Why do some people say that Newton's Second law is the real Law of Motion?

[Rakinniya]

People say that Newton's Second law is the real Law of Motion because both the First and the third law can be proved from the second. If this is true why did Newton state them as separate laws if they are just special cases of the 2nd law? My teacher taught us this.
These are the proofs she taught...

1)1st Law frm 2nd

In the eq
F=ma
when we put F=0 we get Acceleration = 0 ----> the first Law of motion

okay this is fine...

2)Now..3rd frm 2nd..
let two bodies interact
let F1 and F2 be the action n reaction
then according to the second law
F1=dP1/dt F2=dP2/dt


F1+F2 = d/dt (P1+P2)

when no external forces are acting... according to the law of conservation of momentum Momentum must be conserved

P1+P2=const

F1+F2 = d/dt (P1+P2)

F1+F2=0

F1= -F2 ----->third Law

But the problem I find here is ... We prove the Law of conservation of momentum by applying the third law ... Then isn't it silly and incorrect to use the momentum law to prove the third law...? ...

.. Please tell me if there is some other valid proof...or any changes that can be made in this one

 

[tsw99]

Linear momentum is conserved because of Newton's 3rd law. I think it is on every physics textbook

3 laws are essential, they are not proved from each other.

 

[pabloenigma]

Linear Momentum Of a System Of particles is conserved because Of 3rd Law.For a Single particle,2nd Law Suffices.
Conservation Of Momentum and Angular Momentum for a system of particles rest On third Law.I don't think Its derivable from the 2nd.Once upon a time I saw such a derivation in a very bad school textbook.But it doesn't make any sense.

 

[Rakinniya]

Thank you...It's not there in my textbook...our teacher taught it ...but I found it in a guide as well...and its de on some sites also...ofcourse it doesn't make any sense ...but what about the First Law?...Is it or not just a special case of the second...?

 

[Ambidext]

I see it as more of a "check" rather than a proof. physics can get very ambiguous sometimes. When you come across Daniel schroeder's thermal physics you'll see what i mean. :p

 

[Dickfore]

1)1st Law frm 2nd

In the eq
F=ma
when we put F=0 we get Acceleration = 0 ----> the first Law of motion

This is true, but, perform the following experiment:

Inside a standing vehicle, put a ball on a horizontal surface. It should remain still. It means all the forces acting on it are in equilibrium. Then, the vehicle starts accelerating. You notice the ball starts to roll. If it starts to move, its velocity surely changes, therefore it acquires some acceleration. However, nothing changed with the forces acting on the ball before and after the acceleration. If before the acceleration Ʃ F = o, so should it be after the acceleration. But, now we have a contradiction: The sum of all the forces is zero, and the ball is accelerating. This means First Newton's Law is incorrect. Because you derived it from the Second Law, it means that one is incorrect as well?

2)Now..3rd frm 2nd..
let two bodies interact
let F1 and F2 be the action n reaction
then according to the second law
F1=dP1/dt F2=dP2/dt


F1+F2 = d/dt (P1+P2)

when no external forces are acting... according to the law of conservation of momentum Momentum must be conserved

P1+P2=const

F1+F2 = d/dt (P1+P2)

F1+F2=0

F1= -F2 ----->third Law

But the problem I find here is ... We prove the Law of conservation of momentum by applying the third law ... Then isn't it silly and incorrect to use the momentum law to prove the third law...? ...

.. Please tell me if there is some other valid proof...or any changes that can be made in this one

You are right, you performed a circular argument. Third Newton's Law is related to Conservation of Linear momentum.

 

[Rakinniya]

Dickfore ... I don't really understand the mistake you are pointing out... and about what you said
"Then, the vehicle starts accelerating. You notice the ball starts to roll. If it starts to move, its velocity surely changes, therefore it acquires some acceleration."

Here you are talking about the velocity of the ball w.r.t the ground right??..so at first the ball is not at rest it is moving with respect to the ground... and though the train accelerates...the ball rolls because it has to stay its earlier state of uniform motion(cuz no force is acting on the ball)...so at that instant it is not accelerating with the train......

