Newton's second law is so intuitively obvious

[SlowProgress]

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.

 

[bp_psy]

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?

 

[SlowProgress]

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?

 

[wotanub]

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.

 

[SlowProgress]

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?

 

[ModusPwnd]

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.

 

[wotanub]

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]

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.

 

[SlowProgress]

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

 

[zoobyshoe]

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.)

 

[zoobyshoe]

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.

 

[arildno]

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.

 

[harrylin]

[..] 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.

 

[harrylin]

[..] 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]

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]

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]

"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..

 

[harrylin]

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

 

[harrylin]

[..] why are only EXTERNAL commands relevant for body motion? [..].

Not exactly. See: https://www.physicsforums.com/showthread.php?t=527460

 

[Andrew Mason]

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

 

[D H]

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.

 

[zoobyshoe]

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.

 

[Mandelbroth]

\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]

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]

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."

 

[arildno]

I wouldn't say such systems (geometrically defined, rather than strictly based on material particles) are necessarily goofy, even if the boundary of the geometry is goofy.

Rather, just as you can perfectly well calculating the angular momentum of a train moving from New York-Washington, relative to a moment point whizzing about the Notredame Cathedral in Paris, such geometric systems contains BOTH dynamical information, and kinematic information deriving from the arbitrarily choosen geometry/moment point.

Sometimes, clever choices of geometrical systems might simplify the problem, relative to a material systems approach.
I made a tutorial on this some years ago:
https://www.physicsforums.com/showthread.php?t=72176

 

[A.T.]

When laws are counter intuitive, people complain that they make no sense.
When laws are quite intuitive, people complain that they are too trivial.

 

[SlowProgress]

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.

"Why is it intuitive?"

I think of it as intuitive because a push of any object is a lot easier than the same force applied to something heavier. One might readily deduce that the mass and acceleration produced are proportional. Maybe not the m1/m2 = a2/a1 part (which implies that the magnitudes of mass and acceleration are inversely proportional; so that if you double the mass, the acceleration will decrease by half.) That is the only part of the law that is unintuitive, since the increase could have been by a certain ratio, and not just a mere increase and decrease by the same number. However, I suppose the let-go of a pendulum with a weight of different magnitudes could verify the mass-acceleration proportionality. This would be all the proof that is needed to verify Newton's laws, which would render the second one of which I speak intuitive. I could have postulated proportionality of same magnitudes before having any mathematical proof, but I wouldn't have the mathematical adeptness to prove it.

From the results of empirical analysis, I could take the proportionality of mass and acceleration and multiply them together to create a number which we can use as a unit of force, which can be maintained by increasing one (suppose mass for example) and decreasing the other (acceleration) respectively. This is the undeniable aspect I speak about.

"Why is it only EXTERNAL forces that contribute to the object's motion?"

Why would it be anything other than an external force that contributes to an objects motion?

"Why is, for example, what we call "mass" a scalar quantity or even comparable to other "masses"?"

Why is a "mass" a scalar quantity? If I have two rocks composed of the same substance, and they have different weights on the same large body like Earth, we can assume that their masses are different only by a scalar quantity. Under the assumption that the density remains average through the rocks, we can even statistically determine its relative mass by its size. Why would a vector be needed for mass?

"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."

Please note the "pardon my ignorance" I said after it was all said and asked.You've made me even more confused than I started out.

I will look into the pendulum experiments that Newton carried out to prove his laws. I'm still uncertain if this law stems from rigorous, undeniable mathematical proof or if it is merely concluded from empirical observation.

Also, please, don't express anger or frustration by the way i phrase my statements. I only structure them this way to elicit a response that counter-argues the statements I make.

 

[bp_psy]

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).

Well you argue that something is intuitive because it follows from something that is arguably even less intuitive.I do not think that 'all inertial frames of reference are equivalent'' is that intuitive or self evident it is more like a big leap in inductive reasoning. You can derive the second law from the principle of least action but that doesn't make in any way more intuitive either. In my opinion one would not be justified to say that the laws in Principia are self evident .Newton did not do that,he made clear that they were postulates based on observation and not results from some logical argument. If I recall correctly the last part of the laws chapter was about experiments.

 

[SlowProgress]

By the way, I love that "experts" are arguing over interpretations of something already established. Gives me and my scatterbrained mind some confidence.

 

[gmax137]

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.

This, I think, is the best answer to the OP. For 2,000 years people believed "intuitively" that motion implies force (per Aristotle). Most people still do.

Newton said force implies change in motion.

