B Gravitational Force acting on a massless body

[Dale]

Sigh.

https://phys.libretexts.org/Bookshe...Harmonic_Motion/11.02:_Simple_Harmonic_Motion

https://phys.libretexts.org/Bookshe...Waves_in_One_Dimension/12.06:_Advanced_Topics

https://phys.libretexts.org/Bookshe...cs_of_Waves_(Goergi)/05:_Waves/5.04:_New_Page

I know you are well aware of these examples so go ahead with your excuse that, for whatever reason, these examples don’t apply. But these examples all clearly and explicitly involve theoretical massless objects in Newtonian physics.

So your blanket assertion that they don’t exist in Newtonian physics is wrong. You need to make a much more limited claim, something like “massless objects <insert excuse from above> in Newtonian physics”. The broad claim that there are no massless objects in Newtonian physics is refuted.

 

[valenumr]

Ok. Then show an example in a textbook where they write down the equation of motion of a zero-mass object. I don't know a single example. The reason is simply that such a thing cannot even be defined mathematically.

Well, ideal pendelum. I think the point (edit for clarity) [is that] mass is neglected as factor to simplify things, but it's not the same as proper treatment of a massless object.

 

[italicus]

@Dale

I agree with @vanhees71 ;, in Newtonian mechanics, not relativistic, we very often consider some objects as having trascurable mass wrt other parts of the system concerned, and say : let’s put it to zero. For example, dealing with the Atwood machine we in the simplified version trascure the mass of the pulley; but in a more accurate treatment we don’t; this doesn’t mean that the pulley is “massless’ (n the first hypothesis).

 

[vanhees71]

Sigh.

https://phys.libretexts.org/Bookshelves/University_Physics/Book:_University_Physics_I_-_Classical_Mechanics_(Gea-Banacloche)/11:_Simple_Harmonic_Motion/11.02:_Simple_Harmonic_Motion

https://phys.libretexts.org/Bookshelves/University_Physics/Book:_University_Physics_I_-_Classical_Mechanics_(Gea-Banacloche)/12:_Waves_in_One_Dimension/12.06:_Advanced_Topics

https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/The_Physics_of_Waves_(Goergi)/05:_Waves/5.04:_New_Page

I know you are well aware of these examples so go ahead with your excuse that, for whatever reason, these examples don’t apply. But these examples all clearly and explicitly involve theoretical massless objects in Newtonian physics.

So your blanket assertion that they don’t exist in Newtonian physics is wrong. You need to make a much more limited claim, something like “massless objects <insert excuse from above> in Newtonian physics”. The broad claim that there are no massless objects in Newtonian physics is refuted.

None of this examples treat the dynamics of a massless system. All are in the category I mentioned from the first posting on this subject on: You neglect the mass of parts of the setup (here the springs) against other masses. Again: There is no physically interpretable dynamics of massless objects in Newtonian physics.

 

[Motore]

The OP already got the answer, but I am posting here another one:
https://physics.stackexchange.com/a/561207

Of course the OP was reminded that the photon cannot be treated accurately using Newtonian dynamics, but a massless object (like a ultra small ball) can be. Here massless means negligible mass and that is what @Dale was referring to all along in my opinion. So a small massless ball does not exist in reality as does not a massless spring, but we can (and we do) use them (as referenced by @Dale in post #31).

 

[Dale]

There is no physically interpretable dynamics of massless objects in Newtonian physics.

So then make that specific narrow claim, not your overly broad claim that there are no massless objects in Newtonian physics. There manifestly are such objects even if they fail some specific criterion that massive objects fulfill.

However, I disagree with even this more narrow claim. With a massless spring we determine the spring’s motion and internal energy. What dynamics are missing? What more dynamics do we get for a massive object?

 

[vanhees71]

I rest my case. It's obviously impossible to convince you about mathematical facts.

 

[ergospherical]

With a massless spring we determine the spring’s motion and internal energy. What dynamics are missing? What more dynamics do we get for a massive object?

This is pushing the analogy too far - if two particles are said to be connected by a massless spring it means only that they interact via a force which depends affine linearly on their separation. The actual details of how this force arises or the mechanism of energy storage is irrelevant to the problem. It's not an actual spring with its own dynamics, so much as a heuristic device to help you visualise the interaction.

 

[italicus]

@Dale

“So, an object attached to an ideal, massless spring”…

This is clearly an “ideal” situation, in which the spring mass is negligible, but not exactly zero. So negligible that you let it tend to zero, to simplify dynamics.
Any material body has a mass. If you apply a Force to a non-existing massless body, its acceleration would be infinite.

