Reason for separate concepts of gravitational vs inertial mass

[rem1618]

My mechanics prof today said when setting GMm/r² = ma, the canceling of the small m is actually a bit nuanced because you have to assume the gravitational mass is equal to the inertial mass (though it's supported by experiments). I'm so used to seeing mass as just mass so I'm having a bit of trouble understanding the need for separating the two.

He also said mass can macroscopically be defined by the acceleration caused by an applied force. Is this where the separation came from, the fact that one mass is defined by the gravitational force, and one by the inertial force?

If they weren't equivalent, what implications would there be? Assuming F = ma is universal, would the gravitational force be a "subforce"? Or perhaps the gravitational mass is a "submass"? He also brought up people who were looking for the 5th fundamental force, but I didn't understand the context so I don't know what it was really about.

 

[UltrafastPED]

We have two different equations which can be used to define mass - Newton noticed this long ago.

The first is inertial mass from F=ma. It has nothing to do with gravity - it even works in zero-g environments.
The second is Newton's Universal Law of Gravitation which describes the force between two masses. This also works pretty well until you get into situations which require General Relativity.

Physicists have toyed around with this ever since then. The equivalence of these two methods lead Einstein to his famous "Equivalence Principle": http://en.wikipedia.org/wiki/Equivalence_principle

 

[mfb]

If you cancel in m in GMm/r² = ma you get GM/r² = a. On the surface of earth, G, M and r are constant, so a is the same for all objects. All objects have to fall down with the same acceleration.

Now imagine a world where this is not true. For example, a world where some elementary particles fall down, and some do not. Clearly this violates your equation, so something has to be wrong with it.
You can fix this if you introduce two different masses - one for gravity (=something a scale would show) and one for the relation between force and acceleration (=something an accelerating car would feel). We do not need this in our universe, but that is just an experimental result.

 

[jeremyfiennes]

If you cancel in m in GMm/r² = ma you get GM/r² = a. On the surface of earth, G, M and r are constant, so a is the same for all objects. All objects have to fall down with the same acceleration.

Now imagine a world where this is not true. For example, a world where some elementary particles fall down, and some do not. Clearly this violates your equation, so something has to be wrong with it.
You can fix this if you introduce two different masses - one for gravity (=something a scale would show) and one for the relation between force and acceleration (=something an accelerating car would feel). We do not need this in our universe, but that is just an experimental result.

It is a question of definition. The basic MKS units are the metre, kilogram and second. Force is defined in terms of these: F= m.a. And the value of the gravitational constant is then G=F.r2/(m1.m2) So the value of G is based on inertial mass.

 

[mfb]

Well, you would have to put other numbers in that formula to get a gravitational constant. Or introduce many gravitational constants for different particles, but that gets ugly.