# Newton's second law is so intuitively obvious

Hi, question here.

Newton's second law is so intuitively obvious to anyone not even hearing or reading his law in words that it makes me wonder: how was it such a great revelation that many years ago?

Pardon my ignorance, but I'm very curious.

I kind of think it's the least intuitive of the three. At that level why not F=ma^2+5Nm/s^2 or F=m^2sqrt(a) or F=ma^1.005.Direct proportionality is probably the most natural but other forms are not excluded. The direction is also not very certain why wouldn't the direction of a be little different than F?

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Oh, so you mean it's not intuitive in that it was previously unknown if it was directly proportional over mere proportionality (alternatively, and respectively, an equal growth in magnitude for both factors over a mere increase by an unknown ratio)?

In that respect, I understand, it really isn't intuitive since it is hard to prove.

How then did Newton prove that they are DIRECTLY proportional and not just proportional?

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No what he meant was the fact that the force was directly proportional to the mass and acceleration was obvious, as you said, but what was not obvious was the nature of the proportionality. For example, why doesn't the force increase as the square of the mass as opposed to linearly?

However, I have a different answer for your question. Newton's second law was a triumph because it showed that the newly invented mathematics of calculus was useful. It's even said that newton invented calculus because of his observations. Newton's second law is a differential equation. I imagine (I'm not a history major) when calculus was invented by Newton and Leibniz, people were like "okay, so what? That's cool and all but why is it useful?" Then came the bang: Newton didn't actually write down $F = ma$ he wrote $\frac{dp}{dt} = m \frac{d^{2}x}{dt^{2}}$. The concept of the derivative was new, and he showed that calculus could actually describe the real world.

I think the situation is analogous to Heisenberg, Pascal and Dirac figuring out that linear algebra is relevant to physics and how modern physicists are discovering how group theory is useful in describing the universe. It kinda makes me wonder which branch of mathematics will all of the sudden become relevant to physics.

No what he meant was the fact that the force was directly proportional to the mass and acceleration was obvious, as you said, but what was not obvious was the nature of the proportionality. For example, why doesn't the force increase as the square of the mass as opposed to linearly?

Right, that cleared up a lot of my confusion. But the question still remains. How did Newton go about proving its validity? I imagine it can really only be done by empirical experimentation, which alone is difficult to do because it's difficult to determine if you are applying an equal force every time.

Also, how can you square "delta (d)" or "change in [variable]" in d^2x/dt^2?

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He was able to derive Kepler's laws using it. Before Galileo it was thought that F=mv per Aristotle, that is how many people intuitively think things work.

What do you mean by prove? Show that it works? Well, what is a theory without an experiment? I'm not sure exactly what experiments Newton did, but there are many simple experiments that can be done to show that his law holds in the classical limit.

Also, as ModusPwnd above me has touched on, a good way to prove the strength of your theory (without experiment) is to show that other known results come out of it.

D H
Staff Emeritus
But the question still remains. How did Newton go about proving its validity? I imagine it can really only be done by empirical experimentation, which alone is difficult to do because it's difficult to determine if you are applying an equal force every time.
You can't "prove" things in science the way one can prove theorems in mathematics. Newton's second law is just that, a scientific law. It can't be proved. It can merely be tested against reality. And that testing is what Newton and his predecessors did. It was the start of the scientific method.

Newton's second law comes from Galileo. Newton attributed his first two laws to his predecessors, calling out Galileo as particularly important. Newton's additions to the work of his predecessors were stating the second law in mathematical form, and his third law. Newton's third law is his unique contribution.

You can't "prove" things in science the way one can prove theorems in mathematics. Newton's second law is just that, a scientific law. It can't be proved. It can merely be tested against reality. And that testing is what Newton and his predecessors did. It was the start of the scientific method.

Newton's second law comes from Galileo. Newton attributed his first two laws to his predecessors, calling out Galileo as particularly important. Newton's additions to the work of his predecessors were stating the second law in mathematical form, and his third law. Newton's third law is his unique contribution.

What kind of "testing"? It's difficult to "test against reality" if you'll get a different outcome every time!! Thanks for the responses. I'm happy to get some insight.

Also, I think I have misdefined "directly proportional" and "proportional". I always thought of directly proportional to be equal numerical increases, and proportional just sayinh that there is a general increase, with unknown ratio.

