### Re: Why do less dense things float and more dense things sink?

Date: Tue Jan 30 20:06:58 2007
Posted By: Joel Chapman, Undergraduate, Mechanical Engineering, NC State
Area of science: Physics
ID: 1169513280.Ph
Message:
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The whole reason that buoyancy works the way it does is because of the
balance of forces.  On any object, the change in momentum equals the force
on the object multiplied by the amount of time that the force was applied.

deltaP = Fnet * deltaT

For simplicity's sake, let's look at a block which is more dense than
water.  This block is 1 meter x 1 meter x 1 meter, and let's assume it is
twice as dense as water (water is 1000kg/m^3, so this block is 2000kg/m^3).

Fluid pressure at any depth acts normal to the surface and equals

fluid.density * gravity * depth   (fluid density is denoted by the greek
letter rho)

So, let's put this block's top at a depth of 3 feet, which puts the bottom
at a depth of 4 feet.  The pressure caused by the water at the top of the
block is

1000kg/m^3 * 9.807 m/s^2 * 3 meters

The pressure across the bottom is

1000kg/m^3 * 9.807 m/s^2 * 4 meters

Which means that the pressure across the bottom is greater than the
pressure across the top by the amount

1000kg/m^3 * 9.807 m/s^2 * 1 meter

This is true no matter where you put the block.  The difference between the
top and the bottom for this 1 x 1 x 1 block will always be that.

This value is 9,807 N/m^2 (that's Newtons per square meter)

Since each surface is 1 m^2, that correlates to a difference of 9,807
Newtons of force between the bottom and the top, with the bottom having the
higher force.  If no other forces acted on this block, the block would rise
because the net force is acting upward.

However, the block has mass.

For this block which is double the density of water, the force acting on it
by gravity is

rho * g * V

That's density times gravity times volume.

Which is 2000 kg/m^3 * 9.807 m/s^2 * 1 m^3

So the downward force caused by gravity is 19,614 N.

We determined earlier that the upward force caused by the water pressure
came out to a net upward force of 9,807.

So our net DOWNWARD force is 9807 because 9807 + (-19614) = -9807

No matter what the shape of the object is or it's size, it always works out
this way.  An object which is more dense than the fluid it is in has a net
downward force.

And you can see that for an object which is less dense, the net force will
be upwards...

For a 1 x 1 x 1 block of density 500 kg/m^3, the force acting on it by
gravity is

500 kg/m^3 x 9.807 m/s^2 x 1 m^3

Which is 4603.5 Newtons of force downward.

However, the net force caused by fluid pressure is the same as it was for
the other block, a net upward force of 9807 N.

9807 + (-4603.5) = 4603.5

Which means that the net force is upward.  The block will move upward.

This also works out for any shape object.

From this, it is also apparent that for an object with density equal to the
fluid it is in, it will sit still.  The downward force caused by gravity
would be equal to the upward force caused by the difference in pressure,
and the result would be 0 force.

References:

R.C. Hibbeler "Engineering Mechanics:  Statics, Tenth Edition" pg 484-487

Frank M White "Fluid Mechanics 5th edition" pg 89-93

The Fluid Mechanics book does the solution based on integrations and
differential areas, so I simplified it down a bit so that you could "see"
what was going on.

```

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