 

[spaghetti3451]

People say that Newton's Second law is the real Law of Motion because both the First and the third law can be proved from the second. If this is true why did Newton state them as separate laws if they are just special cases of the 2nd law? My teacher taught us this.
These are the proofs she taught...

1)1st Law frm 2nd

In the eq
F=ma
when we put F=0 we get Acceleration = 0 ----> the first Law of motion

okay this is fine...

Newton's first law is, in fact, a special case of the second. I understand that you are confused as to why we need a 'Newton's First Law' if it can be derived from 'Newton's second law'. In other words, you want to boil down the rules of mechanics down to the most basic, fundamental postulates/laws/assumptions and since you see that Newton's second law is the most fundamental, you want to rip the first law of its status as a fundamental law. That is all fine, and you are right to say that Newton's second law is the most fundamental postulate of Newtonian mechanics. I am going to refer you to this link https://www.physicsforums.com/showthread.php?t=165100. It has further discussions about the problem you mentioned. The discussion there should help you develop your own point of view regarding the issue.

2)Now..3rd frm 2nd..
let two bodies interact
let F1 and F2 be the action n reaction
then according to the second law
F1=dP1/dt F2=dP2/dt


F1+F2 = d/dt (P1+P2)

when no external forces are acting... according to the law of conservation of momentum Momentum must be conserved

P1+P2=const

F1+F2 = d/dt (P1+P2)

F1+F2=0

F1= -F2 ----->third Law

But the problem I find here is ... We prove the Law of conservation of momentum by applying the third law ... Then isn't it silly and incorrect to use the momentum law to prove the third law...? ...

.. Please tell me if there is some other valid proof...or any changes that can be made in this one

Actually, Newton's Third Law is correct if and only if the law of conservation of motion is valid. One cannot exist without the other. Once you will have studied quantum mechanics and relativity, you will understand that the law of conservation of momentum is the more fundamental principle. The law of conservation of momentum has been taken to be one of the fundamental postulates of all of physics and Newton's laws have been modified to take into account some of the bizarre experimental results that have been noticed in the late 19th century. These experiments could not be explained using Newtonian mechanics so physicists had to resort to modifying Newton's laws themselves (to form what is called quantum mechanics and relativity) to explain the observations. However, even in the newer modern theories, the law of conservation of momentum is still valid.

 

[spaghetti3451]

This is true, but, perform the following experiment:

Inside a standing vehicle, put a ball on a horizontal surface. It should remain still. It means all the forces acting on it are in equilibrium. Then, the vehicle starts accelerating. You notice the ball starts to roll. If it starts to move, its velocity surely changes, therefore it acquires some acceleration. However, nothing changed with the forces acting on the ball before and after the acceleration. If before the acceleration Ʃ F = o, so should it be after the acceleration. But, now we have a contradiction: The sum of all the forces is zero, and the ball is accelerating. This means First Newton's Law is incorrect. Because you derived it from the Second Law, it means that one is incorrect as well?

Pardon me if I'm wrong, but the vehicle is a non-inertial reference frame when it is accelerating, isn't it? And we all know Newton's laws of motion are not valid in a non-inertial frames? :-)

 

[Dickfore]

Here you are talking about the velocity of the ball w.r.t the ground right?

No, relative to you, while you are sitting still in the vehicle.

 

[D H]

Newton's first law is, in fact, a special case of the second.

No, it's not, for several reasons.

The most important reason is that Newton's second and third law are valid only in an inertial frame of reference. The modern interpretation of Newton's first law is that serves as a test of whether a frame of reference is inertial. Newton's first law tells us whether his other laws are even applicable.

It's also important to consider the historical background. Newton's first law went directly against the grain of Aristotelian physics. Newton was establishing a background for his other laws. (Aside: Only Newton's third law is Newton's. Newton explicitly attributed the first two laws to his predecessors in his Principia.)

A third reason is that there are some pathological cases such as Norton's dome where Newtonian mechanics from the perspective of Newton's second and third law only are non-deterministic. Add in Newton's first law and voila! those non-determinstic solutions disappear. Newton's first law in fact does add something that is not in Newton's second law.

 

[Rakinniya]

No, relative to you, while you are sitting still in the vehicle.