Also good to know. Thanks

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First off, Newton did not express the second law as F=ma. He never expressed it as a formula at all.

This is the second law as he wrote it:

"The alteration of motion is ever proportional to the motive force impressd; and is made in a direction of the right line in which that force is impressed.

If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to, or subtracted from the former motion, according as they directly conspire with, or are directly contrary to each other, or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both."

To understand his wording you have to understand that he and his contemporaries had no word for momentum. The word "motion" covered the concept of momentum, by which I mean that "motion" might refer to uniform displacement in space, or, it might refer to "quantity of motion," the product of mass and velocity that we call "momentum."

Newton defines all his terms in the very first chapter of Principia. "Quantity of motion" is the second definition:

"Definition II

The quantity of motion is the measure of the same arising from the velocity and quantity of matter conjunctly.

The motion of the whole is the sum of all of the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple."

Anyway, the second law might best be expressed by the formula Δρ=FΔt. In other words, Newton's actual second law was the 'definition' of change in momentum.

I haven't been able to determine exactly when or why Newton's actual second law got replaced in textbooks by F=ma, but it seems to have been somewhere between the 1830's and 1900. (Euler explored F=ma, but it's not clear he thought it should replace Newton's statement.)

You can't "prove" things in science the way one can prove theorems in mathematics. Newton's second law is just that, a scientific law. It can't be proved. It can merely be tested against reality. And that testing is what Newton and his predecessors did. It was the start of the scientific method.

Newton's second law comes from Galileo. Newton attributed his first two laws to his predecessors, calling out Galileo as particularly important. Newton's additions to the work of his predecessors were stating the second law in mathematical form, and his third law. Newton's third law is his unique contribution.

Newton says:

""Hitherto I have laid down such principles as have been received by mathematicians, and are confirmed by abundance of experiments. By the first two Laws and the first two Corollaries, Galileo discovered that the descent of bodies observed the duplicate ratio of the time; and that the motion of projectiles was in the curve of a parabola…"

However, he is confused. Galileo did not know or understand the second law because Galileo never quite hit on the correct definition of momentum. He waved his hand very close to it many times without getting it. DesCartes was the first person to understand that the product of mass and velocity was important. He, however, didn't seem to understand it was a vector. The first evidence of anyone understanding momentum, mv, as a vector comes from papers written in 1657 by Huygens. He was a prominent member of the Royal Society and apparently shared his insights with the others. "Quantity of motion" as the product of mass and velocity and as a vector was known to all of them before Newton joined.

Newton says:

"By the same [that is, the first two laws], together with Law 3, Sir Christopher Wren, Dr. Wallis, and Mr. Huygens, did severally determine the rules of the impact and reflection of hard bodies, and about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves as to those rules."

He is crediting the three people mentioned as having, all three, independently discovered the rules of elastic collisions (hence: conservation of momentum). And he says they did that with knowledge of all three Laws. Law 3, therefore, preceded Newton. He received it from Wren, Wallis, or Huygens, or all three. It's not clear where they got the idea of the third law, but Newton says it was part of their aim in conducting experiments with pendulums to confirm it. (Their experiments are too extensive to describe here, but what they were doing was letting one pendulum hit another pendulum and recording the effect.) Newton recreated all their experiments for himself, confirmed the previous results, and mentions twice that the experiments also confirm the 3rd Law.

All this history, and a detailed explanation of the pendulum experiments, in contained in the "scholium" of the chapter, "Axioms, or Laws of Motion," right near the beginning of the Principia.

Anyway, everyone should agree Galileo was responsible for the first law. Newton also ascribes the second law (as he expressed it) to Galileo, but that can't be right since Galileo had no solid conception of momentum. A scrupulous search might turn up some first statement of it in the papers of Wren, Wallis, Hooke, or Huygens, I'm not sure. But it's clear Newton doesn't claim credit for it. And neither does he claim credit for the third Law, pretty much granting it to all three who discovered the rules of elastic collisions.

In conclusion: Newton didn't originate any of Newton's 3 Laws.

And, to the Opening Poster, the "testing" was the pendulum experiments.

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arildno
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Why is it intuitive?

For example:
Why is it only EXTERNAL forces that contribute to the object's motion?
Why is, for example, what we call "mass" a scalar quantity or even comparable to other "masses"?
Couldn't different materials have different types of sensitivities to forces that cannot be easily reduced to the single concept of "mass"?

