Archimedes principle: iron floating in mercury, find the submerged percent.

Mercury is more dense than iron, and that means an iron block floats in mercury!

In this problem, we compute the submerged percent of an iron sample floating in mercury. We start with a force diagram of the floating iron block, with the weight, mg, pointing down, and the buoyant force F_B pointing up. The buoyant force is equal to the weight of the displaced fluid, in other words, the weight of the submerged volume of the iron block.

We give a quick reminder of the definition of density as mass divided by volume, and that can be turned around to say that mass is density times volume - a relation that will be useful a couple times in our calculation.

Since the iron block is in equilibrium, F_B=mg, and now we can replace the buoyant force with the details using Archimedes' principle: the buoyant force is equal to the density of mercury times the submerged volume of the block times g. On the right hand side, the weight of the iron block is given by the density of iron times the volume of the iron block times g.

Solving for the submerged volume divided by the total volume of the iron block, we obtain the percent submerged for a block of iron floating in merury.