s} \]

Calculating this:

\[ v = 1.2\pi \, \text{m/s} \approx 3.77 \, \text{m/s} \]

### Step 4: Calculate the centripetal acceleration (a_c)
Now we can use the centripetal acceleration formula:

\[ a_c = \frac{v^2}{r} \]

Substituting the values we have:

\[ a_c = \frac{(1.2\pi)^2}{0.3} \]

Calculating this:

\[ a_c = \frac{1.44\pi^2}{0.3} \approx \frac{1.44 \times 9.87}{0.3} \approx \frac{14.23}{0.3} \approx 47.43 \, \text{m/s}^2 \]

### Final Answer
The acceleration of a point on the tip of the blade is approximately \( 47.43 \, \text{m/s}^2 \).

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