Norton's Dome

[swansont]

But no one is suggesting an instantaneous violation of the law, you're simply assuming there must be one if the mass moves while no additional external force is introduced (besides gravity, since it has no effect at the mass's initial rest location).

 

But the gravitational force and normal force cancel. There is no net force on the object. It is at rest, and must remain at rest, without some other force on it. That it does not remain at rest, without such a force, is a violation of the first law.

[md65536]

But the gravitational force and normal force cancel. There is no net force on the object. It is at rest, and must remain at rest, without some other force on it. That it does not remain at rest, without such a force, is a violation of the first law.

The forces cancel only at the one point. You're looking at "causes", I'm looking at "durations" in which the first law may be violated, and there is none. The "some other force" is present everywhere on the dome except the one point. I repeat: In any possible duration in which the mass is not at rest, it is acted upon by an external force. The law is not violated.

 

(Going back to my original argument, which was maybe not clearly thought out... this is why I say they analyze the thought experiment at an instant, because there is no violation in any duration.)

 

 

 

I think what is happening here is that the thought experiment is being analyzed in a purely mathematical way, and because of that we purposefully ignore the physics that are easy to ignore, the stuff that's not intuitive to assume, like uncertainty or randomness etc. YET we mistakenly keep the physics that we intuitively assume is true even in the purely mathematical view, such as causality. But I don't think you need to assume causality; Newton's first law doesn't state that the force must come before the change in momentum, in a strictly causal manner.

 

For example, a bizarre interpretation in which only a force can move the mass off the point, and in which the first law is not violated (to my understanding)... if you look at the moment that the mass has moved off the point, it becomes subject to a net force because there is a net force everywhere off that point. In quantum mechanics, I don't think you are able to say for certain that one event preceded the other on the smallest time and distance scales, so you can have a situation where the force that is there when the mass has moved off the point is the force that moves the mass off the point in the first place! This is certainly bizarre because it violates our common sense understanding of causality, but such is the nature of quantum mechanics... yet it doesn't violate the first law. The absurdities that can happen in a moment with quantum mechanics do not apply in a classical duration. --- I'm not saying that this is what is really happening when/if the mass moves off the dome, but I'm saying this is the type of assumption that is making you see a violation of the first law where there isn't one.

[swansont]

The forces cancel only at the one point. You're looking at "causes", I'm looking at "durations" in which the first law may be violated, and there is none.

 

That would be because it's instantaneous.

[md65536]

That would be because it's instantaneous.

There's no instant where the mass is not at rest and not subject to a net force.

Every instant that the mass is at rest, it is subject to no net force.

Every instant that the mass is not at rest, it is subject to a net force.

Unless you're saying there's an instant where the mass is in motion but hasn't yet moved. Or rather I think you mean that there's an instant where the mass has been moved off its initial point but is not yet subject to the net force that applies at its new position.*

 

You're saying there's no force to move the mass off its balance point. But the force is everywhere around that point. If the mass has moved, it is subject to a net force that is already part of the system. I don't accept your objection that an additional force is needed to knock the mass off its balance, because that is an answer to the question of "why" does the mass not remain at rest, and the first law only concerns "what" is observed.

 

I don't know how you can say that the law is instantaneously violated, yet the law can't be applied instantaneously. That seems like a way of saying "it is so but it can't be proven". Can your objection be expressed mathematically? When I consider the math, I see no violation.

 

 

 

 

 

* tl;dr...

I think this is the heart of the matter. You're saying that the force that knocks the mass off its balance must come before the force that is applied at the mass's new location. I'm saying that's merely classical causality, and is not encoded in the first law. The causal condition can be violated without the first law being violated.

 

I've looked at the first law to see where I might be wrong and I don't see the causal condition that you're using.

Edited June 5, 2015 by md65536

[J.C.MacSwell]

There's no instant where the mass is not at rest and not subject to a net force.

Every instant that the mass is at rest, it is subject to no net force.

Every instant that the mass is not at rest, it is subject to a net force.

That's all frame dependant. Newton's Laws are not (inertial) frame dependant.

[swansont]

 

You're saying there's no force to move the mass off its balance point. But the force is everywhere around that point. If the mass has moved, it is subject to a net force that is already part of the system.

 

That just it, though. You have give a conditional statement. It has to move off of the point to feel the force. But to do so, there must be a force, since it's at rest, and must remain at rest as long as there is no net force on it. So what force causes it to move off of the point?

[MigL]

And the only forces which are random and can account for the variable time spent at the instability point are quantum effects ( much like radioactivity ).

 

It makes no sense to discuss a thought experiment which does not take into consideration much of modern physics.

We all know classical physics is incomplete.

 

Who are you guys, Einstein, Rosen and Podwolsky ?

[md65536]

That just it, though. You have give a conditional statement. It has to move off of the point to feel the force. But to do so, there must be a force, since it's at rest, and must remain at rest as long as there is no net force on it. So what force causes it to move off of the point?

