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.
I think you can do it with a straightforward modification of their method.
After finding the center of mass, use the pizza box as a square to draw a chord perpendicular to the radius, intersecting the radius at the center of mass.
Cut the semicircle along the chord so you have two pieces, one slightly heavier than the other.
Sharpen the other end of the pencil, balance the pencil on your finger to find its center of mass, then put a piece of the semicircle on each end. The pencil tips down towards the heavier half, so take a small slice off that half, and move it to the other end of the pencil.
Repeat until the pencil is balanced again…if you slice in an organized way, you should be able to put the pieces of the bottom portion of the semicircle back together and measure its length, even if you overcompensate along the way and have to move some mass back to the top portion of the semicircle.
Or decide how close you need to get to measuring the 1/4 mark and diced if the difference between center of mass and half the weight matters for your application.
Yes, that’ll work. Nice!
Isn’t that basically “cut the thing into two pieces, weigh them, if different, cut a sliver from the big piece to add to the small piece, try again”? The only thing the center of mass is give you a reasonable starting point.
You also have to make sure that the center of masses of the two pieces are at equal distance from the fulcrum.
Ted, I’m surprised you didn’t use the auto enthusiast’s favorite word: torque.
I prefer the beer can solution. Take a can of beer (or soda if you prefer), transfer the liquid into a cup until tipping the can on its side shows that the liquid level reaches the center. Drink the beer in the cup. Transfer the beer from the can into the cup. With the assistance of a second identical cup, divide the liquid in half. Drink one cup, pour the other cup back into the can. Tip the can on its side. There is your 1/4 mark. Now drink the rest of the beer. (Note: Assumes perfectly cylindrical beer can, which is not the case in practice due to indentations in the bottom. But hey, if you guys can assume a perfectly spherical horse, I can assume a perfectly cylindrical beer can.)
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