Problem 3 - the frequency at which wheels turn
One type of locomotive used to pull trains has wheels 1250 mm in diameter and its maximum speed is 200 km/h. At what frequency do the wheels turn when the locomotive is traveling at maximum speed?
Given:
d = 1250 mm – diameter of the wheel of the locomotive,
v = 200 km/h – speed of locomotive.
We are looking for:
f = ? – frequency at which the wheels turn
Solution.
We must realize that one full turn of the wheel moves the locomotive by a distance equal to the circumference C of the wheel.
C = 2πr, with radius r = d/2 .
The speed is defined as the ratio of distance traveled to the time required to travel this distance. I case of the turning wheel we have:
Distance traveled equal C, the time for traveling this distance is equal to period T defined in “Circular motion” as the time required to make a full rotation.
Therefore the speed of the locomotive is
v = 2πr(1/T) (1)
as
1/T = f (inverse of period is a frequency) and r = d/2,
v = 2π(d/2)f (2)
This is the speed of the locomotive expressed trough the diameter of the wheel and frequency at which it turns while the locomotive is traveling.
In Eq.2 only the frequency f is unknown and this is what we are looking for. So
f = v/(πd). (3)
Now we rearrange this equation, substitute numbers given in the problem and change millimeters to meters, kilometers to meters, and hours to seconds.
The answer is: the wheels turn at a frequency of 14.14 revolution per second when the locomotive is traveling at 200km/h