Hi

I was wondering why things float (leading up to buoyancy) from a physics point of view. My viewpoint is that something floats when it is less dense than water. But i was wondering how floating is explained from a forces point of view.

I have looked at Archimedes principle and this says 'Any floating object displaces its own weight of fluid.' I was not aware that floating objects displaced any fluid. I naively thought a floating object simply sat on top of the liquid. I do have more questions but I may as well wait until someone clears that misconception up for me :)

thanks

Let rho2 = density of a material 2 with property that rho2 < rho1

Consider a cylinder of water having a base of 1 m^2 and height h.

It's volume is V=h*1=h

What is the weight of the water in this cylinder? It is W1=rho1 * V = rho1*h

Pressure = force/area

What is the pressure of the water on the bottom of the cylinder? P=F/A=W/1 = rho1*h

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The pressure at d meters below sea level is rho1*d

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Take a cylinder made up of material having density rho2 (same base of 1 m^2)

It's weight is W2 = rho2 * h < W1

Submerge it completely under water and let go.

There are 3 forces on the cylinder.

1) Its weight, -W2

2) The pressure of the water on top of the cylinder pushing down = -Ptop = -Ft*A = -Ft

3) The pressure of the water on bottom of the cylinder pushing up = +Pbot = Fu*A = Fu

Total force on cylinder of of height, h, is -W2 -Ft + Fu = -W2 + (Pbot - Ptop)

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Here is the tricky part that maybe Archimedes realized.

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Whatever is the top water pressure on the top part of the material 2 cylinder, obviously the water pressure at the bottom of the cylinder is greater, since if I had the same cylinder filled with just water, then the weight of that cylinder is (see above):

W1=rho1 * V = rho1*h

The pressure at d meters below sea level is rho1*d

If the top of the material 2 cylinder is at depth d1, and the bottom is at depth d2, then the difference in water pressure is (rho1*d1 - rho1*d2) = rho1*(d1 - d2)

But (d1 - d2) is just the height of the cylinder

The difference in water pressure between top and bottom surfaces is just rho1*h

Total force on cylinder of of height, h, = -W2 + (Pbot - Ptop) = -W2 + rho1*h

= - rho2 * h + rho1 * h = h * (rho1-rho2) > 0

since rho2 < rho1

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Now that you see there is an upward force when completely submerged, how high will the cylinder move?

When the total force on the cylinder is 0 it is in equilibrium

Since the top of the cylinder is no longer under water, Ptop = 0

If the cylinder is in equilibrium, and is b meters below sea level, then the pressure difference is just Pbot = +rho1 * b

So, we have in equilibrium net force = 0 = -W2 + rho1 * b = - rho2 * h + rho1 * b

Or,

rho2 * h = rho1 * b

b = h * (rho2/rho1)