Change in Momentum from mass contribution to the moving object

[DaTario]

TL;DR Summary

How to use Newton's second law to treat differently the two cases below in which a system has its mass increased?

Hi All,

in the figure below we see two cases in which a wheelbarrow moves with speed v, containing a certain mass of water. At a given time t0 a drop of water is added to the system by a dripper. In case 1, the drop enters the system, which increases in mass, having the same speed as the cart. In case 2 the drop is at rest relative to the ground on which the cart moves. Considering Newton's second law expressed in terms of the derivative of linear momentum, how do we treat each case to understand the respective consequences of the increase in mass?
Obs.: consider possible existence of an external and constant force acting on this system.

Best wishes

 

[PeroK]

You need to show us your best attempt at this problem before we can help.

 

[DaTario]

You need to show us your best attempt at this problem before we can help.

Hi Perok, I have just assembled these figures (plane, dripper, etc) to mount the exercise. I have a general doubt concerning the way mass enters or leaves a system. It seems that elements of mass can enter a system with some initial velocity, for instance. My question has to do with this. How do we manipulate the formalism to make it manifest the entrance of mass with or without some given initial velocity. Specifically the formalism implied by the second law, with the derivative of the linear momentum included.

 

[PeroK]

Hi Perok, I have just assembled these figures (plane, dripper, etc) to mount the exercise. I have a general doubt concerning the way mass enters or leaves a system. It seems that elements of mass can enter a system with some initial velocity, for instance. My question has to do with this. How do we manipulate the formalism to make it manifest the entrance of mass with or without some given initial velocity.

Let me make a general observation. All over the Internet, including in some otherwise reputable sources, you will see Newton's second law expressed as:$$F = \frac d {dt}(mv) = \frac{dm}{dt}v + m\frac {dv}{dt}$$This is not correct. The changing mass, which can only be achieved by mass physically entering or leaving the system. This problem may have been set to illustrate the problem with that equation.

 

[DaTario]

I would like you to believe that I have just mounted this problem, based on a doubt I have had for years. In which sense you say that this very famous way to express Newton's second law is not correct?

 

[PeroK]

I would like you to believe that I have just mounted this problem, based on a doubt I have had for years. In which sense you say that this very famous way to express Newton's second law is not correct?

Wikipedia has a good analysis of this:

https://en.m.wikipedia.org/wiki/Variable-mass_system

 

[DaTario]

Thank you. It seems to be a nice material. Is it correct to say that the concept associated with the verb 'accrete' is somewhat related to the concepts involved in inelastic collisions?

 

[Ibix]

In this context, one could say that your droplet of water is accreting to your cart, yes, because the two things join together.

In my experience it is not a word that is used except as a technical term in some scientific contexts.

 

[DaTario]

Thank you, Ibix and wrobel. I confess now I am interested in Perok's explanation on why presenting the often called 'the more complete and comprehensive version of Newton's second law' is not correct.

Why is it not correct to tell students that $\vec F_{ext} = \frac{d}{dt}\vec p$ ?

 

[PeroK]

Thank you, Ibix and wrobel. I confess now I am interested in Perok's explanation on why presenting the often called 'the more complete and comprehensive version of Newton's second law' is not correct.

There are several threads on here dealing with this. The main issue is that you then have to define a force in cases where there is no external force using the normal definition $F = ma$.

As an extreme example, consider a large number of balls with a blue light inside. When the light is switched off, the ball is white.

Now, we consider the momentum of the system of blue balls and the system of white balls. Each time a light is switched on or off the mass of the two systems change, and by the erroneous definition of force each system exerts a force on the other. But, with the usual definition, there is no force applied to any ball nor to either system.

 

[DaTario]

Are you taking into consideration the light as a content of energy that can be translated into mass content by Einstein's equation?

 

[DaTario]

In the context of Einstein's Relativity this equation seems to be useful when we try to explain why a constant force does not make a particle reach velocities higher than c.

 

[Ibix]

Are you taking into consideration the light as a content of energy that can be translated into mass content by Einstein's equation?

No. He's just arguing that there is a system of white balls and a system of blue balls. The mass of one system is $m$ and the mass of the other system is 0, and he can choose which system has the non-zero mass by turning a light on or off and hence changing the balls' colour. Thus he is changing the systems' boundaries and moving mass between them without actually moving anything, just changing the light. Changing the masses of the systems like that would imply that the system of blue balls (which doesn't even really exist when the light is off) exerts a force on the system of white balls if you take $\frac{d}{dt}(mv)$ as your definition of force.

