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

[Michael Edmiston <edmiston@bluffton.edu>]

Yes, I agree with with what Bob LaMontagne said (below) but I would say 
quite a bit more.

It seems many of us on this list are most accustomed to figuring torque the 
first way Bob stated... the point about which the torque is figured is 
chosen, and the point of attachment is chosen.  In that mode of calculation, 
we have specified the r vector and we don't need any spatial fix for the F 
vector.  We only need the F vector's magnitude and direction and we do the 
cross product r x F.  But note again that this requires specification of the 
point of attachment.

On the other hand, if we again specify the point about which the torque is 
figured, and then choose the magnitude and line of action of the force, then 
we might say that neither r nor F are fixed, but they are semi-fixed.  The 
force vector itself can be drawn with it's head and tail anywhere along the 
line of action.  The r vector has its start at the point about which the 
torque is figured, but it can have any angular orientation that allows its 
arrowhead to fall on the line of action anywhere within the object.  That 
is, the point of attachment can be anywhere along the line of force, and the 
torque will be the same.

The point is... once the line of action is specified, and the point about 
which the torque is figured is chosen, then r does not become a specific 
vector.  There are infinite r vectors (each with different length and 
magnitude) that will yield the same torque.  I have to admit I was a bit 
stunned when I realized this.  I had never thought of it before.

Does anyone care?  Yes.  Suppose we want to rotate an object through some 
angle with a constant torque.  We can't fix the point of attachment unless 
we intend to vary the direction or magnitude of the sliding force.  But we 
can achieve constant torque if a constant-force linearly-moving drive shaft 
attaches to the rotating object with a radially-sliding point of attachment. 
Perhaps we might be more interested in a rotary motor that drives a linearly 
moving shaft, and the motor provides constant torque, and we want the shaft 
to exert constant force regardless of whether it needs to move 1 cm or 3 cm. 
This is just the reverse of the first example, and we need a sliding point 
of attachment.

This is some pretty neat engineering.  If you are having difficulty 
understanding this, let me know and I can post some pictures on my web 
space.

In case you haven't guessed, I think John Mallinckrodt's statement  is wrong 
(that you can't get the physics correct without specifying the point of 
attachment).

Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton University
1 University Drive
Bluffton, OH 45817
419.358.3270
edmiston@bluffton.edu


--------------------------------------------------
From: "LaMontagne, Bob" <RLAMONT@providence.edu>
Sent: Tuesday, September 07, 2010 2:19 PM
To: "Forum for Physics Educators" <phys-l@carnot.physics.buffalo.edu>
Subject: Re: [Phys-l] bound vectors ... or not

> I don't feel this is a misguided discussion at all. A torque requires an r 
> vector and an F vector. It certainly doesn't matter what the location of 
> those vectors happen to be. However, the vector r is not just picked out 
> of the air. It is generated by two locations: the point where the torque 
> is taken about, and either the point of application of the force or a 
> point along the line of action of the force. Once we have used those 
> points to define r it becomes like any other vector and has only direction 
> and length and can be used with F to define a torque. But without those 
> two original points, it has no meaning.
>
> Bob at PC