 

[Dale]

I rest my case. It's obviously impossible to convince you about mathematical facts.

@vanhees71 I have avoided making this personal, please do the same. If the only response you can do is to make it personal, then don’t respond at all.

Note, I have objected specifically to your overly broad claim that massless objects don’t exist in Newtonian physics. They clearly do as evidenced by textbooks that use them.

You have simply stated that they have no physically interpretable dynamics, which is a more narrow claim than your original. And since a massless spring’s motion and internal energy is often calculated, even this more narrow claim is unsupported.

 

[jbriggs444]

I am trying to understand the "debate" here and would prefer to see a claim cast in simpler terms, understandable to a first year physics student who is comfortable with the Newtonian formulation of classical mechanics.

Something like "for a massless particle subject to no external force other than gravity, all trajectories are consistent with the laws of Newtonian mechanics".

 

[ergospherical]

As per usual it's not even an interesting argument but rather trademark PF labouring over irrelevant minutiae

 

[Dale]

This is pushing the analogy too far - if two particles are said to be connected by a massless spring it means only that they interact via a force which depends affine linearly on their separation. The actual details of how this force arises or the mechanism of energy storage is irrelevant to the problem. It's not an actual spring with its own dynamics, so much as a heuristic device to help you visualise the interaction.

Even in the case where it is a stand in for an unspecified force the fact remains that said force is being analyzed by treating it as a massless spring. If you can use a theory to analyze object A by treating it as object B then it is rather a stretch to claim that the theory cannot handle object B or that somehow object B is excluded from the theory. In fact, it indicates the opposite, massless springs are so easy to handle with the theory that it is worthwhile to treat more difficult things as though they were massless springs.

 

[Dale]

It's a mathematical fact that there is no physically interpretable dynamics known for massless particles within Newtonian mechanics.

You have made this claim but with neither any explanation what you mean nor with any evidence to support it. A massless spring can be described in terms of its position, its velocity, and its internal energy, so specifically what physically interpretable dynamics are even missing?

Furthermore, in whatever sense you claim a massless spring doesn’t have dynamics is a far more narrow claim than your overly broad claim that massless objects simply don’t exist in Newtonian theory. They do. That is clearly demonstrated from the professional scientific literature.

 

[Dale]

This is clearly an “ideal” situation, in which the spring mass is negligible, but not exactly zero. So negligible that you let it tend to zero, to simplify dynamics.
Any material body has a mass. If you apply a Force to a non-existing massless body, its acceleration would be infinite.

Yes, obviously there are no real massless springs. The claim I am disputing is the overly broad claim from post 6 that there are no massless objects in Newtonian theory. I.e. that such objects cannot be used in Newtonian theory.

Obviously, I and everyone else understands that such objects are used as approximations to real objects, and the real object has mass. But the point is that such massless objects are compatible with the theory, as evidenced by their frequent use in introductory textbooks on the subject and as evidenced by the fact that we frequently treat other objects or forces as though they were massless springs for theoretical calculations.

To make a blanket claim that there are no massless objects in Newtonian physics is going too far.

 

[George Jones]

None of this examples treat the dynamics of a massless system. All are in the category I mentioned from the first posting on this subject on: You neglect the mass of parts of the setup (here the springs) against other masses. Again: There is no physically interpretable dynamics of massless objects in Newtonian physics.

I, too, say that massless objects do not exist in Newtonian mechanics.

Nonsense. In the many textbooks where massless strings and springs are used they are indeed describing theoretical objects which are theoretically assigned zero mass and then treated according to standard Newtonian theory.

I would agree that massless strings don’t exist in reality, but they are commonplace in Newtonian theory. To state otherwise is to contradict most introductory physics textbooks.

While it is true that first-year texts abound with examples that use "massless" strings/springs, it is also true that these same texts point out that out that the strings/springs are not massless. A random sample of three popular first-year texts (itailcs and bold below are used in the texts):

Halliday and Resnick "In the special case in which the weight of the spring is negligible ..."

Serway and Jewett "In problem statements, the synonymous terms light and of negligible mass are used to indicate that a mass is to be ignored when you work the problems."

Knight: "Often in physics and engineering problems the mass of the string or rope is much less than the masses of the objects that it connects. In such cases, we can adopt the massless string approximation. In the limit $m_s \rightarrow 0$, Equation 7.8 becomes ..."

When I teach first-year physics, I use the term "negligible" before taking $m=0$ in the first few such examples that I present.

 

[Dale]

While it is true that first-year texts abound with examples that use "massless" strings/springs, it is also true that these same texts point out that out that the strings/springs are not massless.