 

[Ibix]

In the context of Einstein's Relativity this equation seems to be useful when we try to explain why a constant force does not make a particle reach velocities higher than c.

No - the so-called "relativistic mass" dropped out of favour decades ago because it causes even more confusion. It clings on in popular science versions of relativity, unfortunately.

Mass (the concept that used to be called "rest mass") is an invariant in relativity. It only changes for the same reasons it does in Newtonian physics - some bit of a system was removed and put somewhere else (possibly in some other form). Relativity just gives you more options for how that can happen and what forms it can take and the extra complexity that masses aren't additive.

 

[DaTario]

Ok, I got it. But in this case we would be just redefining the frontiers of our system. Interesting this purely 'linguistic' way to use the formalism so that a phenomenon seems to have occurred.

 

[Ibix]

Ok, I got it. But in this case we would be just redefining the frontiers of our system. Interesting this purely 'linguistic' way to use the formalism so that a phenomenon seems to have occurred.

I'd say "conceptual" rather than "linguistic". There's nothing mathematically wrong with defining a quantity $\frac{d}{dt}(mv)$. It's just that you can construct experiments (like the balls and lights one) where that quantity does not behave at all like our "if you kick something it starts to move, if you don't it doesn't" concept of force, so it's probably better not to call it force.

 

[DaTario]

No - the so-called "relativistic mass" dropped out of favour decades ago because it causes even more confusion. It clings on in popular science versions of relativity, unfortunately.

Mass (the concept that used to be called "rest mass") is an invariant in relativity. It only changes for the same reasons it does in Newtonian physics - some bit of a system was removed and put somewhere else (possibly in some other form). Relativity just gives you more options for how that can happen and what forms it can take and the extra complexity that masses aren't additive.

Ok, I have heard and read about this rejection of the relativistic mass increase, but formally the relativistic factor is still there multiplying the rest mass. I would also expect gravitational effects related to such 'mass increase' to take place when the particle accelerates. Of course one could also explain that in terms of the energy density at the position of the particle. Although I will not argue in favour of the variable mass, I will just note that the equation $F = \frac{dm}{dt}v + m\frac {dv}{dt}$ had the advantage of showing two possible consequences of the application of a force on a particle: 1) to produce a variation in its velocity and 2) to produce a variation in its mass.

 

[A.T.]

Ok, I got it. But in this case we would be just redefining the frontiers of our system.

Just like you can arbitrarily define which drops become part of the wagon, you also arbitrarily define which balls change from one system to the other.

 

[DaTario]

I'd say "conceptual" rather than "linguistic". There's nothing mathematically wrong with defining a quantity $\frac{d}{dt}(mv)$. It's just that you can construct experiments (like the balls and lights one) where that quantity does not behave at all like our "if you kick something it starts to move, if you don't it doesn't" concept of force, so it's probably better not to call it force.

This example of PeroK reminds me a bit of a notion related to the Einstein's relativity. If two very tall trees fall as in the figure below, their interception can eventually travel faster than light, which does not violate Einstein's postulates.

 

[A.T.]

Why is it not correct to tell students that $\vec F_{ext} = \frac{d}{dt}\vec p$ ?

In principle you can call any term with the right dimensions a 'force' (see inertial forces), but conceptually there is difference to forces that for example can cause deformation.

 

[Ibix]

formally the relativistic factor is still there multiplying the rest mass

Sometimes. Sometimes it's $\gamma^3$, or somewhere in between the two. Or just something different. So the problem of keeping track of which "relativistic mass" you should use in which circumstance is one reason to dislike the concept.

I would also expect gravitational effects related to such 'mass increase' to take place when the particle accelerates.

Not even close, I'm afraid. You need the sixteen components of the stress-energy tensor to describe the source of gravity in relativity. And people making the assumption you did is one of the reasons relativistic mass was deprecated.

2) to produce a variation in its mass.

But applying a force doesn't change the mass. Not in Newtonian physics nor relativistic physics.

And I think relativity is way off topic here - perhaps we should stick to Newtonian dynamics.

 

[DaTario]

Not even close, I'm afraid. You need the sixteen components of the stress-energy tensor to describe the source of gravity in relativity. And people making the assumption you did is one of the reasons relativistic mass was deprecated.

And I think relativity is way off topic here - perhaps we should stick to Newtonian dynamics.

Perhaps you are right with respect to the gravitational events I alluded. With respect to stick to Newtonian physics I also agree. But most of the texts that are in favour of this more general second law equation defend this position based on the fact that in the context of relativity, this equation remains valid, while $F_{ext} = m \cdot a$ don't.

