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.