A ?exible drive belt runs over two ?ywheels whose axles are mountedon a rigid ba

Question

A ?exible drive belt runs over two ?ywheels whose axles are mountedon a rigid base. In the reference frame of the base, the horizontalportions of the belt have a speed v and therefore are subject tolength contraction, which tightens the belt around the ?ywheels.However, in a reference frame moving to the right with the upperportion of the belt, the base is subject to length contraction,which ought to loosen the belt around the ?ywheels. Resolve this“paradox" with by a qualitative argument. Hint: consider thelower portion of the belt as seen in the reference frame of theupper portion.

Explanation / Answer

Let's pose this question in a different way Suppose your are standing on the base to which the pulleys arefixed. The distance between the pulleys does not change. But both the upper and lower sections of the belt are movingat the same relativistic speed and both shrink accordingly. Now suppose the belt is spliced together with 2splices such that in the moving frames the splices pass over the pulleys simultaneously. (These splices serve as reference points for beltmeasurements) In the observer's frame these events cannot be simultaneoussince events that are simultaneous in one frame are never simultaneous in a framemoving wrt to the reference frame. So when you consider simultaneity the problem mustdisappear. However, the problem still has it's complications in that onesection of the belt as measured by the distance between splices will be in two differentmoving frames at the same time.

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