Re: Torque, moment, and couple

[John Denker <jsd@MONMOUTH.COM>]

At 10:06 PM 4/7/00 +0800, romanza wrote:
> The terms torque and moment are often interchangeably used. Is there a
> distinction between them?

I use them interchangeably.

> I have always seen text often referring to torque of a couple,

Couples will be discussed below.

> and moment of a force (less often torque of a force).

You can associate a torque with a force, if and only if you specify a lever
arm.

=====================
General discussion:

We start with the fundamental relationship between the vectors:
        torque = moment = force × distance

where the distance is the separation vector between the datum and the point
where the force is applied, and where × denotes cross product.  The datum
is an arbitrary point from which we choose to measure distances.

The foregoing expression implies the convenient formula:
        |torque| = |moment| = |force| * lever_arm

where |...| denotes the magnitude of a vector, and where * denotes ordinary
scalar multiplication, and where the lever_arm is a component of the
distance, namely the component perpendicular to the force.

There is a deep law of physics that says that the choice of datum doesn't
matter, as long as you use the same datum consistently throughout your
calculation.  To say it another way, Moe can choose one datum and use it
consistently, while Joe chooses another datum and uses it
consistently.  They will disagree as to the numerical values that appear in
their intermediate results, but they will make identical predictions about
all physical observables.

The importance of choosing a datum is in contrast to ordinary straight-line
dynamics, where laws such as
        F = m a

do not depend even superficially on the choice of reference frame;  Moe and
Joe agree on the numerical values of F, m, and a (assuming we restrict
ourselves to unaccelerated reference frames).

It is sometimes convenient (but never necessary) to choose a datum located
at the center of mass.

===
Couples:

An interesting special case arises when we have two equal and opposite
forces applied at two points separated by a distance.  This situation is
called a couple.  In this case, the choice of datum does not matter even
superficially;  Moe and Joe agree on the numerical value of the torque
produced by the couple:
        |couple| = |torque| = |moment| = |force| * lever_arm

where in this case |force| is the magnitude of either one of the forces,
and the lever_arm is a component of the distance between the two points of
application, namely the component perpendicular to the forces.

As a slightly more general statement, consider a case where all the forces
on the system of interest are in overall balance (i.e. the system is not
undergoing any overall straight-line acceleration) but the torques are not
necessarily in balance (because of the places at which the forces are
applied).  If the forces are in balance, the choice of datum does not
matter even superficially.  (Such a case can be, but need not be, described
as a superposition of couples.)