Conceptual difference between gravitational and inertial mas

[CherryWine]

I know that this question has been asked many times before on this forum, but on every existing thread either the question or the answers, or both, were too vague. I understand that inertial mass is defined as the property of an object to resist change of its velocity, that is the mass that appears in Newton's Second Law F=ma. I also understand that gravitational mass on the other hand is defined as the property of an object to attract other objects. I am aware that experimentally it has been shown that gravitational mass and inertial mass are numerically the same. But, let's examine a specific problem. Let there be an object upon which acts just one single force that is gravitational. We can write F=ma. From there we can say that F=GMm/r^2, and further expand GMm/r^2=ma. Now we usually cancel m's on each sides, but how can we do this? If we know that they are conceptually different, how can we cancel them out, regardless of their identical numerical value?

I have seen that inertial mass is dependent on the velocity of the object, so when an object travels at relativistic speeds, its inertial mass is different while gravitational mass stays the same. Now, I don't know GR very well so try to answer my question having that in mind.

Thanks.

 

[pbuk]

If we know that they are conceptually different, how can we cancel them out, regardless of their identical numerical value?

Because experiments (under non-relativistic conditions) have shown them to be equal.

I have seen that inertial mass is dependent on the velocity of the object, so when an object travels at relativistic speeds, its inertial mass is different while gravitational mass stays the same.

Indeed. So under relativistic conditions GM/r² = a no longer holds.