It's Like Riding a Bicycle
One of the advantages of modeling physical phenomena with parametrization
is that we can break up motion into different components.
For example, suppose we want to model the path drawn out by a squashed
piece of gum on the tube of a bicycle wheel. We will assume
- The center of the bicycle wheel travels
at 10 feet per second.
- The bicycle wheel has a radius of 14 inches.
- We measure time in seconds and distance in feet.
Let's first assume that the bicycle travels in a straight line.
To describe the motion, let us first
parametrize the curve in the plane that describes
the path of the center of the wheel: t -> (10t,14/12)
(Why is this correct?).
Meanwhile, on the outside perimeter of the wheel,
the squashed piece of gum turns
in a circular motion. If we use a coordinate system with the origin
at the center of the wheel, the circular component of the
gum's motion is
a circle: t -> (14/12 cos(wt), 14/12 sin(wt)).
The angular velocity, w,
determines the rate at which the gum turns in a circle.
The value for w may be calculated from the bicycle's speed and
the radius of the bicycle tire.
Question #3
- Find w in radians per second.
- Piecing all the information together,
give the final parametrization of the gum in our "viewing" coordinates.
Maple will help you to visualize the motion of the gum if you
substitute the correct formulas for x(t) and y(t)
and then issue the command:
ParamPlot([x(t), y(t)], t=0..3, scaling=constrained);
The curve traced out in the previous problem is called a
cycloid. If we generalize the curve so that we model a bike
reflector on a spoke of the wheel, we obtain a different curve, called
a trochoid.
Question #4
- Parametrize and plot the motion of a reflector that is
1 foot from the center of a bicycle wheel.
- Parametrize and plot the motion of an object 1.5 feet from the center
of the wheel. For example, this might model the pedal of a unicycle
riding across a tight rope.
Next: Moving on to 3D
Up: Introduction
Previous: Circular Motion
Frederick J. Wicklin <fjw@geom.umn.edu>
Jeremy Case
Document Created: Mon Feb 20
Last modified: Tue Feb 27 08:36:55 1996