While I agree with Michael that calling Archimedes' Principle an
artifact is too strong, I have a few quibbles with the examples that he
discusses.
Where Michael uses weight, w, I will use gravitational force, F_g,
because the term "weight" means different things to different
physicists, but the term "gravitational force" is unambiguous. Where
Michael uses acceleration of gravity, I will use gravitational field,
because g is an acceleration only in free fall, but gravitational force
is always the product of mass and gravitational field strength: F_g =
mg. In fact, that is my first major quibble: that F_g always equals mg,
because this relationship defines gravitational field, but it equals
GMm/r^2 only if only two objects are interacting gravitationally. If an
object interacts gravitationally with more than one other object, then
the superposition principle must be used in conjunction with Newton's
law of gravity. (I am assuming that the densities and configuration of
the objects are such that Newtonian gravitation is sufficiently accurate
that general relativity need not be invoked.)
Michael's other example is more problematic, because the relationship
a_max = ug is valid only if both the truck bed and the acceleration are
horizontal. This relationship is thus much less general than either
Archimedes' principle or the definition of gravitational field. I
consider the relationship a_max = ug to be an artifact.
Daniel Crowe
Oklahoma School of Science and Mathematics
Ardmore Regional Center
dcrowe@sotc.org
-----Original Message-----
From: Forum for Physics Educators [mailto:PHYS-L@list1.ucc.nau.edu] On
Behalf Of Michael Edmiston
Sent: Thursday, December 09, 2004 1:11 AM
To: PHYS-L@LISTS.NAU.EDU
Subject: Re: [PHYS-L] Buoyancy question
With regard to the idea that a floating object must displace an amount
of water equal to the weight of the floating object, Ken Fox asked (I
assume rhetorically) why it is true. Ken called this an artifact of the
[true] cause of floating which is a pressure difference.
It seems to me that "artifact" is too strong because it almost makes it
sound like "displaced water equal to the weight of the floating object"
is an accident or serendipity.
However, the algebra shows that the object must submerge sufficiently to
displace water equal to the weight of the floating object in order for
the pressure on the bottom of the object to be sufficiently more than
the pressure on the top to make the upward force sufficient to balance
its weight. In other words, the two statements for why things float
amount to the same thing.
The are other cases where the algebra comes out like this and I don't
think we are inclined to call them "artifacts."
For example, if we say weight is mass times the acceleration of gravity,
is this an artifact? Really the weight is GmM/r^2 where G is the
gravitational constant, m is the mass of the object, M is the mass of
the Earth, and r is the radius of the Earth. The acceleration of
gravity is GM/r^2 because when we apply a=F/m to a falling object we use
GmM/r^2 for F, and when we divide that force by m we end up with GM/r^2
for a. Thus weight is really GmM/r^2 but we call it mg.
Don't get nitpicky about other contributions to weight such as buoyancy
and rotation of Earth, etc. That's not the point here. The point is
that the familiar w = mg is mostly shorthand for GmM/r^2 and I am asking
if that means we shouldn't say weight is mass times acceleration of
gravity because that is just an artifact of the true equation for weight
which requires us to invoke Newton's law of gravity.
Here is another. The maximum acceleration a truck can have before a
non-secured object in the bed begins to slide is ug where u is the
coefficient of static friction and g is the acceleration of gravity. Is
this an artifact? The real reason for this number is because the force
of friction is what must accelerate the unsecured object. The maximum
force of friction is uN which is umg, and that has to provide the ma, so
ma = umg and a = ug. So, when we say the cargo slid because the
acceleration of the truck exceeded ug, is that an artifact?
I agree that in all of these cases we want the students to have a
complete understanding of what is going on. I just don't see buoyancy
as being a special case apart from all the other simplified statements
we make such as w = mg and a(max) = ug.
Michael D. Edmiston, Ph.D.
Professor of Physics and Chemistry
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu