It seems like the units for the constants are determined just by cancelling units in the equation that aren’t wanted in the answer. This seems unscientific to me though so it feels like there must be a more thorough way
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No, you have it pretty much right. An equation necessarily needs the same units on both sides, so physical constants must have the appropriate units to make that happen.
In practice, since we’re getting physical constants by measuring the quantities in one of the equations in which they appear, we’re deriving their units by multiplying or dividing the units of our measurements. For example, E = (Boltzmann constant)T lets us compute the Boltzmann constant by dividing the internal energy of a gas by its temperature, which necessarily leads to units of joules (energy) per Kelvin (temperature).
Basically we have a couple(7 iirc) of fundamental measurements that, while arbitrary in magnitude, are based of things we currently assume constant, such as the speed of light. Examples are the second, the meter and the kilogram. All other measurements, while they might have their own units, are then combinations of these 7 arranged to be easily legible.
For instance, properly measuring power using fundamental measurements leads to an equation with kilos, squared meters and seconds to the power of -3. This is not intuitive at all, and the solution is then to define a unit of power and name it after a famous scientist, e.g. defining a watt as W=kg(m^2)(s^-3).
While you could call this arbitrary canceling, it is in reality a rearrangement of fundamental constants to make physics more understandable using purpose-made units instead of the fundamental ones.
Generally, by finding the slope of line created when you measure the variables.
Finding G, the gravitational constant in F=G*(m*M)/r^2 involves measuring the gravitational Force created at different separations for a given mass.
You graph this, F on one axis, R on the other an you get a clear trend.
Find the “slope” of this value and that’s your proportionality.
What makes it special is that you keep getting the same number regardless of masses you use or separations. It’s found to be constant for all gravitational systems.
The units are defined by the way you calculated the slope. They tag along with the ‘rise’ and ‘run’ terms.
That’s exactly how it is and it can in effect work as a mathematical proof.
Eg speed is distance ( in m) / time in (s) this gives us a unit of m/s
They’re determined by figuring how much the numerical value change if we change our basic units of measurement.
For example, imagine if we change all measurement to km from m, if the numerical value of the constant decrease by a factor of 1000000 then it must have the unit of length squared.
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