As you know, it's not quite true that
F=m
a, but let's for now just think about your deep question for the case of slow-moving objects, where
F=m
a works pretty well. I hope you'll forgive a long-winded, fuzzy answer, since I don't know any good short answer.
The first thing you have to ask about
F=m
a is what it actually tells us about the world, since until we figure that out we can't begin to say why it should be true or false. Here's the problem. Most objects don't come stamped with an "m" value on them. Space and time aren't laid out with a labeled grid of coordinates, so we aren't just given the "
a" values for objects. Worst of all, what's
F?
Let's say we ignore the question of how to determine
a, by assuming that somehow we have a common-sense set of space-time coordinates that we're happy with. The key step to making some meaning out of
F=m
a is then Newton's 3d law- conservation of momentum. We can bounce objects off each other, and by measuring their velocities before and after the bounce, figure out the ratios of their m's. Pick one object to call the unit mass, and now we have a set of m's.
Now we get to the hard part. What are the
F's? Let's say we see some m with an
a. What's to stop us from just inventing an
F to make
F=m
a true? If we could always do that, then
F=m
a would be untestable and meaningless. So we must insist on some rules about the
F's. The third law says that there needs to be an opposite
F on something else, and we can insist that the something else is fairly nearby. More generally, we can insist that the rules for when there should be an
F shouldn't be too weird or complicated. If we can fit what we see within those rules, then we can say that
F=m
a is true. Up to a point, that program works. Once you start including electromagnetic fields and fast-moving objects, it gets too awkward and we need a different set of rules, in which
F=m
a is replaced.
That's a very compressed version of a long discussion. Feel free to follow up.
Mike W.
(published on 06/26/11)
