An auto race takes place on a circular track. A car completes one lap in a time of 26.4 s, with an average tangential speed of 42.9 m/s. Find (a) the average angular speed and (b) the radius of radius of the track.
To find the average angular speed, we use the formula:
Angular speed = Tangential speed / Radius
Let's assume the radius of the track is r. Then, the average angular speed can be expressed as:
Angular speed = 42.9 m/s / r
We are given that the car completes one lap in 26.4 s, which means it travels a distance equal to the circumference of the track. The circumference of a circle is given by the formula:
Circumference = 2 * π * radius
Since the car completes one lap, the distance traveled is equal to the circumference, so:
Circumference = 2 * π * r = 42.9 m/s * 26.4 s
Solving for r in the above equation:
2 * π * r = 42.9 m/s * 26.4 s
2 * π * r = 1131.36 m
r = 1131.36 m / (2 * π)
Thus, the radius of the track is r = 179.907 m.
Now, substituting this value of r into the formula for angular speed:
Angular speed = 42.9 m/s / r
Angular speed = 42.9 m/s / 179.907 m
Simplifying this expression:
Angular speed ≈ 0.238 rad/s
Therefore, the average angular speed is approximately 0.238 rad/s and the radius of the track is approximately 179.907 m.