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?
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.