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Re: [Phys-l] bound vectors ... or not





-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu
[mailto:phys-l-bounces@carnot.physics.buffalo.edu] On Behalf
Of John Denker
Sent: Tuesday, September 07, 2010 6:47 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] bound vectors ... or not

On 09/07/2010 02:47 PM, Edmiston, Mike wrote:

www.bluffton.edu/~edmistonm/ConstantTorque.pdf

Very nice.

these are actual devices that have utility in the
engineering of real
products.

Another place where the same line (so to speak) of reasoning
shows up is in aerodynamics. Replace the linear force
actuator by an engine+propeller.
Consider the top drawing as a front-engine design and
consider the bottom drawing as a rear-engine design. The
pitching moment will be the same in either case.

Being able to glance at situations like this and appreciate
the invariances is a valuable skill.

==========

There are about ten good ways to think about this.
One way that appeals to me is to represent the torque as a
bivector. The magnitude of the torque is the area of a
parallelogram, which is invariant with respect to shearing
the parallelogram. YMMV, but for me this is very visual and
very intuitive.

I don't know what you mean by shearing the parallelogram in this context
but neglecting the width of the slot, the parallelogram in the case at
hand is just a rectangle and its area is not invariant but decreases as
theta varies from 25 degrees to 90 degrees and then increases from 90
degrees to 155 degrees.

It's way better than cross products.
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