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PULLEY TEASER Draw as many 2-dimensional pulley systems as your inventiveness allows.

ANSWER: A movable pulley has 3 rope segments acting on it, 2 of them due to a single rope passing over the sheave. The 2 segments over the sheave exert a net force 2T on the pulley in the same direction, so the third rope segment must have tension 2T in the opposite direction, to achieve force equilibrium. Therefore that third segment cannot be part of the same rope passing over the sheave, for its tension is only T throughout. LOVE TEASERS? Take our Pulley Teaser challenge, above. Or, for something completely different, check out the classic Love Puzzle, from The Book of Five Hundred Puzzles and Curious Paradoxes, published in 1859.

TIMELESS TEASER

Drill 3 holes in a rectangular piece of wood. Thread 2 wooden hearts or beads onto a string. Loop and tie the cord as shown. The challenge is to get the hearts or beads onto the same loop.

ANSWER: Draw the left heart along the string through the loop in the middle until it reaches the back of the center hole, pull the loop through the hole, and pass the heart through the 2 loops that will then be formed. Then draw the string back through the hole as before, and the heart may be easily passed to its companion.

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Two-dimensional systems are those where the pulleys can only move up or down, and all rope segments are parallel. Classify them as workable or unworkable. Without doing a force analysis, can you spot the common geometric feature of the unworkable ones? (Failed machine designs are always due to an attempt to violate the geometry of the universe.)

CONCLUSION: An idealized pulley system is unworkable if even one movable pulley is acted upon by only one rope.

2F. At the next pulley the downward force is 4F, and at the next it is 8F, and by the time we get to the last pulley, the downward force must be 32F = L. So the mechanical advantage is 32, and there are nowhere near that many ropes or even rope segments in the system. (There are 5 ropes and 10 segments.) One thing seldom addressed in textbooks is how to do estimates (back-of-envelope calculations) comparing efficiency of different systems. Suppose that each pulley, moving or not, has a force due to friction, proportional to the weight its axle directly supports. Suppose also that each pulley that moves up and down has a non-negligible weight. Now what could possibly be the superiority of the Spanish burton over a block and tackle with the same ideal mechanical advantage? The block and tackle would require 32 pulleys compared to the 6 of the Spanish burton, and the block and tackle would have 16 pulleys moving, compared to 5 of the Spanish burton (moving at different speeds, of course). But the Spanish burton has geometric problems, as well as problems with rope stretch. Figure E is misleading, because the pulley spacing, bottom to top, must be 1, 2, 4, 8, and so on, at all times. This system is seldom seen with more than 2 or 3 movable pulleys. Da Vinci took things to extremes, often drawing pictures of things that weren’t practical. I’ve raised some questions that you can easily answer by building such systems and testing their performance. Small pulleys can be obtained at science supply stores, or from toy construction sets. Add some stout, non-stretchy cord and some weights or small spring scales, and you can have a lot of fun learning about simple machines. You can also devise puzzles such as “Given N pulleys and N ropes, what’s the greatest mechanical advantage you could achieve using all of the pulleys? What’s the greatest efficiency you can get from them?” No matter how ingenious you are, you probably won’t find an unworkable system that doesn’t have a fool’s tackle hidden within it. (Some mathematician may be able to prove or disprove this as a theorem.) You can buy “simple machines” educational toy kits with the necessary parts. But to get full benefit from them, children need to have some guidance and be challenged by “What if?” and “How?” questions that stimulate measurement and quantitative analysis.

Donald Simanek is an emeritus professor of physics at Lock Haven University of Pennsylvania. He writes about science, pseudoscience, and humor at www.lhup.edu/~dsimanek.

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