STATEMENT OF PROBLEM

Consider a wheel of radius r = 10 cm and a stick of length l = 40 cm. (In general prescribe l >= 2r for this problem.) The stick is attached to the edge of the wheel with a pin which permits it to freely move while the wheel roles along a flat surface (say the x-axis). In fact, as the wheel turns along the flat surface one end of the stick merely drags along the surface while the other end (attached to the pin) rides up and down.

Suppose we start the wheel with the stick attachment pin at the origin
(0, 0) and center at (0, 10), positive x-axis to the right, and the stick laying on the x-axis to the left, with its left hand end-point at (-40, 0).

Further, suppose the wheel roles to the right, without friction, at a rate of t radians/sec, i.e. it takes 2 Pi sec to complete one revolution at which point we find the stick-pin-wheel configuration looking exactly the same (pin on the x-axis - only at (2 Pi 10,0) now, stick on the x-axis from pin to point (2 Pi 10 - 40, 0), and the wheel center at (2 Pi 10, 10).

1. Write a function to describe the position of the tip of the stick not attached to the wheel as the wheel is rolling on a plane, i.e. in terms of the parameter t, the angle of rotation of the wheel. We suggest a step by step approach in (a) - (e) below.

Here are some questions concerning the dragging tip of the stick.

2. Plot a graph of the horizontal displacement of the dragging tip of the stick. We know the vertical position is constantly y = 0 as it drags along the flat surface of the x-axis. A reasonable interval over which to plot this function is [0, 4 Pi], i.e. two full rotations of the wheel. Does this graph make sense? Recall where it should start, where it should be at t = 0, at
t = 2 Pi, at t = 4 Pi.

Explain some of the features of this curve in terms of the motion of the dragging tip of the stick. Does the dragging tip of the stick ever go backwards? How would you characterize its horizontal motion?

3. Plot a graph of the horizontal velocity on the dragging tip of the stick. Explain some of the features of this curve in terms of the motion of the dragging tip of the stick.

4. Plot a graph of the horizontal acceleration on the dragging tip of the stick. Explain some of the features of this curve in terms of the motion of the dragging tip of the stick.

5. For what time(s)/angle(s) is the the horizontal velocity on the dragging tip of the stick a maximum and a minimum? What are these maxima and minima values of the velocity?

6. State other observations, conjectures, issues, or questions you have concerning the motion of the dragging tip of the stick.

7-12
Now do the problem with length of the stick (l = 40) equal to the length of the diameter (d = 2 r = 40).

There is something odd about some of the graphs of horizontal displacement, velocity, and acceleration. If the domain of the functions is [0, 4Pi] find and explain any discontinuities in these plots? What is going on in the motion at these points of discontinuity?

13 (For the dimensions used in 7-12 only.) You may see a place where the derivative of a discontinuous function actually appears to exist at the point of discontinuity. What theorem on derivatives and continuity seems to be violated? Explain why this violation is only apparently so and not in fact true.