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(Edit: can someone please help me migrate this to physics stack? I think they would be more interested in helping me out with this problem. Thanks.)

I have a bicycle with one of those O-locks on it and too often when I park the bike and I want to lock it, the lock hits one of the spokes of the rim. This can be frustrating and surprises me that it occurs so often. I mean, the spokes are so thin and not that many really so one would think that this should not happen that often (like every day or so). And so every time this happened I reminded myself to calculate the probability of this happening. I just did it (after a year) and now I want to know what you guys think about my calculation, is it OK?

This is not a textbook example so there is no answer to look up or anything, that's why I need your feedback.

I have modeled the situation as shown in the figure below, where I have included one spoke only.

Let the O-lock have diameter $d_{L}$ and the spoke have a diameter $d_S$. Let $R$ denote the "radius" from center of wheel to the point where the O-lock comes and goes (this is approximately equal to the radius of the rim). Then, we have that the corresponding angles are given by $w_S=d_S/R$ and $w_L = d_L/R$ so that the spoke will hit the lock (the shaded disk in figure below) when $\theta$ is in the interval $[0, w_L+w_S]$ modulo $2\pi$ (See figure: $R$ is fixed so the point $(R,\theta)$ is on a circle). Then the probability for hit (wheel with one spoke) is given by $$ P(1) = \frac{w_L+w_S}{2\pi}\times 1 = \frac{1}{2\pi R}(d_L+d_S).$$

Generalizing to $N$ equally spaced spokes we get $$ P(N) = \frac{N}{2\pi R}(d_L+d_S). $$

Example: plugging in some typical numbers; $N=36, R\approx 0.3$m, $d_S\approx 2\times 10^{-3}$m , $d_L\approx \times 10^{-2}$m we find

$$P(36) \approx 0.23$$ which kind of agrees with my everyday experience of this problem.

I guess one could obtain a (Monte-Carlo)-value for $\pi$ this way.

Could someone tell me what I have done wrong above? Or perhaps derive the correct expression for the probability?

Cross section of the O-lock (the disk) and the cross section of one spoke in the plane. $(R,\theta)$ denotes the position on one "side" of the spoke.

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