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Can two units having the same dimensions always be used interchangeably? For example $s^{-1}$ and $\frac{rad}{s}$ have the same physical dimension, does that mean we can measure frequency $\nu$ with either one? Another question, can we measure all dimensionally equivalent physical quantities with the same units, so for example, can we measure, torque, torque times theta, and energy all in joules?

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Can two units having the same dimensions always be used interchangeably?

This is an interesting question, and a lot hinges on what you mean by “interchangeably”.

If you simply mean the mathematical operation of substitution then I would say “yes”, at least in SI. In SI a $\mathrm{J}$ is just a shorthand for $\mathrm{kg\ m^2\ s^{-2}}$ and a $\mathrm{N\ m}$ is also just shorthand for $\mathrm{kg\ m^2\ s^{-2}}$. So by substitution mathematically you can always exchange them.

However, physics is more than just math. Physics is an experimental science, so experiments are essential. An experiment that measures torque is different from an experiment that measures energy. So although they have the same SI dimensions, they are not experimentally interchangeable, and the answer would be “no”.

Interestingly the converse doesn’t always hold. For example, the same experiment which measures a SI volt can also measure a CGS statvolt, even though they have different dimensions. So having different dimensions doesn’t preclude interchangeability.

One final issue is that units are used for communication. While mathematically the substitution $\mathrm{J} = \mathrm{kg\ m^2\ s^{-2}} = \mathrm{N\ m}$ is valid, readers will find it jarring to read a torque written in $\mathrm{J}$. It will impede communication to interchange them that way. This is a social and convention issue, and may not follow any rigorous principled rules.

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$s^{-1}$ and $\text{rad}\>s^{-1}$ are not equivalent, note that $1s^{-1}$ implies that one "something" is happening every second, whereas $1 \text{rad}\>s^{-1}$ implies specifically that one radian is passing per second.

Consider for example the clock speed of a computer processor, the frequency with which the clock "ticks" can be measured in Hertz ($=s^{-1}$), as in $100\text{Hz}$ means the procesor is doing $100$ ticks per second. On the other hand, what would it mean for it to be ticking at $100$ radians per second? Clearly the latter choice of units does not make sense here. Illustrating the fact that they are not physically equivalent units.

As for the second part of your question, I believe that is answered here.

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  • $\begingroup$ Referring to the link in your comment Torque is measured in either joule or joule/radian, so how do you explain that? $\endgroup$
    – Jack
    Nov 18, 2023 at 13:44
  • $\begingroup$ This implies to me that dimensionally equivalent units are equivalently acceptable units of measure. $\endgroup$
    – Jack
    Nov 18, 2023 at 13:46
  • $\begingroup$ And the fact that torque and energy can be measured in joule says that the answer to my second question is also yes! $\endgroup$
    – Jack
    Nov 18, 2023 at 13:48

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