The best way I can describe what I mean in the title is with examples. When you look at the Bernoulli equation, specifically whn written as $\frac12 \rho v^2 + \rho gh + p = \rm{constant}$, or our process of analyzing heat engines with $p$-$V$ diagrams, we can understand pressure as the amount of work that can be done by a given volume of fluid, since $\rm{pressure=\frac{force}{area}=\frac{force×length}{area×length}=\frac{energy}{volume}}$.
This is also seen in the ideal gas law: $$\require{cancel} p(\mathrm{pressure})V(\mathrm{volume})=n(\mathrm{\cancel{moles}})R\left(\mathrm{\frac{energy}{\cancel{moles}×\cancel{temperature}}}\right)T(\cancel{\mathrm{temperature}})$$ $$\implies$$ $$pV\ (\mathrm{pressure×volume})=nRT\ (\mathrm{energy})$$
And there are arguably weirder ones. Energy and torque have the same dimensions, force×distance or mass×distance$^2$×time$^{-2}$. Angular momentum, torque×time, then has the same dimensions as $\rm{energy×time}$. And it just so happens that the Planck constant $h$, measured in Joule seconds, is used to relate energy and frequency for light, and the reduced Planck constant, which is just $h/ 2\pi$, is a quantity of angular momentum, so $$\hbar(\mathrm{angular\ momentum = torque×time})\ \cdot2\pi=h(\mathrm{energy×time}).$$ Now I don't think it's a good idea to take from this that energy and torque are actually the same, and I"m not even sure $h$ and $2\pi\hbar$ can actually be considered "equal" quantities in the most literal sense, since they are used for different things. But this is still an interesting connection. Are there others like this, where quantities that have similar or identical dimensions can be connected conceptually? And how useful is all of this?
