The guys at Car Talk got a math-physics based puzzler wrong this week. You can hear or read the question and their answer at their Web site.
The puzzler in question is labeled week of Jan. 17 as of now, but it should really be the week of Jan. 15 — in fact, the original question aired here in Richmond on the 14th. Anyway, it’s the one about how to tell when a cylindrical gas tank is 1/4 full.
Here’s what I wrote to them:
Sorry, but you got the answer to last week’s puzzler (i.e., the puzzler from Jan. 14, 2011) wrong. You assumed that the center of mass of an object has the property that there are equal amounts of mass on both sides, but that’s not true.
To convince yourself of this, think of the following example: take a 100-pound weight, and a 200-pound weight, and join them together with a long bar. The weight of the bar is small, say 1 pound. Now where’s the center of mass of this funny asymmetrical dumbbell? It’s somewhere between the two masses, about 2/3 of the way along the bar. But there’s certainly not the same amount of weight on the two sides of that point: there’s over 200 pounds on one side, and just over 100 pounds on the other.
In the case of a semicircle, the difference isn’t as dramatic as that, but there still is a difference. The center of mass is about 42.4% of the way out from the center (4/3pi, if you must know). The correct answer (i.e., the point where half the mass is on one side and half the mass is on the other) is only 40.3% of the way out.
Unfortunately, I don’t know a good way to answer the original question: I can’t think of a way to get the 40.3% value without some annoying calculus.
Update: Ray admits he got it wrong.