Reconciling Units in Classical System Analogies: Why Does Torque Have Units of Energy?

Question

In classical physics we often cast an analogy between translational and rotational systems

Force < > Torque

Energy < > Rotational Energy

Momentum < > Angular Momentum

and considering SI units we have [Force] = N, [Torque] = N-m, [Energy] = [Rotational Energy] = N-m (Joules), [Momentum] = N-sec and [Angular Momentum] = N-m-sec.

Physically this analogy seems to make sense, but if you ponder the units in a simplistic way, questions come up like:

Why does torque, which is an analogy of force have the same units as energy, but force does not?

and

If there are differences in units between the analogy for force and torque, why not also a difference between energy and rotational energy?

Is there a simple way to reconcile these questions, or do you have to step outside classical physics?

Answer 1

This is a side-effect of treating angles as dimension-less.

For translational systems, we have

\begin{align*} [\text{linear momentum}] &= [\text{action}][\text{length}]^{-1} \\ [\text{force}] &= [\text{linear momentum}][\text{time}]^{-1} \\&= [\text{energy}][\text{length}]^{-1} \end{align*}

Correspondingly, for rotational systems, we have

\begin{align*} [\text{angular momentum}] &= [\text{action}][\text{angle}]^{-1} \\ [\text{torque}] &= [\text{angular momentum}][\text{time}]^{-1} \\&= [\text{energy}][\text{angle}]^{-1} \end{align*}

If $[\text{angle}] = 1$, obviously $[\text{torque}] = [\text{energy}]$, even though these quantities are rather different, both from a physical as well as geometrical point of view.

In contrast, translational and rotational energies both contribute to total energy and it doesn't really make sense to introduce a distinct unit for each type of energy.

Answer 2

Torque is a cross product, and work is a dot product. So one big difference is that torque is a vector and work is a scalar. Another way to think about it is that work is a force being applied over a length interval, where only the force applied in the direction parallel to the displacement counts toward the work performed.

On the other hand, torque is best thought of as a force applied _at_a distance away from an axis of rotation. Only the part of the force applied perpendicular to the lever arm distance counts toward torque. These really are two very different concepts, and despite the apparent match-up in units, are not analogous at all. Since in torque, the distance that you are using only states how far away the force is from the axis of rotation, and not how much the rigid body actually rotates, you can see the mismatch. Think of a static situation where a rigid body is experiencing balanced torques. Since the object is not moving, obviously no work is being done, but there are torques on the object (although admittedly not a net torque).

Answer 3

Why does torque, which is an analogy of force have the same units as energy, but force does not?

The magnitude of (net) force is, in a sense, work done per unit displacement thus we can think of it as, e.g., Joules per meter.

In a similar sense, the magnitude of (net) torque is work done per unit angular displacement, e.g., Joules per radian.

But the unit of work is simply Joules so, conceptually, it isn't really correct to say that torque has the same units as work (or energy).

However, they do have the same dimensions since angular displacement is dimensionless quantity.

From the Wikipedia article "Torque":

For example, the quantity torque may be thought of as the cross product of force and distance, suggesting the unit newton metre, or it may be thought of as energy per angle, suggesting the unit joule per radian."

Answer 4

We could standardise Torque at 1 meter and it would then only be a force. It is in fact only a force, but one that can be varied with r (radius). As a vector it has direction, but we do not use this by convention. As engineers, we want to divide this force, or multiply it. Gears, pulleys, chains. The units are the units most suitable for our calculations.

Do not assume that because there is torque, there is rotation.

Answer 5

Torque and work are different physical quantities,so it makes sense to use different units. Since torque is a vector and work is a scalar, one idea would be to use "$\mathrm{N\times m}$" for torque, instead of "$\mathrm{N \cdot m}$". $\mathrm{N\times m}$ would be consistent with torque, or cross product, while work is dot product! Moreover, $\mathrm{N\cdot m = J}$. But it is never easy to change units, as history teaches us.