Re: Tension, scalar or vector?

[EMAIL PROTECTED] (Peter R. Oakfield) wrote in message
news:<[EMAIL PROTECTED]>... 

> Hi. I am in a big argument regarding tension, like in a cord or
> string. Some knowledgeable people tell me it is a vector; others say
> it is a scalar. The implications are important. I think it is a
> scalar, because it has no unique direction and I cannot imagine the
> meaning of negative tension. Who is right? Could someone please help
> me?

Tension is a scalar. You can always write the force of tension
along a rope as T*t. Where t is the unit vector tangen to the rope.
Since force is a vector and t is a vector, then T, which is the magnitude
of the tension or just "tension", must be a scalar.

Another quick way to see that is to note that tension enters as
a scalar Lagrange multiplier in the action formulatino of the dynamics
of an inextensible rope/string/chain. It enforces the condition of
inextensibility.

The caveat here is that t is not unique, -t works just as well. The
tangent vector is uniquely defined only once a parametrization is
chosen for the curve describing the rope. This is probably related
the fact that geometrically the tangent to a curve is not a vector
but a line! Some times it's also called a "director". Mumble muble...
traceless symmetric tensor representation of the rotation
group ... mumble mumble.

Hmm, negative tension... I don't see a reason why this can't exist,
at least in theory. Consider for example an inextensible rope
positioned perfectly vertically on the ground on one end, and supporting
a weight on the other end. 
                                         weight
                                       *
                                 rope  |
                                       |   ground
                                    --------

If negative tension is allowed, then this system is in equilibrium.
Since the tension is negative, the rope exerts a force of tension
on the ground at one end, and a force of tension that supports the
weight on the other end. This situation can be seen as the limit of
the same configuration of a flexible spring, in the limit as the
spring stiffness goes to infinity.

However, this configuration is obviously unstable to the slightest
disturbance. I would imagine that any configuration of a rope with
negative tension is unstable, as the rope would prefer to bend itself
than be in that situation. So in real life they do not occur. That
would explain why it is hard to imagine.

Hope this helps.

Igor