Padraig Dunne is the head of the School of Physics at UCD.
Imagine you are made of water, nothing more, nothing less. You have your head, complete with features, your arms, your legs, your torso, all enclosed in a form that is identical to your current form. You enter a swimming pool, filled with water and lie there motionless. What can you learn about buoyancy from such a thought experiment?
Well, if you are motionless, there is no change in your motion, and Newton’s first law tells us that if you add up all the forces acting on your body the sum equals zero. If we consider the forces acting vertically, that is upwards or downwards, they also add up to zero. As the swimming pool is on Earth, we know that you have a weight, let’s call it W(↓), acting downwards, as weight always does. Newton’s first law implies that there must be another force acting to cancel W(↓), but acting upwards. Let’s call this force B(↑), the buoyant force. So now we can say that B(↑) + W(↓) = zero.
This means that the buoyant force is just your weight if you were made of water – which is the same as the weight of the water that your body displaces. Now the water in the pool does not “know” what you are made of, so B(↑) remains as described above as long as you are fully immersed. Of course, normally, you have air in your lungs, so the weight of water your body displaces is greater than the weight of your body, and as B (↑) is greater than W (↓) so you move upwards towards the surface, reaching a state of no motion (usually partially afloat) where a new buoyant force balances out your weight (still with air in your lungs).
This can be summed up as a simple statement of Archimedes’ Principle: “A body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces”. I can’t be certain that this is the approach Archimedes took, but it might be….