 

[gmax137]

This is a nice read

http://www.thecatalyst.org/physics/chapter-two.html

 

[AlephZero]

I suspect the reason the OP things the second law is intuitive is because he/she absorbed the mindset that makes it seem intuitive, without ever really stopping to think about it. If Newton and his successors hadn't done what they did, the absorbing process would never have happened.

There's an analogy with the avant-garde composer John Cage, who wrote a famous (or infamous) piece of music consisting of exactly 4 minutes 33 seconds of silence. When somebody complained to him that "anybody could have written that", his reply was "No. Anybody else could have copied that, but only after I did it first."

 

[SlowProgress]

I suspect the reason the OP things the second law is intuitive is because he/she absorbed the mindset that makes it seem intuitive, without ever really stopping to think about it. If Newton and his successors hadn't done what they did, the absorbing process would never have happened.

There's an analogy with the avant-garde composer John Cage, who wrote a famous (or infamous) piece of music consisting of exactly 4 minutes 33 seconds of silence. When somebody complained to him that "anybody could have written that", his reply was "No. Anybody else could have copied that, but only after I did it first."

I fail to reach the point you are trying to make. That has no relevance to what this thread is about.

I say it is intuitive because I see it as intuitive. I've stopped to think about it and cannot think about it and break it down any further. I am coming to this forum for clarification, but all I am getting are comments like yours instead of an explanation of what makes it so unintuitive. This, coming from an "advisor". Alarming.

Perhaps we are mis-defining intuitive. I think I have mis-defined a few concepts here already.

I do not define intuitive as a solo acknowledgment. I define intuitive as something that is understood in such clarity that you can explain the concept in your own words; I can visualize the concept in my own mind from a different perspective, not from the one I observe and assume by deduction from my senses here on the surface.

All I am saying is that the idea of m1/m2 = a2/a1 should have been thought long before 300-400 years ago. It's one of those ideas that can be thought but not easily proven--only until someone gets up and carries out experiments.

 

[SlowProgress]

other potential, plausible mechanisms for motion Newton (but also many of his predecessors) had to clear off the table.
You didn't.

Also, what other mechanisms were considered?

Looking at a collision, only a few things are considered: the weight of the vehicle and the speed at which it was moving.

 

[zoobyshoe]

I fail to reach the point you are trying to make. That has no relevance to what this thread is about.

His point is that, it's extremely easy to hear Newton's 3 Laws and see they make obvious sense and accept them. However, it turns out it was very hard for mankind to arrive at the three laws, as evidenced by the fact of ± 2000 years between when Aristotle started seriously thinking about motion and when Newton compiled them.

The odds are that, in the absence of knowledge of the three laws, you would not ever, ever be able to intuit them, even one of them. Why do I say that? Because only a smattering of people between Aristotle and Newton got close to a rigorous statement of anyone of them.

The view looks perfectly clear and unobstructed when you're standing on the shoulders of a giant, yes. But remember: that doesn't make you a giant.

Find some rural village in the third world somewhere and ask the locals to determine and state the laws of motion. I don't think anyone from Afghanistan to the Amazon would be able to figure even one of them out.

 

[SlowProgress]

His point is that, it's extremely easy to hear Newton's 3 Laws and see they make obvious sense and accept them. However, it turns out it was very hard for mankind to arrive at the three laws, as evidenced by the fact of ± 2000 years between when Aristotle started seriously thinking about motion and when Newton compiled them.

The odds are that, in the absence of knowledge of the three laws, you would not ever, ever be able to intuit them, even one of them. Why do I say that? Because only a smattering of people between Aristotle and Newton got close to a rigorous statement of anyone of them.

The view looks perfectly clear and unobstructed when you're standing on the shoulders of a giant, yes. But remember: that doesn't make you a giant.

Find some rural village in the third world somewhere and ask the locals to determine and state the laws of motion. I don't think anyone from Afghanistan to the Amazon would be able to figure even one of them out.

That was all I needed to know. Something is so clear to me that I cannot believe anyone else cannot arrive at the same idea.

Thank you for covering up a lot of what I was waiting for after all of the implicit accusations of egotism.

 

[SlowProgress]

It just seems odd, anyone who wants to injure another through a flying projectile knows to exert enough force so that it travels faster than the force required to make it move relatively slower. This acceleration component is already covered. He knows that the heavier the ball, with the same speed as the ball he originally threw, the greater the damage inflicted on someone, resolving the mass component. I just can't believe this is so enigmatic to other people. I think I will hold that we all know this law, but don't have the words to explain it,

 

[Andrew Mason]

Well you argue that something is intuitive because it follows from something that is arguably even less intuitive.I do not think that 'all inertial frames of reference are equivalent'' is that intuitive or self evident it is more like a big leap in inductive reasoning. You can derive the second law from the principle of least action but that doesn't make in any way more intuitive either. In my opinion one would not be justified to say that the laws in Principia are self evident .Newton did not do that,he made clear that they were postulates based on observation and not results from some logical argument. If I recall correctly the last part of the laws chapter was about experiments.

