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Re: [Phys-l] basic laws of motion +- vectors +- angular momentum



I'm not sure number 2 is correct. Consider a uniform rod with two forces, equal in magnitude, applied at the ends and perpendicular to the central axis of the rod.

If the forces act in the same direction, then angular momentum will be conserved about the center of mass, but linear momentum will not be conserved.

If the forces are applied in opposite directions (a force couple) then linear momentum will be conserved about the center of mass, but angular momentum will not be conserved.

Bob Carlson


--- On Tue, 12/9/08, John Denker <jsd@av8n.com> wrote:

From: John Denker <jsd@av8n.com>
Subject: [Phys-l] basic laws of motion +- vectors +- angular momentum
To: "Forum for Physics Educators" <phys-l@carnot.physics.buffalo.edu>
Date: Tuesday, December 9, 2008, 12:51 PM
Hi Folks --

I'm conflicted about the following:

1) As I see it, Newton's third law implies conservation
of
momentum ... and vice versa. They're equivalent. So
far
so good.

Conservation of momentum means conservation of *linear*
momentum.

2) Conservation of angular momentum implies conservation of
linear momentum ... but not vice versa. Hint: consider
angular momentum about an axis infinitely far away.

3) Combining (1) and (2), I don't see any way, starting

from the usual statement of Newton's laws, to derive
conservation of angular momentum ... not without
additional assumptions.

4a) In a previous discussion of this topic, somebody said
we should start with point particles, and assume all
forces are central forces. Then larger objects can be
built out of point particles plus forces of constraint.
I guess that would be logical, but I'm not convinced
it
is physically correct. For example, consider the
magnetic force between two current-carrying wires, or
the force on an electron in a betatron, or anything
involving the Poynting momentum. It's not at all
obvious that these forces are pointlike or central.

4b) An alternative would be to just introduce conservation
of angular momentum as a fourth law of motion. The third
law would retain its usual form and apply only to linear
momentum. As such, it would be demoted to a mere
corollary of the fourth law ... in much the same way as
the first law is merely a corollary of the second law.

I'm happy to retain the first law and the third law
because of their historical importance and pedagogical
usefulness ... but that doesn't prevent us from
superseding them with more powerful and systematic
laws (the second and fourth).

4c) Of course if we were serious about formalizing
things properly, we wouldn't start with Newton's
laws. We would start by writing down a Lagrangian.
That would tell us what's conserved and what's
not.
Excellent reference:

http://groups.csail.mit.edu/mac/users/gjs/6946/sicm-html/index.html

http://groups.csail.mit.edu/mac/users/gjs/6946/sicm-html/book-Z-H-3.html

=================

Here's another way of looking at the whole issue. This
may explain why it is hard to think clearly about this
topic.

We need to carefully distinguish
-- natural, physical reality
-- our intuition
-- our formal models and equations

Just because something is true physics-wise doesn't
mean
it is logically derivable formalism-wise. Gödel had
something to say about this.

A) In this case, we formalize force as a vector. Formally,
vectors have direction and magnitude, period.

B) Meanwhile, I suspect that many of us, and many students,
have an intuitive force-like notion that involves
direction,
magnitude, and *point of attachment*.

There is a huuuge difference between (A) and (B), as we
can see from the following diagrams:

<----------
action (A)

---------->
reaction (A)

=================================================


<---------- ---------->
action (B) reaction (B)


You can see that diagram (A) conserves momentum and
upholds the usual vector-based version of Newton's
third law, but violates conservation of angular momentum.

Meanwhile, diagram (B) upholds conservation of angular
momentum as well as conservation of plain old linear
momentum.

The point is that in terms of vectors, strictly speaking,
there is no difference between diagram (A) and diagram (B).
Vectors have magnitude and direction, period.

If you want to talk about something having magnitude,
direction, and point of attachment, you need a bivector:

torque = force /\ lever_arm

So ... as far as I can see, so long as the third law is
formalized in terms of forces i.e. vectors, the formal
law does not capture the full physical reality.

Or am I missing something?

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