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

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!

Thank you for your interest.

Sincerely,

Jim Guinn
Georgia Perimeter College