Is Joule equivalent to Joule/radian?

Question

The work a constant torque does when it moves a particle through an angel $\theta$ is given by $\tau \theta$. The work is measured in Joule, so the torque should be measured Joule/radian, but the torque also is measured in Newton metre which equals in turns to Joule. So, does that mean Joule=Joule/radian?

Answer 1

A radian has no physical dimension, it's a pure number. So, if you do dimensional analysis,

\begin{equation} \dfrac{[torque]}{[angle]} = \dfrac{[torque]}{1} = [torque] \ . \end{equation}

Recalling the definition of torque and the definition of work, it's easy to realize that they have the same physical dimensions,

\begin{equation} [energy] = [torque] = [force][length] = \dfrac{[mass][length]}{[time]^2} [length] = \dfrac{[mass][length]^2}{[time]^2} \ . \end{equation}

Note. Among all the possible ways to measure angles, radian is considered as "the most natural way" to measure angles.

Answer 2

A radian is not a unit in the same sense that a second or a metre is a unit. The last two are defined using special standardising items (e.g caesium light sources) whereas the radian is defined as the ratio of arc length to radius when these are equal and it is independent of what unit these lengths are measured in, provided the same is used for each. This is related to time and distance being quantities with dimensions, whereas angle is dimensionless.

One could even argue that the radian is redundant as a unit; we usually say that the ratio of a circle's circumference to its radius is $2\pi$ instead of saying $2\pi$ rad.

But you have drawn attention to a case where 'radian' serves a purpose ... With the intention of saving confusion the convention is to leave the unit of torque as N m rather than writing it as J. The latter is reserved for work and energy, where the relevant distance is in the same direction as the force, rather than at right angles to it. But, as you've said, the work done by a torque $\tau$ acting through an angle $\theta$ is $\tau\theta$, so if we left $\theta$ as a unitless number, we'd be forced to say that $\tau=\text{work}/\theta$ has the unit J. Because this would go against the convention mentioned earlier, we fall back on the radian and write the unit of torque as J rad$^{-1}$. It is an alternative to N m.