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 rotating along the outside rim of a
fixed wheel of radius . Trace the motion of a point on the edge
of the outer wheel for a few different values of and . 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, ...it 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.
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