A car travels at a constant speed around a circular track whose radius is 3.1 km. The car goes once around the track in 280 s. What is the magnitude of the centripetal acceleration of the car?

To find the magnitude of the centripetal acceleration of the car, we can use the formula:

Centripetal acceleration (a) = (v^2) / r

Where:
v = velocity of the car
r = radius of the circular track

We can find the velocity (v) by dividing the distance traveled around the track by the time taken.

Distance traveled around the track = circumference of the circle = 2 * π * r
Time taken = 280 s

Therefore, the velocity (v) = (2 * π * 3.1 km) / 280 s

Now we can substitute the values into the formula to find the centripetal acceleration:

a = [(2 * π * 3.1 km) / 280 s]^2 / 3.1 km

Simplifying this equation will give us the magnitude of the centripetal acceleration.

To find the magnitude of the centripetal acceleration of the car, we can use the formula:

\(a_c = \frac{v^2}{r}\)

Where:
- \(a_c\) is the centripetal acceleration
- \(v\) is the linear velocity
- \(r\) is the radius of the circular track

First, we need to find the linear velocity of the car. We know that linear velocity is given by the formula:

\(v = \frac{{2\pi r}}{{t}}\)

Where:
- \(v\) is the linear velocity
- \(r\) is the radius of the circular track
- \(t\) is the time taken to complete one full revolution

Plugging in the values, we get:

\(v = \frac{{2\pi \cdot 3.1}}{{280}}\)

Simplifying, we find:

\(v \approx 0.0697 \, \text{km/s}\)

Now, we can substitute this value of linear velocity into the formula for centripetal acceleration:

\(a_c = \frac{{(0.0697 \, \text{km/s})^2}}{{3.1 \, \text{km}}}\)

Simplifying further, we get:

\(a_c \approx 0.0016 \, \text{km/s}^2\)

Therefore, the magnitude of the centripetal acceleration of the car is approximately \(0.0016 \, \text{km/s}^2\).