# Basic buoyancy question: Man in a boat with a stone

This comes from a brain teaser but I'm not sure I can solve it:

You are in a rowing boat on a lake. A large heavy rock is also in the boat. You heave the rock overboard. It sinks to the bottom of the lake. What happens to the water level in the lake? Does it rise, fall or stay the same?

I'd say that the water level drops, because when you drop the stone in the lake, the water level rises according to the volume of the stone, BUT the water level decreases by a volume of water weighting the stone's weight, when you take it off the boat.

Is that correct?

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I think your own answer is nicer than any of the listed ones so far. Like you say, on the boat, the rock displaces its weight in water. In the lake, the rock displaces its volume in water. The latter is less displaced water than the former, because the rock is denser than water. – Steve B Jun 18 '12 at 21:50

## 2 Answers

This diagram is my attempt to show the situation first when the rock is in the boat and secondly when you've chucked the rock over the side.

The mass of the boat of $M$ and the mass of the rock is $m$. The density of water is $\rho_w$ and the density of the rock is $\rho_r$. In the first case Archimedes' principle tells us that the volume of water displaced is:

$$V_{disp1} = \frac{M + m}{\rho_w}$$

In the second case the volume of water displaced is:

$$V_{disp2} = \frac{M}{\rho_w} + \frac{m}{\rho_r}$$

where the second term is just the volume of the rock. If we take the difference of these two we get:

$$V_{disp1} - V_{disp2} = \frac{m}{\rho_w} - \frac{m}{\rho_r}$$

I think it's safe to assume that $\rho_r$ > $\rho_w$, i.e. the rock sinks in water, and in that case $V_{disp1} - V_{disp2}$ is positive i.e. more water is displaced when the rock is in the boat, so the water level falls when you chuck the rock overboard.

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Personally, I think the displacement is side to side rather than up and down. The water level stays the same.

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My thought is, "Suggestions and personal theories are not encouraged here". BTW, I didn't down-vote. It maybe the reason for down-vote :-) – Ϛѓăʑɏ βµԂԃϔ Oct 25 '12 at 19:02
I have often found it the case in science that any opinion contrary to [insert other person here] is not generally accepted. Either read from the text book, or take my word for it! Thinking for yourself is strictly forbidden! Science is the next great religion. – aserwin Oct 25 '12 at 19:28
It's fine to think for yourself, so long as there is actual thinking involved. The answer is that the water level falls, since the stone displaces water according to its weight when it's on the boat, and according to it's volume when it is in the water. If you think about it, you see that this is right, and the fact that you didn't see that this is right is evidence that you didn't think about it. So you can't complain that thinking for yourself is forbidden, since you obviously didn't think. Not my downvote, but I agree with it. – Ron Maimon Oct 25 '12 at 23:45
For both of you guys, the answer is - "Very True" :-) – Ϛѓăʑɏ βµԂԃϔ Oct 26 '12 at 1:11
@Ron Maimon: that works if you do the experiment on a closed container where the water is not allowed to expand... but that wasn't the question. The question involved a lake. – aserwin Oct 26 '12 at 13:34
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