A racing completes three rounds on a circular racing track in one minute . If the car has a uniform centripetal acceleration of `pi^(2) m//s` then radius of the track will be

Options

A. 100m
B. 10m
C. 1000 m
D. 125m

Explanation

To find the radius of the circular racing track, we can follow these steps:

### Step 1: Determine the angular velocity (ω)
The racer completes 3 rounds in 1 minute. We need to convert this into radians per second.

- **Number of revolutions per minute (rpm)**: 3 rpm
- **Convert revolutions to radians**:
- 1 revolution = \(2\pi\) radians
- Therefore, 3 revolutions = \(3 \times 2\pi = 6\pi\) radians

- **Convert minutes to seconds**:
- 1 minute = 60 seconds

- **Calculate ω in radians per second**:
\[ \omega = \frac{6\pi \text{ radians}}{60 \text{ seconds}} = \frac{\pi}{10} \text{ radians/second} \]

### Step 2: Use the formula for centripetal acceleration
The formula for centripetal acceleration (\(a_c\)) is given by:
\[ a_c = \omega^2 \cdot r \]
where:
- \(a_c\) = centripetal acceleration
- \(r\) = radius of the circular path

We know:
- \(a_c = \pi^2 \text{ m/s}^2\)
- \(\omega = \frac{\pi}{10} \text{ radians/second}\)

### Step 3: Substitute ω into the centripetal acceleration formula
Substituting the value of ω into the formula:
\[ \pi^2 = \left(\frac{\pi}{10}\right)^2 \cdot r \]

### Step 4: Simplify the equation
Calculating \(\left(\frac{\pi}{10}\right)^2\):
\[ \left(\frac{\pi}{10}\right)^2 = \frac{\pi^2}{100} \]
Thus, the equation becomes:
\[ \pi^2 = \frac{\pi^2}{100} \cdot r \]

### Step 5: Solve for r
To isolate \(r\), we can multiply both sides by 100:
\[ 100 \cdot \pi^2 = \pi^2 \cdot r \]
Now, divide both sides by \(\pi^2\):
\[ r = 100 \text{ meters} \]

### Conclusion
The radius of the track is \(100\) meters.

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