Then its worng isn't it??...It is an accelerating frame of reference...provided we don't fall down

 

[Rakinniya]

so which is more fundamental? Newton's third law of Motion or The law of conservation of linear momentum? Both can't be...

 

[Dickfore]

They're equivalent within Classical Mechanics. However, when you consider particles interacting with a field, the field carries momentum as well. Total momentum is conserved, due to homogeneity of space, but Third Newton's Law may not hold anymore.

 

[Dadface]

No, it's not, for several reasons.

The most important reason is that Newton's second and third law are valid only in an inertial frame of reference. The modern interpretation of Newton's first law is that serves as a test of whether a frame of reference is inertial. Newton's first law tells us whether his other laws are even applicable.

.

The above comment surprised me and seemed to be paradoxical.After a quick search I read something similar in a Wiki article.But I am still surprised.The problem as I see it is that the second law expresses the first law in that when F equals zero dv/dt=zero(in other words the object must be at rest or moving with constant velocity).If my interpretation here is correct we have the second law serving as a test to tell us whether the second law is applicable.

 

[Dickfore]

If my interpretation here is correct we have the FIRST law serving as a test to tell us whether the second law is applicable.

I think you made a typo. If it was supposed to read the way I corrected it, then you're right.

 

[Dadface]

I think you made a typo. If it was supposed to read the way I corrected it, then you're right.

Thanks for pointing out what you thought was a mistake but it's not a typo.The main point I'm making is that the second law expresses the first law.

 

[D H]

The above comment surprised me and seemed to be paradoxical.After a quick search I read something similar in a Wiki article.But I am still surprised.The problem as I see it is that the second law expresses the first law in that when F equals zero dv/dt=zero(in other words the object must be at rest or moving with constant velocity).

Don't look at it that way then!

I, along with others (you have found some of the others), prefer to look at Newton's 1st as distinct from the other two laws. Newton made his first law distinct from his second for some reason. Why? This is perhaps a bit of appeal to authority, but Newton was dang smart. He certainly was smarter than am I, smarter than almost everyone at this forum, and quite possibly the smartest person on the planet, ever.

Newton's 1st does a nice job of resolving some pathological cases in Newton's 2nd law. I've raised the specter of Norton's dome in various other threads. Here's a brief synopsis of a recent discussion:

Can you write the ODE?

[tex]\frac {d^2 r(t)}{dt^2} = \sqrt r[/tex]Given initial conditions r(0)=0, r'(0)=0, one solution is the trivial solution r(t)=0. It also has non-trivial solutions
[tex]r(t) = \begin{cases} 0 & t<t_0 \\ \frac{(t-t_0)^4}{144} & t\ge t_0 \end{cases}[/tex]
This "bowl" is called Norton's dome.

At first glance, I'm not sure I agree that the non-trivial solutions satisfy Newton's First Law. Although it is an interesting point -- I'd never really given much thought to why the First Law might have been stated separately from the Second. But here might be one case where it might potentially see some use.

 

[Fredrik]

No, it's not, for several reasons.

The most important reason is that Newton's second and third law are valid only in an inertial frame of reference. The modern interpretation of Newton's first law is that serves as a test of whether a frame of reference is inertial. Newton's first law tells us whether his other laws are even applicable.

It's also important to consider the historical background. Newton's first law went directly against the grain of Aristotelian physics. Newton was establishing a background for his other laws. (Aside: Only Newton's third law is Newton's. Newton explicitly attributed the first two laws to his predecessors in his Principia.)

A third reason is that there are some pathological cases such as Norton's dome where Newtonian mechanics from the perspective of Newton's second and third law only are non-deterministic. Add in Newton's first law and voila! those non-determinstic solutions disappear. Newton's first law in fact does add something that is not in Newton's second law.

I had never heard of Norton's dome, so thanks for mentioning that. I found a nice explanation of it on youtube.



The non-trivial solution that's presented at 8:40-9:00 has ##dr^2/dt^2=1/12(t-T)^2##, so at t=T, i.e. the moment when the ball starts rolling, the acceleration is =0. So it doesn't look like Newton's first can rule out the non-trivial solutions.