Furthermore:
Why is it just accelerations, rather than also velocities and/or jolts (third derivatives) that "forces"/external pushings directly impart to objects?

Before declaring that something is so "obvious" that even you yourself might have formulated, you really ought to have shown even a sliver of understanding of what other potential, plausible mechanisms for motion Newton (but also many of his predecessors) had to clear off the table.
You didn't.

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[..] Then came the bang: Newton didn't actually write down $F = ma$ he wrote $\frac{dp}{dt} = m \frac{d^{2}x}{dt^{2}}$. The concept of the derivative was new, and he showed that calculus could actually describe the real world [..].
A little correction here: Newton phrased it in words, in English translation:

The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

That phrase corresponds to Δp/Δt ~ F. Of course, that doesn't alter your argument.

[..] everyone should agree Galileo was responsible for the first law [..] .
I searched for evidence for the inherent claim that Galileo defined straight motion relative to "the fixed stars" or compatible with that definition; but instead it appears that he defined straight motion relative to the Earth, which he maybe already assumed to be circling around the Sun. If you know proof that Galileo was responsible for the first law equivalent to the meaning given to it by Newton, that would be interesting!

SteamKing
Staff Emeritus
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Right, that cleared up a lot of my confusion. But the question still remains. How did Newton go about proving its validity? I imagine it can really only be done by empirical experimentation, which alone is difficult to do because it's difficult to determine if you are applying an equal force every time.

Also, how can you square "delta (d)" or "change in [variable]" in d^2x/dt^2?

You're not squaring anything. d^2x/dt^2 is calculus notation for the second derivative of x with respect to time. If 'x' is the position of the particle as a function of time, the second time derivative is also known as the acceleration of the particle.

D H
Staff Emeritus
Why is it intuitive?
Exactly. Just look at Newton's first law for something that is anything but intuitively obvious. Those nice rock walls that separate fields all over Europe: Those aren't there because they're decorative. They're there because European soil is amazingly good at growing rocks. Every spring farmers are confronted with a new crop of rocks that need to be removed from their fields. They would haul a bunch of rocks to a sledge, hitch up a team of horses to the sledge, have the horses drag the sledge to the closest point at the edge of the field, and unload the rocks. Some of those rocks are too big to carry to the sledge; they need to be dragged or pushed. What happens if the farmers stop pushing the rock? It comes to a rest. What happens if the horses stop pulling the sledge? It comes to a rest. What happens when the rocks are unloaded? They come to a rest, in that nice rock fence. Those rocks and they sledge: Their natural state of being is a state of rest. That is what was intuitively obvious. That the natural state of being is to keep on moving at a constant velocity unless acted upon by some external force is highly unintuitive.

The second law is even more unintuitive.

First off, Newton did not express the second law as F=ma. He never expressed it as a formula at all.

This is the second law as he wrote it:

"The alteration of motion is ever proportional to the motive force impressd; and is made in a direction of the right line in which that force is impressed.
That's a bit of a red herring, zoobyshoe. Of course he didn't express it as a formula using his calculus. He intentionally avoided doing that in his Principia. He expressed (almost) everything in his Principia using synthetic geometry. People immediately saw that Newton's Principia was all about his calculus without ever using calculus explicitly. Newton even made that argument himself, in court, in his nasty priority dispute with Leibniz over who invented the calculus. People began writing the equivalent of F=kma (but not F=ma) shortly after Newton published his Principia.

There's a profound difference between F=kma (force is proportional to the product of mass and acceleration) and F=ma (force *is* the product of mass and acceleration). Newton saw forces as something distinct from mass or acceleration. To Newton, F=kma (or the synthetic geometry equivalent as written in the Principia) is a true law of nature. The modern view, F=ma, is that this is a definitional statement rather than a law of nature.

arildno
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"The second law is even more unintuitive."

Not to speak of all those philosophical unituitivities that WE can see "riddled" the brains of the "unenlightened" but are by no means "obviously" false (even if they ARE false).

For example, we all have "inner selves" others can't see, smell or touch or anything, but that WE feel is critical in making our bodies move, imparting some inner command to move.

Why can't it be that that other, non-human objects ALSO have such soulstuff, imparting in their own mysterious ways commands/forces to impel the visible body forwards?

