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Re: [Phys-l] Definition of upthrust or buoyancy

-----Original Message-----
From: [mailto:phys-l-] On Behalf Of M. Horton
Sent: Tuesday, October 19, 2010 8:20 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] Definition of upthrust or buoyancy

I agree that the original question may not have been asking what it
to ask. So, I'll ask what I think the original questioner was
intending to
ask because I've long had the same question. (Wow, that was

Where does the buoyant force come from?

I consider the buoyant force to be the net force exerted on an object by
a fluid in which the object is suspended.

To find the buoyant force, for each infinitesimal area element on the
submerged surface of the object, define an infinitesimal area vector
having magnitude equal to the area of the area element and direction
perpendicular to the area element an outward, away from the object,
toward the fluid. Now, for infinitesimal area element, multiply the
negative of the area element vector by the pressure (considered a scalar
in this model) and sum the results. That vector sum, is what I call the
buoyant force of the fluid on the object. Archimedes law does this
vector integral for you in cases where every bit of the submerged
surface of the submerged object is in contact with the fluid.

If I replace the word fluid with "agent" to mean the entity in question
that is exerting the force, then I can replace the word "buoyant" with
"normal" and the same argument applies to what I mean by the expression
"the normal force" exerted on an object by a surfaces with which it is
in contact. The same idea applies to the force exerted by a rope tied
to a frictionless hook. For a frictional force, the direction of the
force on an area element is not completely determined by the orientation
of the surface element so I associate the direction with the stress
rather than the area element (that can be done for the case of the
pressure aka. normal stress exerted by a fluid as well but the
orientation of the area element does determine the direction of that
stress so I choose, at least this time, to think of the pressure as a

It would be better to call what I call the buoyant force the net contact
force exerted by the fluid on the object. Likewise, the normal force
exerted by a crate by the floor on which the crate is resting should
probably be called the perpendicular-to-the-floor compenent of the
contact force exerted by the floor on the crate, and the centripetal
force exerted by a piece of string on a ball tied to the string when the
ball is moving in a circle centered on the other end of the string
(which is fixed), the force exerted by the string on the ball. If
everybody did that, then my students would probably not draw a free body
diagram of the ball depicting both a string tension force and a separate
"centripetal" force acting on the ball, on a free-body diagram of a rock
hanging at rest near the surface of the earth from the end of a string
they would probably not include a normal force in addition to a string
force and a gravitational force, and they would probably not include
both a PA force and a buoyant force on an object submerged in a fluid.

However, we are stuck with the expressions buoyant force, normal force,
and centripetal force; such terms appear in my students' work whether or
not they have ever heard them from me.

Given that I do use the expression buoyant force, I use the definition I
have put forth in this discussion because I like to associate forces
with interactions. I consider the buoyant force to be one of the two
forces in the interaction between the submerged object and the fluid.
The other force in that interaction is the net force exerted by the
submerged object on the fluid. In John Denker's example of the object
on the scale under water, with the dewetted interface between the scale
and the object, what he calls the buoyant force is the vector sum of the
net force of the fluid on the object, and a large fraction (all but 1 N
upward) of the normal force exerted on the object by the surface of the
scale. I think such use of the term "buoyant force" tends to reinforce
the misconception that the buoyant force is some new kind of force like
the new kind of force students tend to think the centripetal force is.

From a practical matter, if a salvage operator thinks that the force
displayed by the underwater scale is indicative of how hard she has to
pull up on the object with a cable to dislodge the object from the
scale, and chooses her cable strength accordingly, she is going to be
sadly mistaken.

In the end, this seems to be one of those big-endian little-endian
discussions. The way you define "the buoyant force" is of no great
relevance. I like Carl Mungan's take on it--if it is to be decided at
all it should be decided on pedagogical grounds. I'm guessing we on
this list won't even be able to come to a consensus on it.

One of the cons with my definition is that for the case of a dart stuck
to a window by a toy dart gun, assuming no air between the glass of the
window and the synthetic rubber of the suction cup, the buoyant force is
a vector with a huge component toward the window and a tiny upward
component. In other words, the buoyant force is almost horizontal.
Folks like to think of the buoyant force as being upward. Then there's
the suction-cup dart stuck to the ceiling. It is totally submerged in
the air and the upward force exerted on the dart by the air is much
greater than the weight of the displaced air. Folks like to think of
the buoyant force as being (upward but) equal in magnitude to how hard
the earth would be pulling (with the gravitational force) downward on
the amount of air that would be in the space occupied by the dart if the
dart wasn't there.