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# Re: PHYS-L Digest - 1 Feb 2004 to 3 Feb 2004 (#2004-34)

On Tue, 3 Feb 2004, Automatic digest processor wrote:

Quoting Brian Blais <bblais@BRYANT.EDU>:

I was wondering if anyone knew of a calculation showing that the kinetic
energy of an object in a circular orbit is equal to half of the potential
energy, where the calculation does *not* use acceleration or force at all.
Is there an argument for this based purely on energy concepts?

0) Be careful, the result as stated only applies to objects
orbiting in a 1/r potential (although generalizations are
possible to other power laws).

I was assuming a gravitational situation, and wasn't clear in my original
post.

1) Certainly it is possible to derive the viral theorem
without mentioning acceleration. The standard derivations
don't mention it. Indeed you don't need to know the masses
of the particles involved.

Look at
http://math.ucr.edu/home/baez/virial.html
about halfway down, in the section called "the proof".

Actually, this derivation refers to force. Perhaps I should have asked, in
the simple case of a 1/r potential, one small mass orbitting a much larger
one, is there a simple way of obtaining the KE=1/2 PE relationship?

3) Why do you care, anyway? If you know the potential,
you implicitly know the force, and conversely if you know
the conservative force you implicitly know the potential,
plus or minus an arbitrary gauge term.

Good question! I teach an algebra-based, 1-semester intro physics course.
This course has no prerequisites, is not followed by further physics courses,
and may be (perhaps) the last science class the students will ever take. As
such, it doesn't fit into the mold that most textbooks use. Because of
limited time, I have to be very selective in the topics that I can cover.
Others would perhaps disagree with my choices, but I am certainly open to
considering other options. I found that covering energy and momentum, pretty
much in 1-D, was a way that I could link classical physics, relativity and
quantum mechanics without introducing a huge number of new concepts each
time. I found that vectors took too long to cover well, so I don't cover
force at all: everything is in terms of energy. I do cover acceleration, but
I am trying to see if there is any way I can get away without that to. For
example, I can cover near-surface gravitational potential energy, usually
denoted U=mgh, without mentioning that "g" is an acceleration by writing it in
units of J/(m*kg), and describing it as the energy needed to lift 1 kg 1 meter
high. Accleration can also be covered in terms of changes in momentum over
time, which of course is equivalent to the standard kinematic way of
describing acceleration.

I feel that if there are only a few rules, conservation of
energy/momentum/etc., used in many different circumstances, then the students
will gain an appreciation for the *simplification* that physics descriptions
entail.

Kepler's law, which is high on my list of priorities to teach, is usually
derived from acceleration and force, and not with energy. If I had a simple
derivation of the KE=1/2 PE relationship, then Kepler's law follows
straightforwardly. Lacking this, I could decide not to teach Kepler's law,
cover enough of the other concepts (acceleration due to gravity, and force) to
come up with Kepler's law, or assert the KE = 1/2 PE relation is either
empirical or "has a derivation beyond the scope of the class".

thanks,

Brian Blais

-----------------

bblais@bryant.edu
web.bryant.edu/~bblais