Confusion about gravitational acceleration

[kmm]

I understand that gravitational acceleration is independent of mass. However, I've seen a common mathematical description of this that I can't help but find circular. I suspect that there's an error in my thinking that I'm hoping someone can point out for me. It goes like this; $F=mg$ but we know that $F=ma$. If we substitute the first equation into the second, we get $mg=ma$, therefore $g=a$. OK, but what seems wrong to me is that, once you state that the gravitational acceleration is the constant 'g', you're already saying it's independent of mass, BECAUSE it's a constant. And of course 'g' has to equal 'a' here; it is the acceleration. It seems to me that it's merely an experimental fact, so there's no need to justify it mathematically in this way. I feel like there's a subtlety I'm missing here.

 

[Dale]

It seems to me that it's merely an experimental fact, so there's no need to justify it mathematically in this way.

This is correct. It is experimentally observed that the acceleration is independent of the mass. The form of the force law follows from that.

 

[kmm]

This is correct. It is experimentally observed that the acceleration is independent of the mass. The form of the force law follows from that.

Thanks, I wonder why some explain it in this way..

 

[Dale]

Thanks, I wonder why some explain it in this way..

Well, the properties are encoded into the math so it is easy enough to start with the math and derive the properties

 

[russ_watters]

I feel like there's a subtlety I'm missing here.

Try deriving it from f=ma and Newton's law of gravity...

 

[kmm]

Try deriving it from f=ma and Newton's law of gravity...

Ok, in that case, we have $$ ma = G \frac {mM} {R^2}$$ so $$a = G \frac {M} {R^2}$$ which is independent of the mass 'm'. I guess deriving it from Newton's law of gravity, which is more general than the special case of 'f=mg', is actually bit more illuminating. Thanks