A particle is moving on a circular path of radius 0.3 m and rotaing at 1200 rpm. The centripetal acceleration of the particle

Options

A. `4732.60 ms^(-2)`
B. `47.3260 ms^(-2)`
C. `4.73260ms^(-2)`
D. `473.260 ms^(-2)`

Explanation

To find the centripetal acceleration of a particle moving in a circular path, we can follow these steps:

### Step 1: Identify the given values
- Radius of the circular path, \( r = 0.3 \, \text{m} \)
- Rotational speed, \( n = 1200 \, \text{rpm} \)

### Step 2: Convert rotational speed from rpm to radians per second
To convert revolutions per minute (rpm) to radians per second, we use the following conversion:
\[ \omega = n \times \frac{2\pi \, \text{radians}}{1 \, \text{revolution}} \times \frac{1 \, \text{minute}}{60 \, \text{seconds}} \]
Substituting the values:
\[ \omega = 1200 \times \frac{2\pi}{60} \]
Calculating this gives:
\[ \omega = 1200 \times \frac{2\pi}{60} = 1200 \times \frac{\pi}{30} = 40\pi \, \text{radians/second} \]

### Step 3: Use the formula for centripetal acceleration
The formula for centripetal acceleration \( a_c \) is given by:
\[ a_c = r \omega^2 \]
Substituting the values of \( r \) and \( \omega \):
\[ a_c = 0.3 \times (40\pi)^2 \]

### Step 4: Calculate \( \omega^2 \)
First, calculate \( (40\pi)^2 \):
\[ (40\pi)^2 = 1600\pi^2 \]

### Step 5: Substitute back to find \( a_c \)
Now substituting back:
\[ a_c = 0.3 \times 1600\pi^2 \]
\[ a_c = 480\pi^2 \]

### Step 6: Calculate the numerical value
Using \( \pi \approx 3.14 \):
\[ a_c \approx 480 \times (3.14)^2 \approx 480 \times 9.86 \approx 4728.8 \, \text{m/s}^2 \]

### Final Answer
Thus, the centripetal acceleration of the particle is approximately:
\[ a_c \approx 4728.8 \, \text{m/s}^2 \]

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