I was responding to bp_psy's post about it not being clear why the relationship between F and a had to be linear.

I just said that F=ma necessarily follows from Galilean relativity. I did not suggest that this was intuitive. It may or may not be, depending on the individual. It is based on scientific experiments. Galileo observed that his physics experiments worked the same on a uniformly moving ship on calm seas as on dry land. That is just good science. Once that observation is made (and the absolute nature of time and space are assumed) FΔt = mΔv follows mathematically.

AM

 

[zoobyshoe]

That was all I needed to know. Something is so clear to me that I cannot believe anyone else cannot arrive at the same idea.

Thank you for covering up a lot of what I was waiting for after all of the implicit accusations of egotism.

Thank AlephZero. I was just paraphrasing him.

It just seems odd, anyone who wants to injure another through a flying projectile knows to exert enough force so that it travels faster than the force required to make it move relatively slower. This acceleration component is already covered. He knows that the heavier the ball, with the same speed as the ball he originally threw, the greater the damage inflicted on someone, resolving the mass component. I just can't believe this is so enigmatic to other people. I think I will hold that we all know this law, but don't have the words to explain it,

This is exactly right. People often have a remarkably deep 'sense of things' without being able to articulate it. The example that comes to mind is the extremely sophisticated musical instruments people were crafting way back. The instruments preceded the physics explanation of them. That is: the physics was worked out to explain the instruments, the instruments weren't created to demonstrate the physics.

However, once the physics is worked out, it almost always leads to improvements that couldn't have been made prior to the physics.

The rule of thumb, "The faster I throw this rock, the more damage it does," is true, and was probably universally understood by anyone who'd ever thrown a rock, but it is a million miles away from F=ma.

 

[SlowProgress]

Thank AlephZero. I was just paraphrasing him.This is exactly right. People often have a remarkably deep 'sense of things' without being able to articulate it. The example that comes to mind is the extremely sophisticated musical instruments people were crafting way back. The instruments preceded the physics explanation of them. That is: the physics was worked out to explain the instruments, the instruments weren't created to demonstrate the physics.

However, once the physics is worked out, it almost always leads to improvements that couldn't have been made prior to the physics.

The rule of thumb, "The faster I throw this rock, the more damage it does," is true, and was probably universally understood by anyone who'd ever thrown a rock, but it is a million miles away from F=ma.

Thank you for the response.
Pardon my asking again, but this sort of reminds me of the intuition I was speaking about. We all understand it. Isn't this now intuitive for all of us? What component makes the second law not intuitive? I just explained how intuitive it is for all of us, yet somehow you claim I will never intuit it.

Are we misdefining intuition again?

EDIT: Also, the formula sort of speaks for itself like in the scenario I just said. An increase in mass will slow down the acceleration with the same force applied. That is literally exactly what the formula F = ma says, and that is exactly what occurred in the scenario. The only difference is that you've indicated the magnitudes by a numerical measurement.

 

[yenchin]

The book "Physics for Mathematicians: Mechanics I" by Spivak spends a great deal of Chapter 1 just to discuss Newton's laws of motion. You may be interested.

 

[harrylin]

Thank you for the response.
Pardon my asking again, but this sort of reminds me of the intuition I was speaking about. We all understand it. Isn't this now intuitive for all of us? What component makes the second law not intuitive? I just explained how intuitive it is for all of us, yet somehow you claim I will never intuit it.

Are we misdefining intuition again?

EDIT: Also, the formula sort of speaks for itself like in the scenario I just said. An increase in mass will slow down the acceleration with the same force applied. That is literally exactly what the formula F = ma says, and that is exactly what occurred in the scenario. The only difference is that you've indicated the magnitudes by a numerical measurement.

This is not about definition of intuition, I'm afraid that it's deeper as it's applicable to other, somewhat similar concepts. Others have already explained that your example historically led to the belief that F=mv, based on intuition only (lack of serious experiments and the resulting knowledge that modifies intuition). Another comparison: a patent examiner. He/she has to judge after studying a patent if it is obvious or not to someone who has not read it. Most patent applications are obvious after reading it. Do you think that the following argumentation to colleagues is correct:

"You say that it is non-obvious for people who haven't read it. But I understand it. Isn't this now understandable for all of us? What section makes the patent non-obvious? I just explained how obvious it is for all of us, yet some people claim I will never understand it."