I have personally never been a fan of the idea that Newton's second law should be reinterpreted as saying something very different from what Newton actually said. I would say that the first is a special case of the second. I think that a better way to teach mechanics (to an audience with some mathematical maturity) is to start by defining Galilean spacetime (instead of the first law) and then present Newton's second law in the form x''(t)=f(x'(t),x(t),t). Here it would be appropriate to also talk about the existence and uniqueness theorem for differential equations of this type. Mass can be introduced after explaining why we need it, as in example 3 here. I would of course mention the third as well, but I think of it as less significant than the others. It's just a statement about what sort of forces we should expect to encounter. Note that Newton's third is automatically present when we state Coulomb's law or Newton's law of gravity. (If the force that an arbitrary particle exerts on another arbitrary particle is given by the standard formula, then the statement isn't just about what A does to B, but also about what B does to A).

 

[D H]

I have personally never been a fan of the idea that Newton's second law should be reinterpreted as saying something very different from what Newton actually said.

Be careful there. There's a perennial argument over whether Newton said F=ma or F=dp/dt. In fact, he said neither. His Principia is for the most part calculus-free. Saying that Newton's 2nd is F=ma or F=dp/dt is a bit of historical revisionism.

That said, you can take Newton's 1st at face value as eliminating all but the trivial solution to Norton's dome. The ball is initially at rest perched atop the dome and the net force on the ball is zero so long as it remains perched atop the dome. Ergo the ball remains at rest perched atop the dome forever.

I think that a better way to teach mechanics (to an audience with some mathematical maturity) is to start by defining Galilean spacetime (instead of the first law) and then present Newton's second law in the form x''(t)=f(x'(t),x(t),t). Here it would be appropriate to also talk about the existence and uniqueness theorem for differential equations of this type.

That is putting the cart before the horse, miles and miles before the horse!

Most students of science and engineering are taught Newton's laws in high school, well before they have had calculus at all, let alone the mathematical maturity you are talking about. They are re-taught Newton's laws in freshman physics at the same time they are just starting to learn calculus.

 

[pabloenigma]

Well,thanks for the info about Nortons DOme.It was illuminating to know.
But here is an article that begs to differ.What do you guys think about it?

http://www.pitt.edu/~jdnorton/Goodies/Dome/index.html

 

[Fredrik]

Be careful there. There's a perennial argument over whether Newton said F=ma or F=dp/dt. In fact, he said neither. His Principia is for the most part calculus-free. Saying that Newton's 2nd is F=ma or F=dp/dt is a bit of historical revisionism.

Yes, his laws were wordy statements in latin, which have been translated into English, and then into mathematical terms. But the first said something close enough to "F=0 implies a=0" and the second said something close enough to "F=ma". The book that was used for my first two courses in classical mechanics at the university (Kleppner & Kolenkow) claimed that the first law is the assumption that "inertial frames exist". That looks more like a revision of history than anything else.

That said, you can take Newton's 1st at face value as eliminating all but the trivial solution to Norton's dome. The ball is initially at rest perched atop the dome and the net force on the ball is zero so long as it remains perched atop the dome. Ergo the ball remains at rest perched atop the dome forever.

I don't follow you here. I thought your argument was that since there's no force, there's no acceleration, and therefore r=0 is the only solution consistent with the first law in the form "F=0 implies a=0". Is your argument different from this? (This argument doesn't work, because the non-trivial solution has a=0 too).

That is putting the cart before the horse, miles and miles before the horse!

Most students of science and engineering are taught Newton's laws in high school, well before they have had calculus at all, let alone the mathematical maturity you are talking about. They are re-taught Newton's laws in freshman physics at the same time they are just starting to learn calculus.

You're certainly right about how these things are usually done, but I strongly disagree with that first statement. I'd say that what I'm suggesting is to put the horse back where it belongs in front of the cart. The traditional approach makes the first course in mechanics slightly easier, but leaves the students unable to answer simple questions like "is the first law a special case of the first". So someone who really wants to understand how to define a theory of motion, and really wants to know what this theory says, will have to re-learn the basics of classical mechanics once more after that second encounter.

