One of the most elegant ideas in science is Archimedes’ principle.
However, how does such a simple and elegant principle emerge from the messy and chaotic interactions of the atoms and molecules inside a fluid?
For the simulations in this video, let us model the molecules of the fluid as spheres interacting through perfectly elastic collisions, with no other forces involved.
Let us see if we can derive and explain Archimedes’ principle entirely in terms of this model.
Throughout this video, all red objects will have double the density of the white objects.
The walls of the red rectangle have double the density of the white spheres.
The red sphere inside the rectangle also has double the density of the white spheres.
But, this red rectangle has a lower overall density due to the fact that it is hollow inside.
Similarly, a metal ship can float on water because the ship is filled with air,
causing the metal ship as a whole to be less dense than the water.
Let’s consider the interaction of different fluids with different densities.
The red spheres on the left have twice the density of the white spheres.
The violet spheres in the middle have four times the density of the white spheres.
The orange spheres on the right have ten times the density of the white spheres.
The orange spheres on the right, which have the highest density, quickly sink to the bottom.
The violet spheres in the middle are also trying to sink to the bottom, but it takes them longer to do so.
The red spheres on the left are having the hardest time sinking to the bottom,
as they have to wait for the white spheres trapped underneath to slowly bubble up to the top.
Let’s now consider examples just involving the white spheres, but with different shapes for each container.
As we reach an equilibrium steady state condition, notice how the pressure is greater at the bottom than at the top.
At each point, the pressure depends on the height of fluid above that given point, the density of the fluid, and the acceleration due to gravity.
If all three containers are filled with fluids of the same density to the same height, then the pressure at any given height is the same inside all three containers, despite their different shapes.
Unlike the case for closed containers, here we don’t need to know the temperature to calculate the pressure, because the temperature of the fluid is already accounted for in the fluid density.
Let us focus on just a single molecule at the bottom.
Gravity exerts a downward force on this molecule. This is balanced by an upward force by the bottom of the container.
Now consider the molecule directly above this one.
Gravity also exerts a downward force on this other molecule too.
This is balanced by an upward force by the molecule underneath, which now exerts an even greater downward force on the bottom of the container.
In response, the bottom of the container exerts an even greater upward force, so as to support the gravitational forces from both of these molecules.
Let us now consider a third molecule, above our original two.
As gravity exerts a downward force on this third molecule, this increases the pressure on the two molecules below it.
As we continue adding molecules, we see that the pressure at the bottom continues to increase.
We now have a situation where the pressure is greater at the bottom than at the top, as we saw in the simulation.
All molecules at the same height feel the same pressure.
This remains true even if we change the shape of the container.
We are ignoring the pressure of the atmosphere above the fluid, but this will just increase the pressure everywhere by the same amount.
A partially submerged object will feel a net upward pressure by the amount shown.
In equilibrium, this net upward pressure will balance the downward gravitational force on the object.
This net upward pressure will be exactly equal to the downward gravitational forces that would have been felt by the fluid that has been displaced by the object.
And this is precisely what Archimedes' Principle states.
The gravitational forces that would have been felt by the displaced fluid depends on the density of the fluid.
The gravitational force on the object depends on the density of the object.
If the density of the object is half of the density of the fluid, half the object needs to be submerged in order for all the forces to balance.
If the density of the object is greater than the density of the fluid,
then the forces will never balance, and the object sinks to the bottom.
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