Lots of things in nature seem to move on their own account, why are only EXTERNAL commands relevant for body motion?

And so on and on..

What kind of "testing"? It's difficult to "test against reality" if you'll get a different outcome every time!!

I don't know if it's written somewhere what tests Newton did.
A simple way would be to use several weights, a pulley and a long rope. It's easy to verify, for example with a spring, that doubling the weight doubles the "impressed force" (that's also how scales work; apparently the first scales were introduced around that time).

Now connect the two weights to the rope, one on each end. Fix the pulley to a long table and put one weight on top of a toy cart on the table. Let the other weight pull down so that it can accelerate the cart. Let it go and record distance as function of time. You can experiment with different weights on each end.

The complication with this experiment is that a falling weight pulls less than a steady weight. One way to keep it simple is to use for the pulling a total weight that is much lighter than the total pulled weight, so that the pull reduction due to the fall will be small. That also makes the time measurement more precise.

I found a detailed experiment example here:
https://phys.cst.temple.edu/studentservices/lab_schedules/pdf/experiments/lab_8.pdf [Broken]

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Andrew Mason
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I kind of think it's the least intuitive of the three. At that level why not F=ma^2+5Nm/s^2 or F=m^2sqrt(a) or F=ma^1.005.Direct proportionality is probably the most natural but other forms are not excluded. The direction is also not very certain why wouldn't the direction of a be little different than F?

$F \propto a$ is a natural consequence of Galilean relativity: the laws of motion are the same in a ship moving on a calm sea as on land or 'all inertial frames of reference are equivalent, assuming time and space are absolute (measured the same in all reference frames).

$a \propto 1/m$ for the same force flows naturally from the concept that mass represents a quantity of fundamental parts of matter: "The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly. The motion of the whole is the sum of the motions of all the parts, and therefore in a body double in quantity with equal velocity, the motion is double; with twice the velocity it is quadruple."

If you apply these principles, the same push on a body for the same duration must cause the same change in the quantity of motion of that body as any other body of the same mass. And a successive same push for the same time duration has to effect the same change in quantity of motion. So $F\Delta t = m\Delta v$ has to be true if Galilean relativity holds. That is just F = ma

As far as the direction of the force is concerned, we define the direction of the net force to be the direction of the change in velocity.

AM

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D H
Staff Emeritus
$F \propto a$ is a natural consequence of Galilean relativity: the laws of motion are the same in a ship moving on a calm sea as on land or 'all inertial frames of reference are equivalent, assuming time and space are absolute (measured the same in all reference frames).
I don't see that. All that Galilean relativity says is that kinematically, force is a not a function of position or velocity. It might be a function of acceleration, but not necessarily a linear one, or it might a function of jerk, or some even higher order derivative.

$a \propto 1/m$ flows naturally from the concept that mass represents a quantity of fundamental parts of matter: "The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly. The motion of the whole is the sum of the motions of all the parts, and therefore in a body double in quantity with equal velocity, the motion is double; with twice the velocity it is quadruple."

I also disagree with this. Newton was talking about "quantity of motion": linear momentum. You are implicitly assuming Newton's second law to derive Newton's second law here.

That's a bit of a red herring, zoobyshoe. Of course he didn't express it as a formula using his calculus. He intentionally avoided doing that in his Principia. He expressed (almost) everything in his Principia using synthetic geometry. People immediately saw that Newton's Principia was all about his calculus without ever using calculus explicitly. Newton even made that argument himself, in court, in his nasty priority dispute with Leibniz over who invented the calculus. People began writing the equivalent of F=kma (but not F=ma) shortly after Newton published his Principia.
My point was that Newton wasn't thinking in term of F=ma in the second law. The second law as he formulated it was a statement about momentum, not force.

