rix0r.nl

Do you drive a bike regularly? And do you have one of those locks that are attached to your frame and put a bolt through your rear wheel? You know, the kind of lock that Wikipedia calls an O-lock? Here in the Netherlands, nearly everyone has one of these. But if you stop to lock your bike, you’ll notice that it has the tendency to hit one of the spokes in your wheel fairly often. So often that I was wondering:

What is the chance that if you randomly stop your wheel, the lock will jam on a spoke?

Now we’ll need some numbers for this, and I’ll use my own bike as an example to calculate this number. Here’s a model of my bike:

The model has the following parameters:

• Radius of the wheel where the lock bolt crosses it (r).
• Width of the bolt (w).
• Number of spokes in the wheel (n).

For my bike, the numbers are:

• r = 277mm
• w = 9mm
• n = 36

To answer the question, I’ll assume that the bolt is infinitely thin, and each of the spokes is twice the width of the bolt, at radius r. Physically, the chances of the bolt catching on the spokes are the same, but the problem now nicely reduces to:

What percentage of the circumference at radius r of the wheel is taken up by the bolt-wide spokes?

Although you can’t tell from the drawing I make, spokes don’t overlap at the radius of the wheel where the bolt crosses it. This is actually a requirement for my reduction of the problem, but one that all bike wheels will satisfy.

The question is now easy to answer. For my bike, it works out to:

As it turns out, the chance is better than 1 in 3 that my lock will catch on a spoke. Suspicion confirmed!