Assume there is an ice cube in a glass of water. When the ice cube melts, will the water level have risen, fallen, or remained the same? Why?

Asked by: Hugo Polichemi

Answer

Lets see now. Some intuitive center in my brain is screaming 'it will remain the same', but we will try to think it over.

In the first state, we have an ice cube of mass m floating in the water. If it is floating (in equilibrium), it will have to displace enough water to support its weight. How much is that? It is just Volume = m/d , where m is the mass of the ice cube, and d is the density of water.

In the second state, where the ice has melted, it turns into water of volume.... Volume = m/d! exactly the same volume as it displaced before. So the added volume is the same, so the level of the water will not change.

As a matter of fact, as long as objects are floating (i.e. they don't rest on the bottom) they displace enough water to support their mass. Since by turning from solid to liquid, the mass of water does not change (well, maybe it does, due to mass-energy equivalence, but that's _really_ negligible) it will keep displacing the same amount of water.

However, note that this may not apply to everything. If you had solid alcohol floating in water, when it melts, the level would drop, because water and alcohol mix at the molecular level; i.e. water filling spaces among alcohol molecules.
Answered by: Yasar Safkan, Ph.D., Sofware Engineer, Noktalar A.S., Istanbul, Turkey

The water level remains the same when the ice cube melts.

A floating object displaces an amount of water equal to its own weight. Since water expands when it freezes, one ounce of frozen water has a larger volume than one ounce of liquid water. A completely submerged ice cube weighing one ounce, for example, displaces MORE than one ounce of liquid water. The cube will rise until the volume remaining under the surface displaces only one ounce of water.

If you could remove the ice cube and leave a 'hole' in the water where the cube used to float without disturbing the surrounding water, that hole would take exactly one ounce of liquid water to fill. Let the ice cube melt. Since it is now one ounce of liquid water, putting it back into the 'hole' will exactly fill it and leave the remaining water undisturbed.
Answered by: Paul Walorski, Part Time Physics Instructor