I think there are also significant differences between different countries. Our first mechanics course was at the end of the first year, after a soft intro to university-level math, a linear algebra course, and two twice as long calculus courses, one of them really hard. A more mathematical approach would have made a lot of sense right there, but instead it was almost as if we were supposed to forget what we had learned.

 

[Fredrik]

Well,thanks for the info about Nortons DOme.It was illuminating to know.
But here is an article that begs to differ.What do you guys think about it?

http://www.pitt.edu/~jdnorton/Goodies/Dome/index.html

Begs to differ about what?

 

[pabloenigma]

Begs to differ about what?

What it shouts out Bold and clear is that the non-trivial solution in Nortons Dome doesn't really violate Newtons 1st Law,and presents and convincing argument in favor of it.If we agree on this,we are back to square one,whether 1st Law really contains any information not present in 2nd Law

 

[nonequilibrium]

Can someone give an explicit argument for why he believes the first law forbids the non-trivial solutions to Norton's dome?

 

[D H]

I don't follow you here. I thought your argument was that since there's no force, there's no acceleration, and therefore r=0 is the only solution consistent with the first law in the form "F=0 implies a=0". Is your argument different from this? (This argument doesn't work, because the non-trivial solution has a=0 too).

You are reading things into Newton's first law that aren't there. Take Newton's first at its face value: Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon. There is not one word about acceleration in this law. The nontrivial solutions to Norton's dome very much violate Newton's first law. The ball is not "persevering in its state of rest." Newton's first law to me does add something that is not present in the differential equations of motion.

That said, saying that a justification for Newton's 1st is that it helps resolve some of these 21st century pathological cases is historical revisionism. The real reason for Newton's first law was that it was an "in yo' face, Aristotle" kind of statement. It set the stage for the laws that followed.

 

[Fredrik]

in yo' face, Aristotle

You are reading things into Newton's first law that aren't there. Take Newton's first at its face value: Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon. There is not one word about acceleration in this law. The nontrivial solutions to Norton's dome very much violate Newton's first law. The ball is not "persevering in its state of rest." Newton's first law to me does add something that is not present in the differential equations of motion.

If that statement is interpreted as a statement about the situation at a specific time t, then I'd say that there's no way to interpret it as saying anything other than "F=0 implies a=0". See the explanation at the end*. This version of the first law doesn't rule out non-trivial solutions.

If we interpret it as a statement about what's going on during a time interval [t₁,t₂], then it's a stronger statement. "If for all t in [t₁,t₂], we have F(t)=0, then for all t in [t₁,t₂], we have a(t)=0". This also doesn't rule out non-trivial solutions, because if we choose t₁ and t₂ such that T is in [t₁,t₂), then the antecedent of the implication is false, and if we choose t₂=T, then the non-trivial solution says that we have F=0 and a=0 in the entire interval. Note that the non-trivial solution in the video has F=0 precisely when a=0.

Edit: Just to be clear, the non-trivial solution I'm talking about is the r defined by
$$r(t)=
\begin{cases}
0 &\text{ if } t<T\\
\frac{1}{144}(t-T)^4 &\text{ if } t\geq T.
\end{cases}
$$
We could interpret the statement you posted as a statement about all times. "If there is never any force, there is never any acceleration", but this rule doesn't apply here, since there's a non-zero force in this problem.
*) What can "perseveres in its state of rest" mean, other than "continues to have velocity 0"? What can "perseveres in its state of uniform motion" mean, other than "continues to have the same non-zero velocity as before"? So a body that "perseveres" doesn't change its velocity. By definition of acceleration, that means a=0.

If we trim the statement a bit, it says that the body "perseveres" unless there are forces acting on it. I'd say that "A unless B" means "(not B) implies A", and since "perseveres" means "a=0" and the stuff after the "unless" means F≠0, the statement is saying that "F=0 implies a=0".

Edit: Sorry about editing. I hope it didn't cause any confusion. Some edits were necessary since I used the symbol r for two different things.

 

[nonequilibrium]

Ah, DH, I get your point. I see how "Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon" differs from "a = 0, unless there is a force". Until an hour ago or so I thought they were the same, but Norton's dome proves me wrong.

But I wanted to ask, do you then also claim Newton's laws are not time-reversible? It seems like an essential consequence to me.

