Centripetal acceleration of a cyclist completing 7 rounds in a minute along a circular track of radius 5 m with a constant speed ,is

Options

A. `2.7 m//s^(2)`
B. ` 4 m//s^(2)`
C. ` 3.78 m//s^(2)`
D. ` 6 m//s^(2)`

Answer

To find the centripetal acceleration of a cyclist completing 7 rounds in a minute along a circular track of radius 5 m with a constant speed, we can follow these steps:

### Step 1: Calculate the distance covered in one revolution
The distance covered in one revolution (circumference of the circle) is given by the formula:
\[ \text{Circumference} = 2\pi r \]
where \( r \) is the radius of the circular track.

Given:
- Radius \( r = 5 \, \text{m} \)

Calculating the circumference:
\[ \text{Circumference} = 2 \times \pi \times 5 = 10\pi \approx 31.42 \, \text{m} \]

### Step 2: Calculate the total distance covered in 7 revolutions
To find the total distance covered in 7 revolutions, we multiply the circumference by the number of revolutions:
\[ \text{Total distance} = 7 \times \text{Circumference} = 7 \times 31.42 \approx 220 \, \text{m} \]

### Step 3: Calculate the time taken for 7 revolutions
The time taken for 7 revolutions is given as 1 minute, which is equivalent to 60 seconds.

### Step 4: Calculate the speed of the cyclist
The speed \( V \) can be calculated using the formula:
\[ V = \frac{\text{Total distance}}{\text{Time}} \]
Substituting the values:
\[ V = \frac{220 \, \text{m}}{60 \, \text{s}} \approx 3.667 \, \text{m/s} \]

### Step 5: Calculate the centripetal acceleration
Centripetal acceleration \( a_c \) is given by the formula:
\[ a_c = \frac{V^2}{R} \]
where \( R \) is the radius of the circular path.

Substituting the values:
\[ a_c = \frac{(3.667)^2}{5} \approx \frac{13.49}{5} \approx 2.698 \, \text{m/s}^2 \]

### Step 6: Round the answer
Rounding the result gives us:
\[ a_c \approx 2.7 \, \text{m/s}^2 \]

### Final Answer
Thus, the centripetal acceleration of the cyclist is approximately \( 2.7 \, \text{m/s}^2 \).

---