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In classical mechanics, a force is basically defined to be acceleration times mass.
That is an expression for force, not the expression. viz. when a cup sits on a table, there is a reaction force yet no acceleration. When a chain is put under tension, there is a force between each link, but there is no acceleration.
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Yes, I hinted that I had reservations about that statement, since I said "basically". Was just trying to get across the idea that energy is not force. Actually, maybe I would tend to just define a force intuitively as a push or a pull. But I'm not sure your cup example debunks what I said under a suitably flexible interpretation because you could interpret it as potential acceleration. If the table wasn't there, the force would accelerate the cup by an amount inversely proportional to its mass (except the mass cancels out because the force is proportional to mass for gravity). Another objection I could raise is that acceleration behaves the same way. It's a vector and vectors get added and subtracted. If you accelerate upwards by some amount and downwards by the same amount, the accelerations cancel out and the net result is also no acceleration.
As far as Newton's 2nd law, you can take it to "define" force, as in "whatever is required to accelerate this amount of mass by this amount". Under that interpretation, all the physics is contained in Newton's first and third laws and the 2nd law is merely a definition. Or, you can say, we know what a force is--it's a push or a pull or something like that and Newton's 2nd law is telling how much push or pull we need to get some quantity of mass to accelerate.
So, on the whole, you are right--I probably shouldn't say "define" force like that because I think when we say force, we mean the push or the pull, rather than the quantity of push or pull. But, there is a certain sense in which you are "defining" it as ma if you take the former attitude towards Newton's 2nd law.
At any rate, we can probably work out plenty of classical mechanics problems correctly and understand the other concepts, whether or not we can state exactly the right definition of force, and maybe that's what really counts. Too much semantics.>>
A force is the incipient change of system energy, with respect to the incipient displacement. (The 'incipience' means, if you like, that it is a displacement that might happen or is happening. A 'potential' for a change, if you like. F=ma is a force relating only to changes of kinetic energy, and also tends to infer only that a change [of kinetic energy] is happening.)
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Your definition of force is very close to -dV/dx, where V is potential energy. The dx sort of implies a potential change rather than an actual change. F = ma is really referring to NET force. It doesn't infer that a change of kinetic energy is happening because it could be equal to zero. And again, there could be various forces canceling each other out, just like accelerations themselves can cancel out. But each one considered in isolation, would indeed cause an acceleration of F/m.