Yes, it is the lack of a classical cause that's the (only) problem.

 

The force that the mass experiences at any other point is enough to move it off its original point. It does not have to feel the force before it can begin to move, at least not to comply with the first law. The first law only says that the net force and not remaining at rest go together, not that one causes the other and not that one must come before the other. Maybe I'm missing where it says that, maybe you're mixing up what the law actually says with a common sense understanding of what the law means.

 

 

Here's another way to explain my point of view:

In one state, the mass is at rest on the balance point and no net force is applied. The first law is not violated.

In another state, the mass is not at rest and not at the balance point and a net force is applied. The first law is not violated.

Say the mass goes from the first state to the second... what causes that? Frankly it doesn't matter, it is not included in the thought experiment. What if some quantum mechanical effect, like uncertainty of location and momentum, causes the change in state? Is uncertainty of position a force? No. Does it violate the first law??? No. Does it cause the mass to no longer be at rest? No, the forces everywhere else on the dome does.

 

The cause doesn't matter, it's not a part of the first law. If it is, I don't see it. The word "remains" isn't qualified with "until some causal force", it's only "unless" the presence of a force.

It makes no sense to discuss a thought experiment which does not take into consideration much of modern physics.

We all know classical physics is incomplete.

It makes sense to me because nothing material to the thought experiment is omitted. We're not discussing a prediction of whether or not the mass actually does move off the point (the original link mentions 2 solutions, one for each case), and we don't need to discuss "why" it might move.

[swansont]

Yes, it is the lack of a classical cause that's the (only) problem.

 

The force that the mass experiences at any other point is enough to move it off its original point. It does not have to feel the force before it can begin to move, at least not to comply with the first law. The first law only says that the net force and not remaining at rest go together, not that one causes the other and not that one must come before the other. Maybe I'm missing where it says that, maybe you're mixing up what the law actually says with a common sense understanding of what the law means.

 

 

I'm missing where I say that. "before" is not in my explanation. Just that the object remains at rest without a net force on it. What is that net force, at r=0?

 

This just pushes me toward agreeing with studiot that part of the issue is dealing with a singularity, though not because of infinite density. The math can't handle the motion from r=0 to the adjacent point, because mathematically there is no adjacent point. Even though calculus deals with infinitesimals, it doesn't deal with that.

[Robittybob1]

 

I'm missing where I say that. "before" is not in my explanation. Just that the object remains at rest without a net force on it. What is that net force, at r=0?

 

This just pushes me toward agreeing with studiot that part of the issue is dealing with a singularity, though not because of infinite density. The math can't handle the motion from r=0 to the adjacent point, because mathematically there is no adjacent point. Even though calculus deals with infinitesimals, it doesn't deal with that.

I often find myself trying to find the balance point on an unstable object and even when you momentarily find it, it then tips to one side or the other. Be it the natural tremors in the ground from movements of machinery or the wind or even a heartbeat. There are these forces that we are not really aware of that will knock the object from the zero point on top of the dome.

Edited June 6, 2015 by Robittybob1

[michel123456]

[swansont]

I often find myself trying to find the balance point on an unstable object and even when you momentarily find it, it then tips to one side or the other. Be it the natural tremors in the ground from movements of machinery or the wind or even a heartbeat. There are these forces that we are not really aware of that will knock the object from the zero point on top of the dome.

 

None of which are present in the problem.

[J.C.MacSwell]

In the reverse direction there would be just one velocity at some point during the approach that would result in the ball coming to absolute rest at the top. There would be many more where it would "seem" perfectly stationary for some duration before heading off in some direction, the duration and direction being fixed by the mathematics of the idealized case.

 

So Newtonian Mechanics might not be perfectly time reversible after all…but you might have to wait around for an eternity to "prove it".

[md65536]

I'm missing where I say that. "before" is not in my explanation. Just that the object remains at rest without a net force on it. What is that net force, at r=0?

If you don't require the force before the mass is away from r=0 then where is the problem?

 

While the mass is at r=0, there is no net force and the mass is at rest.

While the mass is not at r=0, there is a net force.

 

There is no net force at r=0. The net force is everywhere else. If a force causes the mass to move off of r=0, and no other forces are present in the system, then the mass must not be at r=0 when it has moved.

 

Yes, you're right that the mass remains at rest while no net force is applied!

But there is a net force that is applied in any other case.

 

 

Are any of these statements false?

Edited June 6, 2015 by md65536

[swansont]

If you don't require the force before the mass is away from r=0 then where is the problem?

 

While the mass is at r=0, there is no net force and the mass is at rest.

While the mass is not at r=0, there is a net force.

 

There is no net force at r=0. The net force is everywhere else. If a force causes the mass to move off of r=0, and no other forces are present in the system, then the mass must not be at r=0 when it has moved.

