Question

A piece of ice floating in a glass of water melts, but the level of water in glass does not change. Give reason.

This question has multiple correct options
A

Ice contracts on melting.

B

Ice expands on melting

C

Ice has same density as that of water

D

Ice has density less than that of water

Medium

Solution

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Correct options are A) and D)

It is true because in both the cases weight of water displaced = weight of ice and weight of ice before and after melting is the same. 
This follows from Archimede's principle according to which a body floating in a liquid displaces weight of liquid equal to its own weight.
Water expands about 7% when it freezes, thats why it floats. But it displaces exaxctly its equivalent mass of water when it floats. When the ice melts there is no change in the amount of H2O in the vessel so the water level doesnt rise.
The level does not change because the ice is nearly at neutral bouyancy or barely floating to begin with, and it is colder. 

A tiny amount of the ice often protrudes above the water, so it does not have "exactly" neutral bouyency. But its close. One need only observe that icebergs have exposed surfaces, even in fresh water, that rise above the water surface, same thing happens in a glass. Since part is above water, the ice must be displacing a greater mass of water. So density change must play a part. 

As the ice melts it warms but does not expand, since warming ice above feezing causes contraction, not expansion (water reaches its maximum density at +4C, well above freezing). The warming ice cools the surrounding water thus decreasing its volume. The net result is that floating ice does not raise water levels when it melts, they stay the same. 

So if all the ice floating in the ocean were to melt there would be no change in sea levels

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