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Walking and the Pendulum

Let me show you how this simple result and a little more scaling analysis can be used to derive an interesting and surprisingly simple result about walking. Walking is the most economic way for humans and animals alike to move themselves. To minimize energy consumption in walk the legs swing at their natural pendulum period. This is quite different from running where muscles drive the legs back and forth at greater speed and cost in energy than is the case in walking. The time taken in two steps corresponds to one period of oscillation. Thus two steps take
equation67
Here I am using tex2html_wrap_inline141 since the leg is hardly a simple pendulum so while we should expect the same scaling properties the different mass distribution should be expected to yield a different numerical pre-factor. The distance covered in two steps will grow with the length of the legs, tex2html_wrap_inline105 ie.
equation71
From these equations we derive that the walking speed varies as
eqnarray73
This is a simple yet fascinating expression. It tells you why astronauts on the moon walk so slow. The acceleration of gravity is approximately 10 time less there so relaxed walking speed is three times less. The expression also seems consistent with the general relaxed walking experience. However to really check this out I would be delighted if one of you working on Zoology could plot the squared walking speed for a bunch of animals versus their leg length. Our expression predicts that you should get a straight line.



Collin Broholm
Tue Sep 8 10:16:59 EDT 1998