 

[D H]

That ROFL is a key reason, and quite possibly the key reason, for Newton's 1st. Aristotelian physics was still taught in Newton's time. Aristotelian physics says that an object's natural state of being is to be rest. Roll a boulder down a hill and while it might bounce around a bit, it will soon come to rest at the bottom of the hill. Roll a boulder on a flat surface and it will come to rest right quick. Try to slide or push that resting boulder across the ground and it will resist you, a lot. The boulder "wants" to be at rest. Anything that is solid "wants" to be at rest per Aristotelian physics. Newton wanted to say up front that Aristotelian physics is wrong, wrong, wrong. So he did.

What can "perseveres in its state of rest" mean, other than "continues to have velocity 0"? What can "perseveres in its state of uniform motion" mean, other than "continues to have the same non-zero velocity as before"? So a body that "perseveres" doesn't change its velocity. By definition of acceleration, that means a=0.

Read his words, without math. Persevere means indefinitely (i.e., forever), not just over some finite interval [t1, t2].

You are reading way too much modern math into your interpretation of Newton's laws. Newton did not have those mathematical tools. Algebra was very new in Newton's time. Many of his contemporaries didn't quite trust it. Newton certainly didn't. He had some choice words for those who used algebra in lieu of geometrical reasoning. Newton preferred geometric reasoning over his own calculus, this being one reason why his Principia is for the most part devoid of his calculus.Newton's 1st law is a null hypothesis. All kinds of differential equations are consistent with that null hypothesis. For example, anything of the form [itex]d^2x/dt^2 \propto F^k[/itex] is consistent with Newton's 1st law, as is [itex]dx/dt\propto F^k[/itex]. Newton's 2nd law is just one of an infinite number of forms that is consistent with his 1st law. I like to think of Newton's 2nd as a specialization of Newton's 1st rather than the other way around.

 

[Fredrik]

That ROFL is a key reason, and quite possibly the key reason, for Newton's 1st.

I don't doubt it. I just thought it was funny.

Read his words, without math. Persevere means indefinitely (i.e., forever), not just over some finite interval [t1, t2].

I disagree with that interpretation of the word "persevere" (if it meant "forever", it could never be used about a human), but let's say that we choose to interpret the first law that way anyway, i.e. as "F(t)=0 for all t implies a(t)=0 for all t". It still doesn't rule out the non-trivial solution, because the non-trivial solution doesn't satisfy the antecedent of that implication.

You are reading way too much modern math into your interpretation of Newton's laws.

If we don't, the "laws" are just fuzzy statements that don't say much at all.

Since you're trying to use the first law to refute a solution of a differential equation, you have no choice but to interpret it as a mathematical statement, because only a mathematical statement can imply something about mathematics. I suggested three different mathematical interpretations and found that none of them refutes the non-trivial solution.

 

[Dadface]

I looked at the Nortons dome video and got stuck pretty early on (just after one minute and thirty seconds).The presenter gave a key equation that describes the dome but the equation does not balance in terms of units.Is the whole analysis invalidated or have I overlooked something?

 

[netgypsy]

AXIOMATA, SIVE LEGES MOTUS

[Leges solæ descripta sunt, commentariis prætermissis.]

Lex I

Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus illud a viribus impressis cogitur statum suum mutare.

Lex II

Mutationem motus proportionalem esse vi motrici impressæ, & fieri secundum lineam rectam qua vis illa imprimitur.

Lex III

Actioni contrariam semper & æqualem esse reactionem: sive corporum duorum actiones in se mutuo semper esse æquales & in partes contrarias dirigi.

 

[netgypsy]

Pretty straight forward. You think Isaac is chuckling at this conversation?

 

[Fredrik]

I looked at the Nortons dome video and got stuck pretty early on (just after one minute and thirty seconds).The presenter gave a key equation that describes the dome but the equation does not balance in terms of units.Is the whole analysis invalidated or have I overlooked something?

I hope so, because otherwise no curved shapes can be described mathematically. Probably best to think of everything as dimensionless. Alternatively, you can imagine that there's a (dimensionless 1)*(appropriate unit) multiplying every term.

 

[Rakinniya]