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?
Best Answer
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.