Newton's second law is so intuitively obvious

[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

 

[Andrew Mason]

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.

Landau starts with the principle of least action as a given. It may be interesting to do that but I am not sure it is science. You have to be able to make observations that the action is always a minimum, not make it an a priori assumption.

The principle of Galilean relativity is at least based on observation. And, by applying symmetry arguments, you can show that Newton's second and third laws of motion flow from that principle. And the principle of least action flows mathematically from the laws of motion. The importance that Landau accords to the principle of Galilean relativity indicates that he views it as a foundation for classical mechanics. I definitely agree with that.

AM

 

[T0mr]

Andrew Mason,

I am with you that the principle of least action feels like a priori assumption. However I do not see how Newton's Second Law, F=ma, can be derived directly from Galilean Relativity. I don't remember seeing that anywhere and I am curious.

 

[Andrew Mason]

Andrew Mason,

I am with you that the principle of least action feels like a priori assumption. However I do not see how Newton's Second Law, F=ma, can be derived directly from Galilean Relativity. I don't remember seeing that anywhere and I am curious.

See my post #20.

It is essentially a symmetry argument based on the first law and Galilean relativity (ie. absolute time and the principle that all inertial frames if reference (IFR) are equivalent: that the laws of motion are the same in all IFRs) and a certain concept of mass: that matter is made up of fundamental units of matter and that mass is simply a measure of the quantity or number of such units of matter.

A force F acting on a body m over a time interval δτ will cause a certain change in velocity of δv, as measured in the initial rest frame of m (IFR1). Moving now to the rest frame of m after the change in motion occurs (IFR2 ie. the IFR whose origin is moving at velocity δv with respect to IFR1) we measure the change in velocity of the same force F over a second identical time interval δτ. The equivalence of IFRs requires that there be the same change in velocity δv. If it were not the case, IFR1 and IFR2 would not be equivalent because the same force acting for the same time interval would effect different changes in motion. As measured in IFR1, the change in velocity is 2δv over time interval 2δτ. Repeating that you can see that Δv/Δt is constant. So a constant force causes constant acceleration.

Now, if you repeat the experiment by applying the same force to each of 2 identical bodies (each having the same quantity of matter) alternately for very brief intervals of δτ, the bodies accelerate at exactly half the rate in the first experiment. Because the force is only applied to them every other interval of δτ, it takes 2δτ to achieve a change in velocity of δv. So acceleration is inversely proportional to mass.

So that is pretty much the second law.

AM

 

[jhae2.718]

Those interested in this thread may find Truesdell's Essays in the History of Mechanics to be an interesting read.

 

[T0mr]

It is essentially a symmetry argument based on the first law and Galilean relativity (ie. absolute time and the principle that all inertial frames if reference (IFR) are equivalent: that the laws of motion are the same in all IFRs) and a certain concept of mass: that matter is made up of fundamental units of matter and that mass is simply a measure of the quantity or number of such units of matter.

I think I get for the most part what you are saying. All relative motions between IRFs would be the same because we have absolute time and space coordinates in Galilean Relativity. So as you have said each IRF would observe the other with the same speed but moving in the opposite direction which is the definition of relative velocity. I think I understand essentially what you are getting at by the example of alternating the infinitesimal amounts of time, δt, that acceleration is applied to each mass to effect a change, δv.

I think about force in a sort of similar way. I take it as something we defined to explain why the inertial quality of matter changes. I think scientists like Newton and Galileo understood through experiment that matter had an equilibrium state which was a motion at constant velocity (I take 0 to be a constant too). But they needed a concept to address when there was a change of that equilibrium state, or inertial motion, to another. That change takes place over a period of time and so it was appropriate call something that was a difference in the quantity of motion, mΔv, in some amount of time Δt the force.

Newton explains impressed force:
"Definition IV. An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line.

This force consists in the action only; and remains no longer in the body, when the action is over. For a body maintains every new state it acquires, by its vis inertiae only. Impressed forces are of different origins as from percussion, from pressure, from centripetal force."

So I definitely think its appropriate to look at inertial reference frames to help explain the origin and need for the concept of a force. And for the OP the concept of force may seem obvious now but I doubt at the time of its invention it was.

 

[stevendaryl]

A force F acting on a body m over a time interval δτ will cause a certain change in velocity of δv, as measured in the initial rest frame of m (IFR1). Moving now to the rest frame of m after the change in motion occurs (IFR2 ie. the IFR whose origin is moving at velocity δv with respect to IFR1) we measure the change in velocity of the same force F over a second identical time interval δτ. The equivalence of IFRs requires that there be the same change in velocity δv.

Okay, that's a nice argument, but it implicitly makes the assumption that a force has the same magnitude in all frames of reference. How is that justified?

 

[voko]

Landau starts with the principle of least action as a given. It may be interesting to do that but I am not sure it is science. You have to be able to make observations that the action is always a minimum, not make it an a priori assumption.

I do not believe it is a commonly accepted requirement that everything in a scientific theory must be directly observable. We usually require that something has to be observable, and if it is, it must be in agreement with an appropriate experiment.

Besides, your use of "always" is alarming. No observation or experiment could possibly attest that something is "always" so and so.

 

[harrylin]

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
[..]

All of classical mechanics, in particular relating to force? If so, where is Hooke's force law?

 

[voko]

All of classical mechanics, in particular relating to force? If so, where is Hooke's force law?

Potential energy.

 

[A.T.]

Okay, that's a nice argument, but it implicitly makes the assumption that a force has the same magnitude in all frames of reference. How is that justified?

By assumption of Hooke's law? I guess you have to assume / postulate some parts.