 

Yes, you're right that the mass remains at rest while no net force is applied!

But there is a net force that is applied in any other case.

 

 

Are any of these statements false?

 

No.

 

The mass is at r=0 and there is no force on it, so it must remain at rest. That's what Newton's first law says. Is that false?

[studiot]

At the risk of yet another unwarranted bloody nose in this thread I would wish to comment that it is instructive to consider the issue in the light of the inverted pendulums theorem.

Edited June 6, 2015 by studiot

[md65536]

[accidentally erased this comment with an edit when I meant to add a new reply]

Edited June 6, 2015 by md65536

[studiot]

 

The mass is at r=0 and there is no force on it, so it must remain at rest. That's what Newton's first law says. Is that false?

 

 

No I don't think that's what Newton's first law says and I don't thing that is the situation in Norton's dome either.

 

Edit here is the generally agreed nearest English translation to N1, which was actually written in Latin.

 

 

Law I. 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.

 

Edited June 6, 2015 by studiot

[swansont]

 

No I don't think that's what Newton's first law says and I don't thing that is the situation in Norton's dome either.

 

AFAIK "A mass at rest remains at rest unless acted upon by an external force" is the common phrasing of the relevant part of the first law.

 

edit: that seems to be a faithful phrasing of what you quoted

[studiot]

But the mass in Norton's dome is being affected by (at least) two external forces.

Edited June 6, 2015 by studiot

[Robittybob1]

 

No.

 

The mass is at r=0 and there is no force on it, so it must remain at rest. That's what Newton's first law says. Is that false?

That is true but it is a hard place to physically find that spot, but in theory it is a solution.

But the mass in Norton's dome is being affected by (at least) two external forces.

You'd think Newton meant an unbalanced external force.

[md65536]

The mass is at r=0 and there is no force on it, so it must remain at rest. That's what Newton's first law says. Is that false?

Trying again...

 

Yes, this is false; Newton's first law does not say all of that.

 

Yes, if the first part ("The mass is at r=0 and there is no force on it") is true, Newton's law says that the second part ("so it must remain at rest") is true. If the second part is true, then the first part remains true. The first part is initially true.

 

Yes, there is a contradiction if only one part is true and the other is false.

 

What Newton's law does NOT say is that both parts cannot be false together.

 

And certainly Newton's law does not say that it must remain at rest indefinitely or for some amount of time EVEN in the case that a force is introduced (as is the case if the mass is not at r=0). If the mass is not at r=0, both parts are false. I don't accept that previous applications of the law still apply even if the conditions which would violate the law no longer hold.

 

 

----

Edit: Here's yet another way to explain how I'm looking at this.

 

Assume the first part is true: "The mass is at r=0 and there is no force on it"

What Newton's law says is "So it must remain at rest unless there is a force acting on it."

Since we've already established that there is no force, the "unless" part can be removed, and your statement is true.

 

However, if you change the situation so that the mass is no longer at r=0, the condition that no force acts on it is no longer true. You can not still say that Newton's law still says that "it must remain at rest", because what Newton's law actually says is "it must remain at rest unless a force is acting upon it".

 

The issue is... for how long, or until when, does "remains" apply? By a strict reading of the law, it applies "unless" a force acts upon it. The "remains" requirement disappears if the "unless" is satisfied. It does not have to last.

Edited June 6, 2015 by md65536

[Robittybob1]

 

It's a Newtonian physics issue, so not being exactly at rest doesn't come into play. And the normal force cancels that of gravity, so F=0 at the start.

I agree you can be at rest but any random influence can disturb that equilibrium. They seem to call that "spontaneous" maybe you need to argue whether random and spontaneous are synonymous?

...

Basically, there is a dome shape and the top is a point of unstable equilibrium. There is a solution where an object at the top can spontaneously move off of the top, and once it does, continues accelerating. This is offered as an example of a violation of determinism, as the motion is purportedly spontaneous.

....

 

 

are random experimental errors influencing the outcome "spontaneous"?

[studiot]

 

You'd think Newton meant an unbalanced external force.

 

 

 

I wouldn't, but then I read very carefully what Newton actually wrote.

 

The real solution to this apparent dilemma is in appreciating that all three of Newtions Laws are necessary.

Together they make up complete set that prohibits the sort of difficulty or contradiction that can be otherwise dreamt up.

Unfortunately Newtons Laws are all to often presented with one as a special case of another.

This is just not so.

[Robittybob1]

 

 

I wouldn't, but then I read very carefully what Newton actually wrote.

 

The real solution to this apparent dilemma is in appreciating that all three of Newtions Laws are necessary.

Together they make up complete set that prohibits the sort of difficulty or contradiction that can be otherwise dreamt up.

Unfortunately Newtons Laws are all to often presented with one as a special case of another.

This is just not so.

You present a poor argument. You make a case but don't give us your evidence for the things you argue.