Completely agreed, as mentioned earlier. Real springs are not massless and it is just an approximation.

Nonetheless, in the theory they are in fact treated as having 0 mass, and nothing about the theory breaks when that is done.

 

[gmax137]

Nonetheless, in the theory they are in fact treated as having 0 mass, and nothing about the theory breaks when that is done.

As long as they have exactly zero net force, right? Other wise you get a "divide by zero" error.

I know you know that, so I'm having a hard time deciphering the point of this discussion.

 

[italicus]

As long as they have exactly zero net force, right? Other wise you get a "divide by zero" error.

I know you know that, so I'm having a hard time deciphering the point of this discussion.

Let’s consider a mathematical pendulum. The string of suspension is supposed to have negligible mass, which we pose equal to zero to solve the problem.
Nonetheless, we’re perfectly able to calculate which is the variable tension in the string when the mass swings.
So it seems a little strange that a really massless string can transmit a force to the pin of suspension, doesn’t it?
Many assumptions of basic physics are similar : think of a pulley “ without mass” so without inertia; or a “perfectly flexible, inextensible and massless rope “ that lifts a massive body! Or the definition of “ rigid body”, a plane without any friction …

All useful abstractions, but in real mechanics things are quite different.
I received a lot of observations by people , that weren’t able to grasp the point. The best thing to do Is to clarify that they’re assumptions only, at the very beginning.

 

[gmax137]

I promised myself I would not get drawn into this. But

Let’s consider a mathematical pendulum. The string of suspension is supposed to have negligible mass, which we pose equal to zero to solve the problem.

No, my high-school physics books did not write the equations of motion for the string and then set Mstring=0.

 

[Dale]

So it seems a little strange that a really massless string can transmit a force to the pin of suspension, doesn’t it?

Certainly, but nothing breaks in the theory from it.

 

[gmax137]

So it seems a little strange that a really massless string can transmit a force to the pin of suspension, doesn’t it?

Not much stranger than the Earth transmitting the force of gravity (mg) to the pendulum bob with no string at all.

EDIT: sorry, that's a distraction from the thread.

 

[italicus]

Not much stranger than the Earth transmitting the force of gravity (mg) to the pendulum bob with no string at all.

EDIT: sorry, that's a distraction from the thread.

It’s not a distraction. But you should know that the bob is in the Earth gravitational field. Newton law of universal gravitation applies here. Remember that he was not able to explain the “action at a distance “ of his law: “Hypothesis non fingo” …until a scientist came at the beginning of last century, who changed the point of view radically.
This is a progress of science., that happens not only when we discover and learn something new, but also when we look differently at old questions, and find new and more adequate solutions.

 

[weirdoguy]

Ok, so what EXACTLY breaks down when we cosider massles things in Newtonian mechanics? Any particular example?

 

[italicus]

Ok, so what EXACTLY breaks down when we cosider massles things in Newtonian mechanics? Any particular example?

Simple example: the second law of dynamics.

 

[Dale]

Simple example: the second law of dynamics.

Massless objects don’t violate Newton’s 2nd law

 

[italicus]

Massless objects don’t violate Newton’s 2nd law

Please justify your opinion by physical issues. Here we are speaking of what Physics says, not you or me.

 

[Dale]

Please justify your opinion by physical issues. Here we are speaking of what Physics says, not you or me.

I already showed what “Physics says” by citing several different textbooks. But to be clear the equation $\Sigma F=ma$ for $m=0$ simply means that $\Sigma F=0$ regardless of $a$. $0=0a$ is perfectly valid.

 

[jbriggs444]

I already showed what “Physics says” by citing several different textbooks. But to be clear the equation $F=ma$ for $m=0$ simply means that $F=0$ regardless of $a$. $0=0a$ is perfectly valid.

One can argue that you lose predictivity. Knowing that force is zero and that mass is zero does not allow one to predict acceleration. As you say, this is perfectly compatible with Newton's 2nd law.

 

[italicus]

One can argue that you lose predictivity. Knowing that force is zero and that mass is zero does not allow one to predict acceleration. As you say, this is perfectly compatible with Newton's 2nd law.

When F=0, any mass, as big as you want, has zero acceleration , as stated by the first principle of dynamics, which exists indipendently by the second. Remember the story of the Newton’s “Principia mathematica “.
Anyway, a = 0/0 has no mathematical or physical meaning.
You lose not only predictivity , you lose dynamics principles posing m=0 ; the correct assumption to be made is that, when mass is so small compared to other masses involved in the problem, you “assume “ it to be zero, sic and simpliciter .