Volume of Displaced Fluid equals Volume of Submerged Object

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Physical Law

Let a non-buoyant solid body $B$ be completely immersed in a container of fluid $F$.

Then the volume of $F$ displaced by $B$ equals the volume of $B$.


Proof


Also see


Historical Note

The story goes that Archimedes was asked by King Hiero II of Syracuse to determine whether his new crown was made truly of gold, or whether the goldsmith had cheated him by stealing some of the gold and mixing the remainder with a metal of a lesser value, supposedly silver.

Archimedes was puzzling over this problem when he lay back in the bath and watched the water flow over the sides.

It was then that he suddenly realized that as gold was more dense than silver, it would have a smaller volume, and so displace less water, than a piece of silver of the same mass.

Legend has it that he jumped out of the bath and ran home without bothering to dress, shouting:

Eureka! Eureka!

which means:

I have found it! I have found it!

(It needs to be pointed out that public nakedness was not a forbidden phenomenon in that social milieu, hence this behaviour was not quite as outrageous as it would be if embedded in conventional early twenty-first century Western culture.)

He measured the mass of Hiero's new crown, and created a block of gold and one of silver of the same mass.

By measuring the volume of water that was displaced by each block, he determined that the crown was made of metal which was denser than silver, but less dense than gold, proving that the gold from which the crown was made was impure.