Re: Torque, moment, and couple

["Glenn A. Carlson" <gcarlson@MAIL.WIN.ORG>]

> Subject: Re: Torque, moment, and couple
> Date: Sat, 8 Apr 2000 22:06:01 -0700
> From: Leigh Palmer <palmer@SFU.CA>
>
> [snip]
> How would
> one use the quantity r x (m * v) in any frame other then the one in
> which v = dp/dt?

v = dp/dt??

> One more thing should be mentioned regarding the physical nature of
> torque. Torque is not, strictly speaking, a vector quantity. Vector
> quantities are associated with an intrinsic direction. No one has to
> explain which direction is meant when specifying a force or velocity,
> both of which are vector quantities. Torque, like angular momentum
> and magnetic field, is axial, but not directional. In Gibbsian terms
> it is a "pseudovector".

More accurately, vectors (aka polar vectors; e.g., momentum) are
distinguished from pseudovectors (aka axial vectors; e.g., angular
momentum) by their behavior under improper coordinate transformations
(i.e., coordinate transformations involving inversion).  Polar vectors
are invariant under improper coordinate transformations; axial vectors
are not invariant.

For example, under a 3-d inversion transformation, the coordinate axes
change direction.  But the components of a polar vector also change
sign, so the transformed polar vector is the same as the original
polar vector.  On the other hand, the components of an axial vector do
not change, so the transformed vector is not the same as the original
vector.

The relationships of axial and polar vectors and coordinate
transformations also explains the results of the 1957 experiment of
the beta decay of  polarized Cobalt-60 that demonstrated parity is not
conserved in weak interactions.  Cobalt-60 nuclei were placed in a
magnetic field and cooled to 0.01 K.  The interaction of the cooled
Co-60 nuclei with the magnetic field aligned the nuclear spins
parallel to the field direction.

Under the parity transformation the position vector is reflected in
the origin, i.e., x --> x' = -x. (Note that this is not a coordinate
transformation in the sense described above since it is the vector
itself not the coordinate axes which are transformed.)  The parity
transformation reverses all particle momenta (polar vectors) while
leaving their orbital angular momenta (r x p) (axial vectors) and spin
angular momenta (analogous axial vectors) unchanged.

Parity invariance would require that electrons would be emitted from
Co-60 in the forward direction (parallel with the nuclear spins) at
the same rate as in the reverse direction (antiparallel with the
nuclear spins).  This is not what was observed.

Glenn A. Carlson, P.E.
gcarlson@mail.win.org