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.
162 Make: Volume
23
PULLEY TEASER
Draw as many 2-dimensional pulley
systems as your inventiveness allows.
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.)
C O N C L U SI O N: A n idealize d p ulley syste m is
un w orkableifeven one m ovable pulleyis acted
upon by only onerope.
ANS W ER: A m ovable pulley has 3 rope seg m ents
acting onit, 2 ofthe m dueto a singlerope passing
overthe sheave. The 2 seg m ents overthe sheave
exert a netforce 2 T o n the p ulleyin the sa m e
directio n, s o th e third ro p e s e g m e nt m u st h ave
te n sio n 2 T in th e o p p o site dire ctio n, to a c hie v e
forc e e q uilibriu m . T h erefore th at third s e g m e nt
cannot be part ofthe sa m e rope passing overthe
sheave,foritstensionis 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.
ANS W ER: D ra w th e left h e art alo n g th e strin g
through theloopin the middle untilitreachesthe
back ofthe centerhole,pulltheloop throughthe
hole,and passthe heartthrough the 2loopsthat
willthen beform ed. Then dra w the string back
through the hole as before,and the heart m ay be
easily passed toits co m panion.
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.
