Re: are normal reaction and tension conservative ?

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Sat, 30 Jun 2001, Chetan wrote:

> are normal reaction and tension conservative ?
>
> I think it depends on what forces are causing the normal reaction and
> tension(like when two masses are hanging on the opposite sides of a pulley
> then the tension in th rope is due to gravity which is a conservative force
> and so the tension is conservative).
>
> Is this right?

No, but this is a good question--the type only an uncommonly
thoughtful student would be likely even to think of.

First of all, it is misleading to suggest that gravity might
"cause" a tension or normal force and, somehow, confer its
conservative nature to the tension or normal force.  Tensions and
normal forces are (primarily, but see below) the direct results of
strain, however small, in the rope or in the supporting surface.
It matters not at all why the strain exists. For instance, the
strain in a rope might result from attaching one end to the
ceiling and hanging a body from the other, from two people pulling
on both ends, or from a person holding one end and whirling a body
attached to the other end around a circular path. In each case, if
the strain is the same, the tension is the same.

It may also help to understand the following:  Even to be a
*candidate* for being "conservative," a force must be directly
calculable from a "configuration."  For instance, the
gravitational force is determined from the relative positions of
the interacting bodies and the spring force is determined from the
extension of the spring.

In the type of overly simplified treatment usually found in
introductory physics courses, the question of whether or not
tension and normal forces are conservative rarely arises because
it is generally assumed that ropes are inextensible and surfaces
perfectly rigid.  Where there is no displacement there is no work
and, thus, no need to ask about whether or not the force is
conservative.

Of course, bodies can acquire kinetic energy due to the
application of even these oversimplified tensions and normal
forces.  For instance, consider a body on a table in an elevator
that is accelerating either upward or downward or consider either
of the bodies in an Atwood Machine arrangement.  In these cases,
however, the normal or tension forces are simply external "pushes"
or "pulls" on the body and must be treated as nonconservative
since they bear no relationship to any "configuration" of "the
system."

On the other hand it's easy to relax the "inextensible" and
"perfectly rigid" assumptions.  After all, an extensible rope is a
"spring" and a nonperfectly rigid surface is a "trampoline"--a
more complicated "spring."  Then, one may be able to calculate the
force from the strain and consider the force to be conservative.
Note carefully, however, that the conservative nature is connected
with an ability of the spring or the trampoline *itself* to store
potential energy; that energy should not be considered to be
stored in the supported body or even in the interaction of the
supported body with the spring or trampoline.

Finally, note that real ropes and real surfaces are generally at
least somewhat "dissipative."  They exert forces that are not
completely calculable from their configurations and may exhibit
hysteresis and/or velocity-dependence.  For instance, a rope may
(and usually does) exert a stronger pull at each specific length
while being extended than when it is subsequently allowed to
relax.  As a result, the work done *by* an external agent to
extend the rope is smaller than the work done *on* the external
agent as it is allowed to relax.

John Mallinckrodt      mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm