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Torque and energy have the same dimensionality, and so logically are measured in the same units (joules). However, it seems more natural to call the unit for the former a “newton-metre”, because torque does not feel like a form of energy, even though this elaborately described unit means no more or less than a joule.

I could not think of another example of two quantities physically quite different in character that would be measured in the same units, but I expect that there are others. Any examples, please?

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    $\begingroup$ While it is not discussed in the introductory class, there is another distinction between quantities which is just as fundamental as dimensions. Torque and enegy have different tensor character (energy is a saclar and torque is a pseudo-vector), so you should not think of them as being equivalent and should not conflate their units. A "joule" shold be understood as a scalar Newton-meter, while the un-named unit of torque is a pseudo-vector Newton-meter and these are different things. $\endgroup$ Oct 17 '19 at 16:46
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    $\begingroup$ @dmckee That is a extremely useful comment. I had never thought about it that way. Is there a way to indicate the tensor character of a given unit? $\endgroup$
    – user137661
    Oct 17 '19 at 17:42
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    $\begingroup$ @SV I've never seen a notation for stating tensor character explcitly. $\endgroup$ Oct 17 '19 at 18:01
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    $\begingroup$ There isn't really any such thing as scalar torque. The simplified version of torque that is used to introduce the notion is the same as the real thing (and thus a pseudo-vector), it just has it's direction implicit rather than explicit (and it's limited to cases with constant direction because of that). But most students taking the introductory course aren't yet ready to understand what "tensor character" means, so the whole thing is let pass without comment. $\endgroup$ Oct 17 '19 at 20:09
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    $\begingroup$ If you multiple a torque by an angle, you get energy. So the fact that torque and energy have same units is closely related to the fact that we can measure angles by unitless quantities. So, if torque and energy are a valid answer, then so is also any two quantities that are related to each other by an angle. $\endgroup$
    – JiK
    Oct 18 '19 at 9:03
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Angular momentum and action have the same dimensions, and Planck’s constant quantizes both.

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    $\begingroup$ Interestingly (in light of dmckee's comment on the question) this is another example where the quantities have the same unit-dimensions but different tensor dimensions (pseudo-vector for angular momentum and (I think?) scalar for action). $\endgroup$
    – Dast
    Oct 18 '19 at 11:06
  • $\begingroup$ Thinking of "action" more generally as generalizedMomentum*generalizedDisplacement, then AngularMomentum has units of "action/radian". $\endgroup$
    – robphy
    Oct 18 '19 at 15:16
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The pair of frequency (Hertz) and radioactivity (Becquerel) might fit your requirements. While the former describes a periodic phenomenon, with equal time between events, the latter describes a statistical process, mainly in radioactive decay. Both units translate to one per second.

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A few other examples:

Kinematic Viscosity and Diffusion Coefficients both have dimensions of Length$^2$/Time.

Pressure-driven Permeability has dimensions of Length$^2$, the same as area.

There are undoubtedly many more, and I don't think there's any deep underlying message.

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  • $\begingroup$ Kinematic Viscosity is just a measure of the diffusion of momentum, so I think that's not an example of unrelated quantities. $\endgroup$
    – Rick
    Oct 18 '19 at 13:34
  • $\begingroup$ Energy-density has the same dimension as pressure, and that does have an "underlying message" insofar as I dropped my jaw when finally understanding how solar wind can push, or how the Bernoulli effect becomes evident $\endgroup$ Oct 18 '19 at 16:51
  • $\begingroup$ @Rick : True, they're not totally unrelated. However, thermal diffusivity also has dimensions of Length^2/Time, and isn't a measure of momentum flow. There isn't anything to necessarily conclude or not conclude from equivalences of dimensions. $\endgroup$ Oct 19 '19 at 3:22
  • $\begingroup$ @Hagen von Eitzen : That's a good example. But I'm not saying that there is never any deep connection, just that in general there need not be. $\endgroup$ Oct 19 '19 at 3:29
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Some other examples include specific heat and specific entropy (J/(K*kg)), angular speed and frequency (or for example specific growth rate) (1/s), and concentration and density (kg/m^3).

Not so coincidentally all of these physical quantities with the same units are closely related to each other and can be converted into the other by the means of some numerical factor or factor with dimensions that cancel out. For example converting from torque to work requires multiple by the angular displacement which has units of m_arclength/m_radius.

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Time and Specific Impulse are both measured in seconds.

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There are several unitless quantities:

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In Gaussian units in electrodynamics (https://en.wikipedia.org/wiki/Gaussian_units), there are some coincidences:

  • Capacitance is in $\text{cm}$, just as length
  • Resistivity is in $\text{s}$, just as time
  • Conductivity is in $\text{cm/s}$, just as velocity
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