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I know it will be periodic, and can calculate for integral values of t (0,1,2,3,4,etc) but can't figure out how to find when t isn't a whole number.

@ParthKohli @agentx5 can you help?

Ah wait cycloids...

no, both are rotating clockwise.

http://www.daviddarling.info/images/cycloid.gif (prolate, nested function)

Which of course is going to make it look a whole lot like a sine function

yeah, but a cycloid only describes 1 point. I'm looking for distance between the 2 points.

Or does it... Hmm, might just be additive.

this is what i've figured out:
\[d(0)= 1\]\[d(1) = \sqrt{13}\]\[d(2) = 5\]\[d(3)= \sqrt{13}\]

|dw:1343667310281:dw|

They are pulleys, not gears in this case.
|dw:1343667494092:dw|

t(.5) = t(3.5)
right?

ah...got it:
\[d(t) = \sqrt{(2 \sin (\frac{\pi x}{2}))^2 + (3-2 \cos (\frac{\pi x}{2}))^2}\]

https://www.desmos.com/calculator is my preferable graphing tool...and it works!