Buoyancy and Archimedes' Principle for AP Physics B

Practice problems for these concepts can be found at:

Fluid Mechanics Practice Problems for AP Physics B & C

When an object is submerged wholly or partially in a fluid, the object experiences an upward force called a buoyant force. This leads us to Archimedes' principle, which says, in words:

In mathematics, Archimedes' principle says:

Let's investigate this statement a bit. The weight of anything is mg. So the weight of the displaced fluid is mg, where m is the mass of this fluid. But in fluid mechanics, we like to write mass as ρV. Here, because we're talking about the mass of fluid, ρ refers to the density of the fluid (NOT the submerged mass). And because only the part of an object that is underwater displaces fluid, V represents only the submerged volume of an object.

It is extremely useful to use subscripts liberally when applying Archimedes' principle. Otherwise it's easy to get confused as to what volume, what density, and what mass you are dealing with.

So here's an example.

Let's start by drawing a free-body diagram of the iceberg. The only forces acting on the iceberg are the buoyant force and the iceberg's weight.

Since the iceberg is stationary, it is in equilibrium, so F_B = mg. We can use Archimedes' principle to rewrite the buoyant force as

F_B = (ρ_water)(V_submerged)g.

And we know that the mass of the iceberg equals its density times its volume:

m_iceberg = (ρ_iceberg)(V_iceberg).

We can now rewrite the equation, F_B = mg:

(ρ_water)(V_submerged)g = (ρ_iceberg)(V_iceberg)g.

To find the fraction of the iceberg submerged, we simply take the ratio of the submerged volume to the total volume.

The right-hand side works out to 0.9, because a specific gravity of 0.9 means that the iceberg's density over water's density is 0.9. Therefore, the left-hand side of the equation must equal 0.9, meaning that 90% of the iceberg is submerged. So 10% of the iceberg's volume is visible. Just the tip of the iceberg!

Practice problems for these concepts can be found at:

Fluid Mechanics Practice Problems for AP Physics B & C