Four wheels of different radii are connected with pulleys as shown below. Wheels #2 and #1 have a common axle. If the angular velocity of Wheel #2# is #2# revolutions per second, what is the angular velocity of Wheel #4#?
1 Answer
Wheel 4 turns at 20 rev/s.
Explanation:
It might help you visualize what I will describe if you assume the 3 axles are much farther apart. That means the belts have to be longer, but this is just a mind variation.
If you put a mark on the belt right where it comes off wheel 1, and you rotate wheel 1 thru 1 revolution, the mark on the belt will travel a distance equal to the circumference of wheel 1. (Assume the axles are far enough apart that the mark has not reached wheel 2.) Because the length of the radii of wheels 1 and 2 are 5 and 1 unit respectively, the circumference of wheel 1 is 5X as much as that of wheel 2. That means that wheel 2 will have to rotate thru 5 revolutions in response to the one revolution of wheel 1.
Wheel 3 will also rotate thru 5 revolutions in response to the one revolution of wheel 1. Note that each revolution of wheel 3 causes wheel 4 to rotate thru 2 revolutions because of the value of their radii. Therefore, using the logic of the 2nd paragraph above, wheel 4 will rotate thru 10 revolutions in response to the one revolution of wheel 1.
Therefore, with wheel 1 rotating at 2 revolutions/second, wheel 4 rotates at 20 revolutions/second.
