Question

Suppose you are riding a bike on the plane without slipping. The latter condition of ”no slipping” means that if
P€
is a point in the trajectory of the back wheel of the bicycle, then the front wheel lies on the tangent line to $\gamma$ at point p:

In this problem you will be proving, step-by-step, the following theorem:

Bike theorem: If a bike with a frame of length b made a loop on the plane without slipping, so that the traces of its front and back wheels are disjoint simple closed curves, then the area enclosed between them is independent of the trajectory and equals to $\pi b^2$ .

(a) Prove the theorem for the case when the back wheel’s trajectory is a circle.
Hint: use the ”no slipping” condition to draw the positions of the front wheel and then determine its trajectory. Then use Pythagoras’ theorem to compute the area.

(b) Using the (extended) Green’s theorem, express the area enclosed between the wheels’ trajectories as a difference of line integrals over positively oriented simple closed curves.

(c) Let C be the trajectory curve of the back wheel, parametrized by x(t); y(t); 0 ≤ t ≤ a. Let θ(t) be the angle the bike frame makes with the positive x-axis at time t. Give a parametrization of the trajectory curve of the front wheel C'.

Answer 1

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