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Math Help - Parameterizing a spiral motion

  1. #1
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    Parameterizing a spiral motion

    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..
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  2. #2
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    hmm actually is it
     x=tcos((\pi/ 100) t)
     y=tsin((\pi/ 100) t)
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  3. #3
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    Quote Originally Posted by Linnus View Post
    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".
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  4. #4
    MHF Contributor Calculus26's Avatar
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    for the solution and animation to a similar problem see mosquito on a disk

    on Calculus 2
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