Explanation of density and floatation

[Thundagere]

We know that an object with lower density than the medium it's in will float, and higher density will sink. Could someone provide proof of this in an equation format? I've been trying to use Bernoulli's theorem, but I feel like I'm doing something wrong.
Thanks.

 

[jbriggs444]

Bernoulli's principle relates the pressure in a moving fluid to the velocity of a fluid. It has nothing to do with buoyancy.

Buoyancy works because the pressure of a [stationary] fluid varies according to the acceleration of gravity, the depth of the fluid and the density of the fluid.

I'm going to dumb this down a bit and hope that it makes intuitive sense...

Consider a shape that is partially immersed in water. Like a boat for instance. Ignore the part that extends above the surface of the water. The outline of the boat encloses a certain area on the surface of the water.

Take the integral of the depth of displaced water at each point within this outline times the incremental surface area at that point. [If you're not familiar with integral calculus, just slice the volume of the boat up into a lot of tiny vertical columns and add up their individual volumes]

Claim 1: This integral is equal to the volume of the water displaced by the boat.

Multiply by the density of the water.

Claim 2: This integral is now equal to the mass of water displaced by the boat.

Now multiply by the acceleration of gravity.

Claim 3: This integral is now equal to the weight of water displaced by the boat.

Consider the upward pressure from the water under each point on the boat.

The pressure at each point under the boat is equal to the depth at that point times the fluid density times the acceleration of gravity. Force over a particular incremental surface area on the boat's bottom is given by multiplying this pressure times the incremantal surface area. The _upward_ component of this force is given by multiplying this pressure times the corresponding horizontal area on the surface of the water.

Claim 4: Integrating this gives the total upward buoyant force on the boat.

Claim 5: This integral is exactly the same as the previous integral

Conclusion: The buoyant force on a [partially] submerged object is exactly equal to the weight of the displaced fluid.