Question

Assume that a 400m track is a circle. A 68kg athlete runs the 400m race in 48.26 seconds with a
constant velocity. What is the angular velocity of the athlete in degrees per second? And what is the
centripetal acceleration?

Answer 1

First, we need to find the linear velocity of the athlete running around the track. The linear velocity can be calculated using the formula:

v = d/t

Where:
v = linear velocity
d = distance (400m)
t = time (48.26 seconds)

v = 400m / 48.26s
v = 8.29 m/s

Now, we need to find the circumference of the track in order to calculate the angular velocity using the formula:

ω = v/r

Where:
ω = angular velocity
v = linear velocity (8.29 m/s)
r = radius of the track

Since the track is a circle, the circumference is equal to 400m. Therefore, the radius of the track is:

r = circumference / (2π)
r = 400m / (2π)
r ≈ 63.66m

Now, we can calculate the angular velocity:

ω = 8.29 m/s / 63.66m
ω ≈ 0.13 rad/s

To convert the angular velocity from radians per second to degrees per second, we can use the conversion factor:

1 radian = 57.2958 degrees

Therefore, the angular velocity in degrees per second is:

ω = 0.13 rad/s * 57.2958 degrees/radian
ω ≈ 7.45 degrees/s

Next, we can calculate the centripetal acceleration using the formula:

a = v^2 / r

Where:
a = centripetal acceleration
v = linear velocity (8.29 m/s)
r = radius of the track (63.66m)

a = (8.29 m/s)^2 / 63.66m
a ≈ 1.08 m/s^2

Therefore, the angular velocity of the athlete is approximately 7.45 degrees per second, and the centripetal acceleration is approximately 1.08 m/s^2.