In article , Gary Coffman says...

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.

Exactly correct. Torque is defined as the vector cross product of
a force and a distance, R X F, that is, R(vector) (cross product) F(vector).

The R(vector) in this case is from the center of motion to the point
at which the force is applied, and the F(vector) of course points
in the direction the force is applied, which is tangential in
most practical cases. The Torque vector as a cross product winds
up pointing along the axis of the shaft by convention. You need
to use one of those right hand rule things, put left hand in pocket,
point fingers in direction of R, curl them around in the direction
of F, and your thumb points in the direction of T.

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.

Although from a dimensional standpoint the units of work and torque
happen to be the same, torque is of course not a unit of work.

Angular velocity has units of inverse seconds (the r in rpm is
not a real unit) so you wind up with units of force x distance x 1/second
when you muliply torque times rpm. Which looks sort of like joules/sec
or watts or hp. IIRC that mulitiplication is actually a dot
product.

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.

Yep.

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.

That's right, similar to the 'stuck wrench' analogy I gave before.

Jim

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