Kinematics of scratching a rolling tire

Recently a "U" shaped scratch appeared on my car's front tire. It looks like someone rolled past a curb and scraped it. This has led me to the following problem:

Generate a plot (ideally polar) of a scratch pattern on a tire given tire radius and the ground height of a protrusion. Assume the protrusion is a single point and remains stationary while the tire rolls past. Also assume that the tire rolls straight past the protrusion, which scratches the side of the tire (i.e. hubcap, trim, etc).

If I wasn't so rusty on this material I would have taken a stab at it myself. So, any help in setting up and solving this problem would be greatly appreciated.

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Usually, if there's no question mark in your question, than the question isn't very clear. :-) (Ok, maybe I'm generalising, but I really can't tell what you want.) –  Bruce Connor Dec 28 '10 at 22:39
I understand what you want, but this is not the site for that question. Try math.SE and ask for set difference of the tire and the U-shaped cylinder. If those objects can be described analytically then it's just equation solving. If not, you can still solve for it numerically. –  Marek Dec 28 '10 at 23:33
Also note that what you are looking for is not the round tire but the tire that is compressed by the weight of the car. It will be flattened on the bottom. If you then spin the wheel and move the scratch off the ground its shape will change a little. So I guess this question contains some physics after all. But most of it is math, really. –  Marek Dec 28 '10 at 23:36
The tire deformation shouldn't matter as long as the radius used is the effective tire radius. And thanks for the suggestions-- I'll post this to the math site. –  Andy Dec 28 '10 at 23:53
I was actually going to say, we could potentially broaden our scope to include questions such as this which are basically about applied math involving physical objects. Sometimes the hard part of asking a question of this sort is translating the physical situation into the required math, and that's where a physicist's intuition might be helpful. It could bring us more traffic too ;-) Then again, I suppose applications might be covered on math.SE too and we probably shouldn't have overlapping topics. (@Andy, if you want, you could wait a little while to see if anyone chimes in on this idea) –  David Z Dec 29 '10 at 0:46

Start with the reverse problem. What are the coordinates of a point fixed on the wheel. I am going to use the polar notation $r$ and $\varphi$ to locate a point on the wheel based on some reference axis. Supposed the center of the wheel is $(x_c,0)$ where $R$ is the radius of the wheel and $x_c$ is a function of time (linear). The rotation of the wheel is designated by the angle $\theta$.

The location of a point on the wheel is

$$x-x_{c}=r\,\sin(\theta+\varphi)$$ $$y=r\,\cos(\theta+\varphi)$$

So now the problem is while $x_{c}$ and $\theta$ are varying what is $r$ and $\varphi$ such that $x$ and $y$ are fixed in space. Note that $\theta\,=\,\frac{x_c}{R}$ for a rolling wheel to make everything a function of the independent variable $x_c$.

With a little math you get

$$r(x_{c})=\sqrt{x_{c}^{2}+y^{2}}$$ $$\varphi(x_{c})=\arctan\left(\frac{x_{c}}{y}\right)-\frac{x_{c}}{R}$$

With the assumption(initial condition) that $x=0$ when $x_c=0$. I plot the results for various $\frac{y}{R}$ ranging from $-1\ldots0$ and moving the center of the wheel from $x_c=-R\ldots+R$ and make some cool looking $\gamma$ looking curves. They look like the ribbons people hang to bring awareness to various illnesses. If the tire is profiled such that its width varies with radius, then only part of the shape will imprint leaving only the U shaped part of it.

Below is a graph of the shapes using Asymtpote Vector Graphics.

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Nice work @jalexiou –  user346 Dec 29 '10 at 5:17