Why will the water not overflow when ice melts?

Question

This is an old question but still there's something I could not understand. Here it goes: A glass of water has an ice cube floating in it.The water level just touches the rim of the glass. will the water overflow when the ice melts?

This is how I imagined the scenario:

Now everywhere I see its explanation is given using Archimedes principle like this "Volume of the Ice will be equal to volume of water displaced". But from what I know Archimedes principle states that volume of water displaced is equal to volume of object immersed in water and here the object is NOT completely immersed.

And now wish to get a proper explanation of why when the ice melts , the water won't overflow.

Answer 1

Archimedes' principle says that the buoyant force on any object (partially or fully submerged) is equal to the weight of the water it displaces. It doesn't just apply to fully immersed objects. The only major difference in the application of Archimedes's principle to partially immersed objects is that the buoyant force is determined by the portion of the object's volume that is immersed, rather than the full volume.

So the argument goes like this:

• The ice cube is in equilibrium, so the buoyant force on the ice cube must equal the weight of the ice cube.
• Thus, the weight of the ice cube is equal to the weight of the liquid water it displaces.
• When the ice cube melts, it will turn into liquid water with the same weight.
• Thus, the weight of the melted ice cube is equal to the weight of the water it displaced when it was solid.
• This means that the volume of the ice cube, once it melts, is equal to the volume that the ice displaced when it was solid.
• Thus, the water level does not change.

Answer 2

Now everywhere I see its explanation is given using Archimedes principle like this
"Volume of the Ice will be equal to volume of water displaced".

This wording of Archimedes' law is wrong. The correct formulation should involve weight, not volume:

"Weight of the Ice will be equal to weight of water displaced".

or quoted from Wikipedia - Archimedes' principle:

the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially, is equal to the weight of the fluid that the body displaces.

Thus the ice cube has the same weight as the displaced water. During melting the ice cube changes to the same weight of water (but of course with a different, i.e. smaller volume). And therefore the level of water doesn't change.

Answer 3

Archimedes principle states that a floating object displaces it's weight of the fluid that it is floating in, not it's volume. Since the ice cube is approximately 90% as dense as the 0 deg C water that it is floating in, approximately 10% of the ice cube will be above the water surface before it melts. When the ice cube is fully melted, it contributes its weight to the water in the container that it was floating in, which is exactly equal to the weight of the ice cube. This melt water is more dense than the ice cube, such that the "extra" volume of the ice cube that was floating above the liquid surface is no longer there, and that decrease in volume (due to the higher density) exactly matches the "extra" volume that the ice cube contained. Thus, there is no water overflow from the container, due to the changing density and changing volume involved.

If you repeat this experiment by floating an ice cube in salt water, you will see some overflow, so the results of the experiment depend on one or two details of the experiment. For an ice cube floating in salt water of a given salt concentration, that ice cube still displaces its weight of salt water. With salt water being denser than fresh water, the ice cube floats a bit higher in the salt water than it would float in fresh water. However, since the ice cube is composed entirely of fresh water, once that ice cube melts, it adds fresh water to the container, which dilutes the salt concentration in the container. In that case, a more dilute salt water solution takes up more volume than the original salt water, and a small amount of water will overflow the container as a result.