## Tuesday, June 7, 2011

### MrsDrPoe: Buoyancy

We open up this Thesis Tuesday with a well-known scene from Monty Python and the Quest for the Holy Grail:

Sir Bedevere: Good. So how do you tell whether she is made of wood?

Peasant 1:
Build a bridge out of her.

Sir Bedevere:
But can you not also build bridges out of stone?

Peasant 1:
Oh yeah.

Sir Bedevere:
Does wood sink in water?

Peasant 1:
No, no, it floats!... It floats! Throw her into the pond!

Sir Bedevere:
No, no. What else floats in water?

Peasant 1:

Peasant 2:
Apples.

Peasant 3:
Very small rocks.

Peasant 1:
Cider.

Peasant 2:
Gravy.

Peasant 3:
Cherries.

Peasant 1:
Mud.

Peasant 2:
Churches.

Peasant 3:

King Arthur:
A Duck.

Sir Bedevere:
...Exactly. So, logically...

Peasant 1:
If she weighed the same as a duck... she's made of wood.

Sir Bedevere:
And therefore...

Peasant 2:
...A witch!

So why does wood...or anything else...float? The answer lies in buoyant forces and Archimedes' Principle.

The buoyant force on any partially or fully submerged object is the net (or overall) force placed on the object by the fluid the object is submerged in. If we examine all the forces on a region of fluid around the submerged object, we can determine that the buoyant force acts in the upward direction and is equal to the specific weight of the fluid times the displaced volume of fluid. This is Archimedes' Principle.

Any object submerged in a body of fluid displaces a certain volume water equal to the volume of the object submerged in the fluid. For example, a cylindrical cup is floating in soapy water in the sink. If 2 inches of the 3 inch diameter cup is underwater, 56.52 cubic inches of the cup is submerged. Thus, 56.52 cubic inches is the volume of fluid displaced by the floating cup. We can multiply this volume by the specific weight of the dish water to determine the buoyant force acting on the cup.

Fantastic! But how do we know if an object will float or not? If we apply Newton's Second Law to a stationary floating object submerged in a fluid, we see that only two forces are acting on the object (the buoyant force and the object's weight) and that these two forces must be equal and opposite. So an object will float (fully submerged) if its weight is equal to the specific weight of the given fluid times the total volume of the object or (partially submerged) if its weight is less than the specific weight of the given fluid times the total volume of the object.