Parameterizing a spiral motion

**Linnus** · September 17th 2009, 01:38 AM

An ant crawls along the radius from the center to the edge of a circular disk of radius 100 cm, moving at a constant rate of 1cm/sec. Meanwhile the disk is turning counterclockwise about its center at 1 revolution per second. Parametrize the motion of the ant.

I know it will be something like
x= _____cos (t)
y=______sin (t)
but I don't know how do you find what do you put there..

**Linnus** · September 17th 2009, 03:31 AM

hmm actually is it
$x=tcos((\pi/ 100) t)$
$y=tsin((\pi/ 100) t)$

**HallsofIvy** · September 17th 2009, 04:35 AM

Originally Posted by Linnus

hmm actually is it
$x=tcos((\pi/ 100) t)$
$y=tsin((\pi/ 100) t)$

Well, the angle is given by $tan(\theta)= y/x= sin(\pi t/100)/cos(\pi t/100) = tan(\pi t/100))$ so $\theta= \pi t/100$ and $d\theta/dt= \pi/100$ radians per second which is the same as $(\pi/100)/(2\pi)= 1/200$ revolutions per second, not 1 revolution per second. Do you see how to fix that?

Also $dx/dt= cos(\pi t/100)- t sin(\pi t/100)$ and $dy/dt= sin(\pi t/100)+ t cos(\pi t/100)$ . So $\sqrt{(dx/dt)^2+ (dy/dt)^2}$ $= \sqrt{cos^2(\pi t/100)- 2t sin(\pi t/100)cos(\pi t/100)+ t^2 sin^2(\pi t/100)+ sin^2(\pi t/100)+ 2t cos(\pi t/100)sin(\pi t/100)+ t^2cos^2(\pi t/100)}$ $= \sqrt{1+ t^2}$ . No, that doesn't look like "1".

**Calculus26** · September 17th 2009, 10:02 AM

for the solution and animation to a similar problem see mosquito on a disk

on Calculus 2

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