So Tony Soprano pitches ties a concrete block to Salvatore Bonpensiero and pitches him into the ocean, where he will inform the police no more. Being a big guy, Bonpensiero has a fairly low density compared to your average human being - say, 0.96 grams per cubic centimeter. That's less than water and so he'd have floated were it not for the weight. Assuming Bonpensiero has a mass of 140 kilograms, how much concrete would Tony need to sink him?

Hey, don't look at me. I just grade the finals. The professor writes them.

Anyway, after grading this question a couple hundred times I figured it would be a good thing to write about. It's an illustration of Archimedes' Principle. His principle says that the buoyant force exerted by a fluid is equivalent to the weight of the fluid displaced. After all, before you jumped into the pool there was water occupying the position that your body now occupies. You had to push it out of the way. Newton's laws say the water pushes back.

So how much concrete does Tony need? We need to multiply Bonpensiero's volume by the density of water in order to find the volume of the water he displaced. His volume is just his mass divided by his density. Converting units appropriately I find that his volume is about 0.147 cubic meters, so that's the volume of the water he displaces when submerged.

0.147 cubic meters of water has a mass of 147 kilograms. So you need 147 kilograms of concrete to sink him. Right?

If you think not, you're correct and you've avoided the first mistake some people made on the exam. Remember Bonpensiero himself has a mass of 140 kilograms. If he has 140 kilograms of weight pulling him down and 147 kilograms of buoyant force pushing him up the net force keeping him afloat is 7 kilograms. So you need 7 kilograms of concrete to sink him (obviously 140 would still work, but it would be overkill). Right?

If you think not, you're correct and you've avoided the second mistake some people made on the exam. The concrete itself displaces water and so it has its own buoyant force. The density of concrete is about 2.3 grams per cubic centimeter. (Don't worry, this number was given in the question.) Each cubic centimeter displaces a gram of water, and so each of cubic centimeter of concrete can only provide a net downward force of 2.3 - 1.0 = 1.3 grams. Some quick judo with the units translates that into a statement that 1 gram of concrete has a net downward force 0.565 grams of when submerged. So to get 7 kilograms of downward force in the water you need 12.4 kilograms of concrete. That will sink Tony's poor old friend.

You've noticed I've run roughshod over the difference between mass and weight. This is ok so long as we keep in mind that there's really a constant factor of g implicit throughout this problem. Since at the end we'd divide it out anyway to get the mass, there's no need to put it in in the first place. I'd never tell my students that because of the risk of confusion, but once you understand what's happening it's a convenient simplification.

- Log in to post comments

Now account for the increase in Big Pussy's volume due to internal gases created by decomposition. That really makes it hard, doesn't it?

I think you need more concrete than that, as I'd tend to assume Bonpensiero could swim for at least a few hours with neutral buoyancy, so unless you drop him really far out to sea, he might make it back to shore to share. If it was me, I'd add another big block of something to make sure he sinks right away.

You'd need more to sink him in the ocean. The specific gravity of sea water is 1.027.

#2, apparently they shot him first. I think that's probably downright friendly by Sopranos standards.

The mob uses SI units?

Tony's "Family" is Italian, so of course they use SI units, at least in their import business, maybe not in their trash collection business.

If they were an ethnically WASP "Family", they'd use lbf and lbm, and they would modify Newton by writing F = ma/gc, where gc is 32.17 lbm ft/sec2 lbf.

If they were living in Europe and knew some engineers, they might use kgf and kgm. Newton's Law would still involve gc, but it would have a different value of course, 9.81 kgm m/s2 kgf.

I can't wait to steal that problem. Although this semester I chose to spend more time on Bernoulli at the expense of buoyancy, there is always next year. (I'm glad to see you are covering fluids. My colleagues at nearby Wannabe Flagship have dropped fluids, and thermo, from their course.) I like that this problem requires actual (gasp) algebra to solve correctly, and thinking rather than simply grabbing some old equation.

But my comment here is to encourage you to pay attention to those mistakes. A problem like that one can be used very effectively in a multiple choice format if you find most mistakes generate a small number of wrong answers, then add NOTA to catch the rest. But it is also easy to grade if you keep notes on the wrong answers and what each one is worth in partial credit. (I blogged about that sort of grading approach almost exactly a year ago, on 12/12/07, and similar issues related to exam design on 5/30/08.) I gather you graded every exam with that particular problem, to maintain quality control across sections? Once you figure out what each one is worth, you can become almost like a human scantron machine. Were most of the errors conceptual (the ones you listed), or were they mostly botched algebra?

PS - Follow the link to my blog to see one of my photos of tonight's fantastic full moon.

Arghh. Proof before sending. The article was on 12/20/07, not 12/12 as I wrote.

Ha, I want to steal that problem too! But unfortunately, I gave my test on fluids and Archimedes' Principle over a month ago. Next year!