An auto race takes place on a circular track. A car completes one lap in a time of 22.8 s, with an average tangential speed of 37.3 m/s. Find (a) the average angular speed and (b) the radius of the track.
To find the average angular speed and the radius of the track, we need to use the relationship between linear speed and angular speed.
(a) Average Angular Speed:
The average angular speed (ω) is calculated by dividing the total angle covered by the car in one lap by the time taken to complete one lap.
We can calculate the total angle covered using the formula:
Angle = Distance / Radius
Given that the distance covered in one lap is equal to the circumference (2πr) of a circle, we can rewrite the formula as:
Angle = 2πr / r = 2π radians
Now, we can calculate the average angular speed:
ω = Angle / Time
Substituting the values, we get:
ω = 2π radians / 22.8 s = (π / 11.4) radians/s
Therefore, the average angular speed is (π / 11.4) radians/s.
(b) Radius of the track:
The linear speed (v) of a point on the outer edge of a rotating object is related to the angular speed (ω) and the radius (r) by the formula:
v = ωr
Rearranging the formula, we can solve for the radius:
r = v / ω
Substituting the given values:
r = 37.3 m/s / (π / 11.4 rad/s) = (37.3 * 11.4) / π = 119.76 / π meters
Therefore, the radius of the track is approximately 38.08 meters.
ah come on
2 pi rad/22.8 s
so what is the circumference?
C = 2 pi R = 22.8s * 37.3m/s
= 850 meters around the track
so
R = 850/(2 pi)