Tracing points on planetary gears

A wheel of radius 1 revolves at a constant angular velocity and travels in a straight line on a level surface. Plot the path traced out by a point on the edge of the wheel. You should take in to account the fact that the wheel is not only revolving, but its centre is moving in a direction opposite to the motion of the edge of the wheel at the point where it touches the surface. The equation of motion should be a two coordinate function of the form [X(t), Y(t)]. In order to plot the path of such a function over one complete revolution you can use the Maple command plot([X(t), Y(t)],t=0..2*Pi]);. Trace the motion of a point which is halfway between the edge and the hub of the wheel.

Next, consider a wheel of radius tex2html_wrap_inline9 rotating along the outside rim of a fixed wheel of radius tex2html_wrap_inline11. Trace the motion of a point on the edge of the outer wheel for a few different values of tex2html_wrap_inline9 and tex2html_wrap_inline11. What happens if the free wheel travels along the interior rim of the fixed wheel. Create an animation which shows both wheels, as well as the point on the edge of the free wheel, as they rotate.

Finally, read the excerpt provided from Umberto Eco's novel "The Island of the Day Before" and notice the claim made by Father Caspar that the path traced by the nail in the wheel is "...a trochoid, is like the movement of a ball you throw before you, then it touches ground...". Is this an accurate description of the nail's movement? Recall that a falling object traces quadratic curve. Use Maple to plot such a quadratic curve along side a trochoid and compare the two.


Juris Steprans
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Department of Mathematics and Statistics.
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