Reason for differentiation between inertial and gravitational mass To my knowledge, inertial and gravitational mass are equivalent according to the weak equivalence principle. I believe modern experiments have shown that $\lvert \frac{\mu}{m}-1\rvert\leq 10^{-15}$, but what I still don't quite understand is why there is (or was) a reason to differentiate between inertial mass and gravitational mass.
Was it because Newton measured an anomaly in his pendulum experiments (such that $\omega=\sqrt{\frac{l}{g}}$ wasn't true anymore, but instead was higher [or lower] by a factor of $\sqrt{\frac{\mu}{m}}$)? Or was there a different reason?
 A: Inertial and gravitational mass have different definitions, so only experiments can show, that they are the same.
A: In this anwser:
In the section 'Newtonian mechanics' I give some historical information.
In the section 'Maxwell's equation' I discuss how Maxwell's theory of the electromagnetic field shaped thinking of how (charged) objects are interacting.

Newtonian mechanics
In the context of newtonian mechanics:
In order to be consistent with body of observation data it is necessary that inertial mass and gravitational mass have the same value.
We have that planets of the solar system all have very different mass. The inverse square law of gravity describes the orbits of the planets. If inertial mass and gravitational mass have the same value then for each planet: in the equation of motion the mass drops out of the equation.
In Newton's time it was of course among scholars generally recognized that at the very least inertial mass and gravitational mass are close in value. (Drop two stones with very different size: gravity accelerates them in equal measure, as far as the naked eye can see.)
Newton was aware that a pendulum setup offers a more sensitive assessment of inertial and gravitational mass. (Pendulum swing decays due to friction, therefore Newton constructed pendulums with bobs with equal size (hence equal air friction), but unequal mass.
About the way a pendulum setup is sensitive: take for instance comparison of length from point of suspension to the center of mass of the pendulum bob. Start two pendulums with slightly unequal length. The difference in period is only slightly unequal, but over time that difference accumulates. Over time the accumulation makes the small difference visible to the naked eye.
Newton devoted a lot of effort to verify with the most sensitivity he could achieve that inertial mass and gravitational mass are equivalent. These verification efforts are described in the Principia.

Newton had of course no way of knowing the mass of each planet, but if inertial mass and gravitational mass are equivalent then knowning the mass of each planet is not necessary.
So that is the dependency in the context of newtonian mechanics. The way the planets move can only be accounted for if inertial mass is equivalent to gravitational mass.


The introduction of Maxwell's theory of electromagnetism made it clear that electromagnetism can be well understood as a theory of the electromagnetic field.
In terms of a field theory: in a theory that describes force exerted as mediated by a field: matter is thought of as coupling to the field.
In the case of the Coulomb force:
Both protons and positrons couple to the electric field. And having the same charge the strength of coupling to the electric field is the same for both. But the two have very different inertial mass, and therefore they can be easily differentiated.

Maxwell's equations
Maxwell's theory of the electromagnetic field has showed a way of understanding interactions: to think of the interaction as mediated by a field.
So it was natural to raise the question: does that extend to gravity? Is gravity mediated by a field?
If gravity is mediated by a field then it should not be blindly assumed that the magnitude of coupling to the field and the magnitude of inertia of an object are connected.
In the case of electromagnetism the two are independent from each other; as a possibility it must be acknowledged that in the case of gravity coupling to the gravitational field and inertia are independent properties.

As we know: Einstein made assertion of equivalence of gravitational and inertial mass a matter of principle
To characterize the equivalence Einstein used the german word: 'Wesensgleich'. That word conveys a meaning of gravitational mass and inertial mass not just having the same magnitude, but being intrinsically the same.
What that means in terms of the Einstein Field Equations: theory of gravity and theory of inertia should not be thought of as independent theories. In terms of the Einstein Field Equations it can only be stated as a single theory.

(Clarification: I am referring to 'Einstein Field Equations' in analogy to the name 'Maxwell's equations' for Maxwell's theory of the electromagnetic field.)
