Inertial Mass vs Gravitational Mass

[KnightTheConqueror]

TL;DR Summary

Why fundamentally Gravitational and inertial mass do not have to be equal and just a coincidence if we define the gravitational force using the inertial mass of the objects?

Why does the text saying that the Newton's framework doesn't require the two masses to be equal? If using f = ma give us inertial mass then how is f = Gm1m2/r² a different things? Isn't the law defined as the force is directionly proportional to the product of the masses and we calculated the value of G using the "inertial masses" of the objects and putting them in the equation? Then how is the gravitational and inertial mass having the same value a "co-incidence"?

 

[Ibix]

Newtonian physics supplies no reason that the $m$ in Newton's law should be the same as the one in the second law. It appears to be the case, but there's no reason why. And you can easily imagine other cases - for example, we might find a material tomorrow whose gravitational attraction is half the strength of anything else of that mass. You just change the $m$ in the gravitational force law to $km$ where $k$ depends on the material. That $k$ is always 1 in reality and why that should be is unexplained.

General relativity (and any other metric theory of gravity) does provide a reason - since gravity is spacetime curvature, it can only depend on where you are, not what you are made of.

 

[KnightTheConqueror]

That i understand but I still have a confusion. How did we calculate the value of the universal gravitational constant G? Correct me if I am wrong but isn't the procedure like: We know the force between two objects is proportional to the product of their mass and inversely proportional to the square of the distances between them. And we put the value of force measured in laboratory and mass and distance between them and calculated the value of G. If an additional constant k is added to the equation, replacing m with km, wouldn't that k just get included in the value of G? Because G isn't something which we always knew, we calculated it using the inertial mass we knew about the objects. So by doing this we are assuming or just saying that both are the same things beforehand.

 

[Hornbein]

That i understand but I still have a confusion. How did we calculate the value of the universal gravitational constant G? Correct me if I am wrong but isn't the procedure like: We know the force between two objects is proportional to the product of their mass and inversely proportional to the square of the distances between them. And we put the value of force measured in laboratory and mass and distance between them and calculated the value of G. If an additional constant k is added to the equation, replacing m with km, wouldn't that k just get included in the value of G? Because G isn't something which we always knew, we calculated it using the inertial mass we knew about the objects. So by doing this we are assuming or just saying that both are the same things beforehand.

The constant G was measured by Cavendish, not calculated. You can find a description of his experiment.

 

[Dale]

As @Ibix said, there is no reason in the theory that those masses should be the same. In fact Coulomb's law shows an example of something else: $$F=\frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r^2}$$ This has the same form as the law of gravitation. And there is no reason to suppose that charge is related to mass.

 

[Ibix]

If an additional constant k is added to the equation, replacing m with km, wouldn't that k just get included in the value of G?

Sure, if $k$ is the same for everything (which we strongly believe it is).

If $k$ weren't the same for everything then you'd find objects having different attractions to each other, and you wouldn't be able to encompass that in one universal $G$.

 

[KnightTheConqueror]

Oh you're right. I didn't take it into account that the value of k could be different for different masses. This makes sense, than you