Physics Puzzler: Will the Water Level Rise or Fall?

Say you have a glass of ice water, and you wait around until the ice melts. Will the water level in the glass have gone up, gone down, or stayed the same?

As I showed you in my last video, ice floats on top of liquid water because the density of ice is smaller than the density of water. That's because water expands as it freezes, and so the same mass of liquid turns into a larger volume of solid ice. Then the buoyant force from the water pushing up on the ice is able to support its weight, and the ice floats. We saw that about 90% of the volume of the ice lies below the surface though, and only 10% rises above.

So when the ice that's floating in your drink melts, what's going to happen to the water level? You might think that since the ice will shrink into a smaller volume as it melts, that it will only fill in some fraction of the "hole" that it made when it was sitting under the surface of the water, like in option A, and so the height of the water level would drop. Or maybe it doesn't shrink enough, and since the ice rose above the original water level, when it melts is it still higher than where the water started, and so the level goes up, as in option B? Or maybe the level doesn't change at all, like in option C?

Resolving this question is all about understanding the buoyant force, which is what my last video was about. So if you haven't watched that one yet, you should go watch it first, and then come back here to finish. The key thing we learned was Archimedes' principle: when you sink an object under the surface of a fluid, it exerts an upward force on the object that's equal to the weight of however much fluid was displaced. So let's see how Archimedes' principle clears up this question.

When we float the ice cube on the water, part of the volume falls below the surface of the liquid, displacing whatever water had been there. Call that volume $V_F$. We learned that the upward buoyant force on the ice cube equals the weight of however much liquid was displaced: that's the density of the liquid $\rho_F$, times the volume $V_F$, times $g$: $F_B =\rho_F V_F g$.

Say that our ice cube has mass $m$. Then since it's sitting in equilibrium, its weight $mg$ pulling down must cancel against the buoyant force going up. That means that the ice cube starts to sink under the surface of the water, just until it has displaced a volume of water of that same weight $mg$. So, if you forgot about the ice cube and instead filled the hole it had made with water, it would take the same mass of water $m$.

Now let the ice cube melt. And let's imagine that the water we originally started with is fixed in place, so that we can picture what's going to happen to just the melted ice water. Is it going to partially fill in the hole like in option A, overflow the hole like in B, or perfectly fill it in like option C?

When our ice cube of mass $m$ melts, it turns into a puddle of water of the same mass $m.$ But, that's exactly how much water would have fit in the region that the ice originally displaced! So, the ice melts, and it perfectly fills in the hole it had made with the same mass $m$ of water. The water level doesn't change!

Of course, this is an experiment that you can easily do for yourself at home. Try it and make sure Archimedes was right!