On 04 Oct 2003 02:29:49 GMT, a (Dave Baker) wrote:
Torque bears no relation to any capacity to do work.

Well, it has a relationship. Power is defined as the capacity to do work,
and torque times RPM is power. But power is the independent variable
here. Torque and RPM can be traded off in any fashion as long as their
product equals the input power.

Although the two
quantities are expressed in the same units of force and distance, torque is a
vector quantity and work is scalar one.

Hmmmm. Force is a vector. In the definition of mechanical work given in
Machinery's Handbook, the distance used to calculate work is the total
displacement *in the direction the force is applied*. I think that makes it
a vector too. If I recall my elementary physics correctly, it doesn't matter
what path is taken to the final position when calculating energy states
(and work is alternatively defined as the difference between the beginning
and ending energy states of a closed system). That's consonant with the
summation of an arbitrary number of vectors, so the distance should
indeed be considered a vector quantity.

Now if distance were considered scalar, there'd be no question, a vector
times a scalar is a vector, and so work would be a vector quantity. But if
the distance is considered a vector too, then there are two possibilities.
The cross product of a vector and another vector is a vector. But the dot
product of two vectors is a scalar.

In this particular case of aligned vectors, the *magnitude* of the answer is
the same, but one resultant is vector and the other scalar. So we are left to
wonder why we would apply the dot product instead of the cross product in
calculating work. I'd be interested in hearing others' thoughts on that.

The question of torque is less ambiguous. It is the product of the applied
force, and the length of the lever arm. If the latter is a simple scalar,
a vector times a scalar is always a vector. Also, since the lever arm is at
right angles to the force, if it were treated as a vector, the cross product of
a unit length and a unit force would be zero, which would be inconsistent
with the notions of torque, though it would be consistent with the notion
of torque not being the equivalent of work.

The lever arm's length doesn't fit the definition of the distance used to
calculate work given above (not aligned with the direction of the force).
So we can say that by definition torque is not the same thing as work
regardless of whether we decide work is vector or scalar.

Note too that the torque on a shaft is the same whether the shaft is
turning or stationary. For a torque to do work, the shaft has to turn,
to provide a distance over which the force can act, and then we have
to bring in the notions of RPM and power to give us a rate and capacity
for doing work.

Gary