Built on Facts

Famously there are these exercises called Fermi problems, which are basically questions designed to be solved only by reference to approximation and dimensional analysis.  They aren’t even necessarily physics problems - the canonical example is “How many piano tuners are there in Chicago?”.  The reason why they’re interesting to try to solve is because it develops your intuition for what right answers look like.  If you know what the rough order of magnitude of the answer ought to be, it’s a good first check of your actual formal calculation.  Let me make one up:

I have a pretty standard ceiling fan above my head right now.  If I installed my own electrical system for it, how fast could I spin the blades before they tore off due to the centripetal force?

There’s so little information here it’s ridiculous.  You might think the problem is impossible, and in some sense it is - but if we want a rough estimate we might be able to make some progress.  After all, experience proves that the number is greater than the few-hundred RPM it spins under normal operation and it’s certainly below… I dunno, a few billion RPM.  So we’ve already eliminated two ranges of possibilities, one infinitely large.  It’s a start.  Let’s try to do better.

How much force will it take to break a fan blade?  While there’s no way to get a good answer short of destructive testing I think it’s reasonable to guess that if you mounted a blade vertically instead of the usual horizontally it might be able to support maybe a hundred pounds hanging from it.  Maybe not exactly, but I think it’s reasonable to peg the figure between 50 and 200 pounds.  In Fermi problems rough estimates are all we’ve got.  So 100 pounds it is.  Now the force required for a point mass m in circular motion is F = mv²/r, where v is the velocity and r is the radius of the circle. Now the velocity is rω, where ω is the angular velocity. We can convert that to RPM when we’re done. All that means the force is F = mrω². A fan blade is not a point mass, but we can assume the mass is all at the center of mass. As far as egregious approximations go, it’s not terrible.

I don’t know what the mass or the distance to the center of mass is, but those can both be estimated. Distance to the center of mass is about 18 inches, by my rough guess. The blade itself is pretty much particle board, a quick Google search shows a typical density of 200 kg/m³. The volume is probably about 36in*6in*1/8in. Multiplying and converting units gives a mass of 88 grams.

Solve for ω. We get ω = (F/(mr))^1/2. Plugging all our estimates in gives ω = 105 s^-1. Divide by 2π to get frequency, multiply by 60 to get RPM, you get a final breaking speed of: 1002 RPM. We’ll call it a thousand.

There’s an answer to a Fermi problem. That process applied to more cutting-edge physical situations is often the first step for developing new theories. The next step is to refine the idea and make the wild guesses into something more rigorous. In our example we might calculate the forces throughout the entire blade and find out something much more precise about the strength of particle board. And finally we’d test our theory by actually spinning up the blades and seeing just when they broke.

Still, you might not want to be in the room when you run this test.