 

[zoobyshoe]

Are we misdefining intuition again?

If you suspect this is the issue then the best thing to do is find a dictionary definition to present here as the specific definition of intuition you want the discussion to revolve around.

 

[TurtleMeister]

What component makes the second law not intuitive?

Gravity and friction maybe? If we were born and raised in the ISS then it might be naturally more intuitive.

The view looks perfectly clear and unobstructed when you're standing on the shoulders of a giant, yes. But remember: that doesn't make you a giant.

Haha, I like that. Are you quoting someone?

 

[zoobyshoe]

Haha, I like that. Are you quoting someone?

Well, it's a paraphrase. The original quote is: "If I have not seen as far as my fellow man, it's because some giant is always standing in my way."

(That was the signature quote of a girl who used to post here. [Can't recall her username.] I thought it was hilarious.)

 

[TurtleMeister]

Thanks for the info. I like your version the best. I think your version is more of a paraphrase of Newton's original quote. The unknown members quote seems more like a parody on Newton's.

 

[bhobba]

Well actually Newtons second law is strictly speaking a definition and the first follows from the second so the physical content is in the third law.

BUT its real content is a prescription that says - get thee to the forces. So while a definition, by defining it its saying that's what you should look at.

IMHO the PLA and and the symmetries of Galilean relativity as found in Landau - Mechanics is the more logical approach. The PLA follows from the axioms of QM and its form is determined by Galilean relativity as for example found in Ballentine - QM - A Modern Approach. That way it is easily seen that both QM and CM are really determined by the same thing - symmetries - but what they are symmetries in is different - the PLA for CM, the fundamental axioms of QM for QM.

Thanks
Bill

 

[Andrew Mason]

Well actually Newtons second law is strictly speaking a definition and the first follows from the second so the physical content is in the third law.

The definition of force is really in the first law: "force" is that which must act on a body in order to change the motion of that body. The second law follows from the first and from the observation that the laws of motion are the same in all inertial frames, time and space being absolute. So I would say that F=ma is not a definition but an empirical law. F=ma is defined by nature.

IMHO the PLA and and the symmetries of Galilean relativity as found in Landau - Mechanics is the more logical approach. The PLA follows from the axioms of QM and its form is determined by Galilean relativity as for example found in Ballentine - QM - A Modern Approach. That way it is easily seen that both QM and CM are really determined by the same thing - symmetries - but what they are symmetries in is different - the PLA for CM, the fundamental axioms of QM for QM.

Perhaps you could explain what the "PLA" is.

AM

 

[bhobba]

Perhaps you could explain what the "PLA" is.

That's a reasonable view of force - just not the way I would view it.

PLA - Principle Of Least Action.

Starting from that alone, and the the symmetries implied by Galilean Relativity, you can actually derive all of Classical Mechanics - strange but true - you will find the detail in Landau's book - Mechanics. Now for some shameless pithy admiration for that book:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20

If physicists could weep, they would weep over this book. The book is devastingly brief whilst deriving, in its few pages, all the great results of classical mechanics. Results that in other books take take up many more pages. I first came across Landau's mechanics many years ago as a brash undergrad. My prof at the time had given me this book but warned me that it's the kind of book that ages like wine. I've read this book several times since and I have found that indeed, each time is more rewarding than the last.

The reason for the brevity is that, as pointed out by previous reviewers, Landau derives mechanics from symmetry. Historically, it was long after the main bulk of mechanics was developed that Emmy Noether proved that symmetries underly every important quantity in physics. So instead of starting from concrete mechanical case-studies and generalising to the formal machinery of the Hamilton equations, Landau starts out from the most generic symmetry and dervies the mechanics. The 2nd laws of mechanics, for example, is derived as a consequence of the uniqueness of trajectories in the Lagragian. For some, this may seem too "mathematical" but in reality, it is a sign of sophisitication in physics if one can identify the underlying symmetries in a mechanical system. Thus this book represents the height of theoretical sophistication in that symmetries are used to derive so many physical results.

The difficulty with this approach, and the reason why this book is not a beginner's book, is that to the follow symmetric arguments, one really has to have already mastered vector calculus. Ideally, you should be able to transform coordinate in your sleep, perform integrals without missing a beat, whether they be line, area, or path, and differentiate functions in many dimensions.

Thanks
Bill