Re: Origin of formula z(t) = 4 * w^2 * r * t^2 / Pi^2

Date: Fri Apr 25 13:55:08 2008
Posted By: Jim Guinn, Staff, Science, Georgia Perimeter College
Area of science: Physics
ID: 1198871502.Ph
Message:
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Dear Frank,

I wasn’t the person who first responded to your question, but I can answer
this part for you.  Ben was solving for the motion of the yoyo near the
bottom where it turns around.  Measuring the height from the lowest point
with “z”, he determined that at the time t = -Pi/(2w) the height was z =
+r , at the time t = 0 , the height was z = 0 , and at the time t = +Pi/
(2w) the height was again z = +r .  The question now is to determine
what "z" is as a general function of "t".  Since z=0 when t=0, and it is
symmetric for positive and negative t, you can assume it has a simple
quadratic form, that is, z(t) = a t^2 .  All you need now is the value for
the  constant “a”.  Well, when t = Pi/(2w) we have z = a [Pi/(2w)]^2 = a
Pi^2 / 4w^2 , but we also know that at this time z = r , so we have
aPi^2 / 4w^2 = r , or a = 4w^2 r / Pi^2 , this leaves

z(t) = (4w^2 r / Pi^2) t^2 or

z(t) = 4w^2 r t^2 / Pi^2 ,which was the result Ben got.

I hope that helps, Frank, and good luck with your project!

Sincerely,

Jim Guinn
Georgia Perimeter College

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