Archimedes of Syracuse (287-212) was perhaps one of humanity’s greatest mathematical geniuses. He is remembered for the principle of hydrostatic thrust, the screw pump, and war machines, but his most advanced studies were on conics and mathematical methods for calculating surfaces and volumes (e.g., of sphere). These are considered among the highest achievements of ancient science and, although they lack modern formal rigor, they represent the direct precursors of the concept of limits and integral calculus (independently developed by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century). See, for example, Lucio Russo, Archimedes: A Great Ancient Scientist (Carocci Editore, 2019).
One day, my young son asked me, “Why do ships that are heavy and made of iron float?” I couldn’t explain the density of water to him, but I found a good answer for a child: “Because they weigh less than the water they displace.”
This is exactly Archimedes’ principle of hydrostatic thrust, which in more complicated terms is expressed as: “A body immersed in a fluid experiences an upward thrust equal to the weight of the volume of fluid displaced.” So if this is greater than the weight of the body, it floats.
We can express Archimedes’ principle of hydrostatic thrust (force) F(A) using the following formula:
[1] F(A) = m * g = ρ(fluid) * V(displaced) * g
where m = ρ * V is the mass, g is the acceleration due to gravity, ρ represents the density, and V represents the volume of fluid displaced by the body.
For this reason, any body also ‘weighs’ according to the density of the fluid in which it is immersed. The greater the density, the greater the hydrostatic thrust and the lower the weight of the body.
As an example, if an iron nail (which, due to its shape, does not seem to be able to float like a ship by displacing a lot of fluid) were placed in a container with liquid mercury, which has a density almost double that of iron, the nail would float at the level of the weight of mercury displaced. Calculations show that it would be immersed just over half its volume (about 58%).
In this video, we can see how an iron anvil floats in mercury because the density of iron (7.87 g/cm³) is almost half that of mercury (13.55 g/cm³) (source: Wonder of science on X channel).
Another famous example of buoyancy is that of hot air balloons: they float exactly like a ship because the volume of fluid (atmospheric air) they displace weighs more (is denser) than that contained in the balloon, which is heated with a burner (and is therefore less dense).
An hot air balloon (source: freedome.it)
