Wheeling Around
Math 211  Prof. Leise

Email completed labs to tleise@amherst.edu by noon Monday.

Motivation: Get on your bicycle (or motorcycle, if you’re so inclined) and go in a circle.  Do your front and back wheels trace out the same circle?  Suppose we chalked both wheels.  What chalk paths would the tires leave?  How do you think the relationship between the front and rear wheel paths change as you do smaller and smaller circles?  (See an animation at http://www.rose-hulman.edu/~finn/research/unicycle/bicyclego.html.  If the idea in this lab intrigues you, I encourage you to read the book Which Way Did the Bicycle Go? by Konhauser, Velleman, and Wagon, which we have in the science library: AC Science books / QA43 .K643 1996.)

Exercise 1. Suppose you know the path of the rear wheel (say, due to skid marks or wheel ruts--the rear wheel is more likely to make such marks since it's the driving force).  Can you reconstruct the path of the front wheel and how the person must have been driving?  Write down an expression for the front wheel track F(t) in terms of the rear wheel track R(t).  Assume the distance between the wheel hubs is 1 meter.  Hint:  think about tangent vectors to the rear wheel path.

Type in your expression for front wheel track:

Exercise 2. Plot (by adjusting the command below) the paths of the front and rear wheels if the rear wheel traces out a circle of radius 5 meters.  What changes as you make tighter circles, say down to 2 meters in radius?

Show[ParametricPlot[{Cos[t],Sin[t]},{t,0,2 Pi},AspectRatio->Automatic],ParametricPlot[{2 Cos[t],2Sin[t]},{t,0,2 Pi},AspectRatio->Automatic],PlotRange->All]

Exercise 3. Now suppose you’re a state trooper examining the scene of a motorcycle accident.  There are skid marks on the road that wave back and forth.  If the rear wheel left these marks, how must the driver have been steering?  Let’s break the problem down:

(a) Assume that the rear wheel path is described by R(t)=<t,0.3sin(2t)> for 0≤t≤6.   Find and plot a vector-valued function describing the front wheel path.  Again assume 1 meter between wheels for simplicity.

ParametricPlot[{t,0.3 Sin[2 t]},{t,0,6}]

(b) Find the angle theta θ (as a function of t) of steering for the front wheel to have followed this path.  Plot θ in degrees (you'll need to convert from radians) for 0≤t≤6.  Which angles correspond to turning left and which to turning right?

To make the problem simpler, note that you can find derivatives in Mathematica, and that giving names to quantities makes it much easier to work with them (as shown below).  I highly recommend you follow the example below, or your calculations will get very messy:

Rprimex=D[t,t]
Rprimey=D[0.3 Sin[2 t],t]
RprimeMagnitude=Sqrt[Rprimex^2+Rprimey^2]