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As in, why is newton's second law(for constant mass systems),

$$ F= m \frac{d^2 x }{dt^2}$$

and not something of the sort like

$$ F= m \frac{d^3 x }{dt^3}$$

Why is it that sum of all force can be equated to mass times second derivative of position? Like in all cases the right side of the $F=ma$ equation is same.

My attempts at solving this question:

I saw this stack which led me to this other stack and also this other one, both of which requires a lot more mathematical context than I know already (Lagrangian and Hamiltonian formalism). So, I look for a simpler explanation using mainly physical principles supplemented with nothing more than basic vector calculus to explain why we can equate the sum of all forces as mass times the second derivative of position.

tryst with freedom
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  • Note that the next force in Newton's Second Law is more generally equal to the first time derivative of momentum. Perhaps we should be wondering why Newton's Second Law involves the first derivative of momentum and not its second derivative. – electronpusher Jul 28 '20 at 22:57
  • changing of mass is a complication I did not have in mind while writing the question, I'll add it as a note but I don't want it as my main focus – tryst with freedom Jul 28 '20 at 23:03
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    I've deleted a comment which was a dismissive answer to the question. This isn't a question about dimensional analysis; it's about why we define force the way we do. – rob Jul 28 '20 at 23:15
  • $F=ma$ is more or less a starting point for Newtonian mechanics, I don't think you can "explain it" using anything other than physical situations where the law seems to be obvious. – Akshat Sharma Jul 28 '20 at 23:18
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    Check that physics explain "how", not "why". We might need some philosophy for that (do not neglect philosophy). Understanding why things are like that is complicated. Many simulations have been done changing Newton's second law, and it is weird. I guess it's jsut an empirical law (if you take the "everyday life concepto fo force") or a useful definition, in a physical point of view. – FGSUZ Jul 28 '20 at 23:55
  • See https://physics.stackexchange.com/q/490586/, https://physics.stackexchange.com/q/18588/, https://physics.stackexchange.com/q/4102/ – Tfovid Jul 29 '20 at 07:46
  • @Tfovid those are the original posts that I linked in my question.. – tryst with freedom Jul 29 '20 at 10:24
  • what do you mean by changing second law? @FGSUZ – tryst with freedom Jul 29 '20 at 10:24
  • @DDD4C4U It's like "how would things work if Newton's 2nd law were $F=mv$? It's interesting but weird haha – FGSUZ Jul 29 '20 at 17:33

3 Answers3

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Newton's actual second law is not $F=ma$, but

$F=\dot p.$

And that's more or less what defines force*. Based on the observation that momentum $p=mv$ of an object only changes when it interacts with something, we can see that the change in momentum is an interesting quantity. And since change in momentum is intuitively associated with what laymen would call force, we just call it force as well. So that's a definition, $F:=\dot p$. And since usually the mass of an object doesn't change, this becomes

$F=\dot p=\underbrace{\dot m}_{=0}v+m\dot v=m\dot v=ma.$

It's not true for rockets, though, since they shed mass to accelerate, so $\dot m\neq0$. But that's just an interesting side fact.

To summarize: It can be observed that $mv$ only changes due to external interaction, which makes its change an interesting quantity, and we call that quantity force. That's why force is naturally just the derivative of $mv$, which usually reduced to $ma$.

*Not going into the intricacies of static forces

Vercassivelaunos
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To add a little background on top of the current answers:

As I understand it, prior to Galileo and Newton people believed that you couldn't even have velocity without force; that is, objects maintain their position unless forced. That's because they were used to objects sitting on the ground, which will not budge unless you push them, and normally stop moving the instant you let go.

At some point, someone (I think Galileo?) realized an object can move without being pushed, for example on ice. After you let go, it keeps moving for a while. Yet the ice does not seem to be pushing it, so it must be slowing it down instead. Which means that an unforced object actually tends to maintain its velocity, not its position.

That inspired Newton to formulate a law saying that force is what causes changes in velocity.

But if we replaced acceleration by its derivative, that would be saying that force is what causes changes in acceleration. In other words, an object will maintain its acceleration unless forced to change it. But that would mean that, for example, if you push an object until it's got a certain acceleration, and then let go, it will keep getting faster at that same rate unless something forces it to start slowing down, even if it's just drifting through empty space. That would be a strange world, and it's not what we observe.

Adam Herbst
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Why look further than the first law for a simple explanation?

An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.

A force causes a change in velocity. Velocity is the first derivative of position. If the second derivative of position is not zero, the velocity changes, so there must be an unbalanced force.

  • That is like circular reasoning.. the first law uses force to explain inertia and now you use inertia to explain force – tryst with freedom Jul 29 '20 at 10:28
  • @DDD4C4U Sorry you are not satisfied. Maybe you are looking for answers to different questions? I answered your question about why the second derivative of position and not the third. –  Jul 29 '20 at 12:07
  • to explain unbalanced force, you need to explain force. I think you maybe hinting that velocity can't change unless there is acceleration but why not just use acceleration directly then? – tryst with freedom Jul 29 '20 at 12:34