Question

There is a bug sitting between the threads of your car tire. We start a short commute of 1 mile and the bug remains at the same location between the tread of your tire without getting squished. Assume the tire is 2 feet in diameter ( so the center is exactly 1 foot above the pavement ) and the thread width is negligible so the bug is almost touching the road as the tire rotates.

1. How far does the bug travel on this trip?

2. As the tire rotates, the bug moves, it traces out a curve over the road, and the curve touches the road once per every full rotation. what is the area under thiss curve?

3. What is average velocity of the bug?

4. Does the bug ever occupy the same point in space at two different times.

Assumptions:

You are driving on perfectly flat road and in straight line and the tire is a perfect circle.

Also assume that you are driving at you driving at 60 mp/h or 88 ft/s for the entire trip, so the trip

takes exactly 60 seconds.

Answer 1

Radius of tyre $R=1\,ft$

linear velocity of tyre (velocity of center of tyre) $V=60\,mph=88\,ft/s$

angular velocity of tyre $\omega=\frac{V}{R}=88\,rad/s$

Time taken for one complete rotation of tyre is $T=\frac{2\pi}{\omega}$

Time taken for the trip $t_{0}=60\,s$

As the tyre is rolling without slipping, $V=R\,\omega$

Assume that at time t = 0, the bug is in contact with road. Consider this point of road to be the origin.

Position of bug as a function of time is $(x,y)=(R\omega{t}-R\sin{\omega{t}},R-R\cos{\omega{t}})$

velocity of bug as a function of time is $(V_{x},V_{y})=(R\omega-R\omega\cos{\omega{t}},R\omega\sin{\omega{t}})$

magnitude of velocity is $|\vec{V}|=\sqrt{(R\omega-R\omega\cos{\omega{t}})^2+(R\omega\sin{\omega{t}})^2}=2R\omega\sin{\omega{t}/2}$

Part 1.

Curve traced out by the bug in space is called cycloid. This curve touches the road every $T=\frac{2\pi}{\omega}$ seconds periodically. Distance travelled by the bug during one such period is $\int_{0}^{T}{|\vec{V}|dt}=\int_{0}^{T}{2R\omega\sin{(\omega{t}/2)}dt}=8R$

Hence the total distance travelled by the bug is $s=(8R)\frac{t_{0}}{T}=6722.7\,ft=1.27\,miles$

Part 2.

Area below the cycloid (curve traced out by the bug) in one period is $A=\int_{0}^{T}{ydx}=\int_{0}^{T}{R^2\omega(1-\cos{\omega{t}})(1-\cos{\omega{t}})}dt$

$A=R^2\omega\int_{0}^{T}{(1-\cos{\omega{t}})^2}dt$

$A=R^2\omega\int_{0}^{T}{(1-2\cos{\omega{t}}+\cos^2{\omega{t}})}dt=3\pi{R^2}=9.425\,ft^2$

Part 3.

Average velocity of bug is displacement divided by time. In one time period average velocity is $V_{av}=R\omega=88\,ft/s$ . Over a large number of time periods, the average velocity is again $V_{av}=R\omega=88\,ft/s$

Part 4.

The bug does not occupy the same point in space at two different times.