The reason I pointed out he didn't actually give a formula is that someone might look at what he wrote and argue what he said shouldn't be rendered as specifically as Δρ=FΔt. Someone could argue all he meant was Δρ≈F. If you're confident he didn't mean the latter, it's because of years of post clean-up thinking of the second Law as F=ma. Newton was actually groping hand and footholds out of thick murk, and his reasoning, though ultimately successful, wasn't the straight line we assume today. I believe it was you, yourself, who said in another thread a couple years back, that Physics has a dirty secret, which is that the progress was actually much messier than we realize.
There's a profound difference between F=kma (force is proportional to the product of mass and acceleration) and F=ma (force *is* the product of mass and acceleration). Newton saw forces as something distinct from mass or acceleration. To Newton, F=kma (or the synthetic geometry equivalent as written in the Principia) is a true law of nature. The modern view, F=ma, is that this is a definitional statement rather than a law of nature.
The "synthetic geometry" doesn't show up at all in these first preliminary chapters, Definitions, and Axioms, or Laws of Motion. It makes it's first appearance right after in Book 1. There isn't any 'synthetic geometry equivalent' of F=ma.

Also, no one "recognized" Newtons strange geometry as his calculus, because he hadn't published any of his calculus at the time.

I'm attaching three successive screenshots of a long passage from The Clockwork Universe which should alter your view of calculus in the Principia.

#### Attachments

$\frac{dp}{dt} = m \frac{d^{2}x}{dt^{2}}$.
Nitpicking: Suppose mass changes.

Then we have ##\frac{dp}{dt}=\frac{d(mv)}{dt}=v\frac{dm}{dt}+m\frac{dv}{dt}=v\frac{dm}{dt}+m\frac{d^2x}{dt^2}##.

Andrew Mason
Homework Helper
I don't see that. All that Galilean relativity says is that kinematically, force is a not a function of position or velocity. It might be a function of acceleration, but not necessarily a linear one, or it might a function of jerk, or some even higher order derivative.

Galilean relativity requires that acceleration be constant for a constant force. This can be deduced from the equivalence of all inertial frames and from the absolute nature of time and space.

If one applies a unit of pull or push (eg. applying a standard spring with a standard extension) to a body of mass m for a time τ, an observer in the rest frame of the body prior to the application of the force (call it IFR1) will measure a change of velocity of that body of δ.

If all inertial frames of reference are equivalent then in the new rest frame of that same body, now moving at a velocity of δ relative to IFR1 (call it IFR2), the application of that same unit of force for the same time τ must result in the same change of velocity δ. If this were not the case, inertial frames would not be equivalent.

So in IFR1, the total change in velocity is now 2δ and the duration of the force is 2τ. This necessarily means that Δv/Δt will always be constant if the force is constant.

I also disagree with this. Newton was talking about "quantity of motion": linear momentum. You are implicitly assuming Newton's second law to derive Newton's second law here.

I agree that Newton was definitely using the term "quantity of motion" to mean the quantity of matter multiplied by velocity. That just means that if a given force applied to a body of mass m for a time Δt changes its velocity by Δv, that same event will change the velocity a body of mass 2m by Δv/2. So Δv/Δt is inversely proportional to mass for a given force.

If you assume, as Newton did, that mass simply reflected the quantity of fundamental units of matter this is just a matter of seeing that the motion of a body is the sum of the motion of its parts.

AM

D H
Staff Emeritus
Nitpicking: Suppose mass changes.

Then we have ##\frac{dp}{dt}=\frac{d(mv)}{dt}=v\frac{dm}{dt}+m\frac{dv}{dt}=v\frac{dm}{dt}+m\frac{d^2x}{dt^2}##.
Not so nitpicky. Whether It's F=dp/dt or F=ma is a perennial debate. For a constant mass system, it's a silly argument. F=dp/dt and F=ma are one and the same for a constant mass system.

This is not the case for dynamic mass systems. Here, using F=dp/dt leads to force being frame-dependent; i.e., non-invariant. Using F=ma avoids that problem, but adds some new ones. The (direct) connection with the conservation laws is severed, and choosing a goofy system boundary can lead to goofy results.

For example, consider a stationary rod. Make a system boundary such that the bar is entirely within the boundary at t=t0, entirely outside the boundary at t=t1, and the boundary moves from one end of the bar to the other non-linearly with respect to time. The center of mass of the portion of the bar within the boundary is accelerating. Via F=ma, there's some force acting on the portion of the bar within the boundary. The solution here is simple. It's the standard response to "Doctor! It hurts when I do this!" ⟨Bonk⟩

Despite the shortcomings, ask most any aerospace engineer and the answer will be F=ma. The connection with the conservation laws is easily reestablished by adding thrust as a force, and regarding the goofy system boundary, the solution is the same as the doctor's: "Don't do that then."