 

[PeroK]

I don't know how relativity came into the discussion. Surely the idea of an object changing colour is within the realms of classical physics.

You could have a football player running around and turning his shirt inside out to change its colour!

The other issue with the erroneous definition of Newton's second law is that force is no longer invariant across inertial reference frames:

$$F = \frac d {dt}(mv) = \frac{dm}{dt}v + m\frac {dv}{dt}$$

Note that the $v$, being the instantaneous velocity of the changing mass, is a frame dependent quantity. Hence force becomes frame dependent.

One important point about $F= ma$ is that it is independent of the choice of inertial reference frame.

 

[DaTario]

I don't know how relativity came into the discussion. Surely the idea of an object changing colour is within the realms of classical physics.

You could have a football player running around and turning his shirt inside out to change its colour!

The other issue with the erroneous definition of Newton's second law is that force is no longer invariant across inertial reference frames:

Note that the $v$, being the instantaneous velocity of the changing mass, is a frame dependent quantity. Hence force becomes frame dependent.

One important point about $F= ma$ is that it is independent of the choice of inertial reference frame.

Ok for this three observations. But I think you agree that the last one is just a disadvantage, not an intrinsic error of this definition. I would sumarize my doubts asking you: Is the resultant force on a particle the rate of change of the linear momentum of this particle?

 

[PeroK]

Ok for this three observations. But I think you agree that the last one is just a disadvantage, not an intrinsic error of this definition.

I would say it is an intrinsic error because with that definition you are no longer doing Newtonian physics. Unless that was the intention, then it's wrong.

I would sumarize my doubts asking you: Is the resultant force on a particle the rate of change of the linear momentum of this particle?

Yes. Because under Newtonian physics a particle cannot change its mass.

 

[Juanda]

That took a strange turn into photons and relativity. Just to add what I think is a simpler example with classical physics of why this is not a force.
$$\frac d {dt}(mv) = \frac{dm}{dt}v + m\frac {dv}{dt} \neq F$$

Imagine a rocket moving at a nonzero speed that's expelling fuel symmetrically all around itself. We know the rocket would keep moving at a constant speed because of the conservation of momentum (fuel expelled one way is counteracted by its symmetric equivalent). However, according to the previous equation, the rocket should be under a force that would accelerate it because the mass of the system is changing.

To solve the conundrum the easiest thing is to consider the rocket hull to be a system with constant mass and the fuel inside a different body that will apply a force on the rocket when it accelerates because it needs to work on the fuel's inertia that's still in the rocket too.

As @PeroK said, in Newtonian physics a particle is not allowed to change its mass so the change in linear momentum will always be
$$\frac d {dt}(mv) = m\frac {dv}{dt} = F$$

 

[DaTario]

Thanks also to A.T. and Juanda. Is there any textbook on Newtonial physics that presents this simple but impactful idea: 'in Newtonian physics a particle is not allowed to change its mass' ?

and a possible continuation could well be the following: 'it also applies to the relativity of Einstein.'

I don't remember to have seen this ideia in Halliday & Resnick & Walker, Alonso & Finn, Serway & Jewit, Tipler & Mosca, Keller & Gettys & Skove, Feynman & Leighton & Sands and Paul Hewitt among few others. I honestly think the defense of this idea here by some of you was very convincing. It somewhat reminds me a paper on Am. J. of Phys. (if I can remember well) in which the authors defend the idea that an additional zero-th law of Newton is necessary for the students. "A force ceases to produce acceleration at the very moment it ceases to be applied".

 

[PeroK]

Thanks also to A.T. and Juanda. Is there any textbook on Newtonial physics that presents this simple but impactful idea: 'in Newtonian physics a particle is not allowed to change its mass' ?

Newtonian physics is built on three conservation laws (for a closed system): conservation of mass, momentum and energy.

A corollary of which is that a particle cannot change its mass. Note that, in particular, it cannot "decay".

and a possible continuation could well be the following: 'it also applies to the relativity of Einstein.'

In SR, these three conservation laws are unified in the conservation of the energy-momentum (four-vector). The energy being the zeroth (time) component, momentum the three spatial components; and the mass being its magnitude - which is also an invariant. This is represented by one of the most important equations in physics:$$E^2 = p^2c^2 + m^2c^4$$Where $m$ is the rest mass of particle. This is invariant, but not conserved. In particular, a particle can decay into two or more particles with less total rest mass.

 

[DaTario]

This is represented by one of the most important equations in physics:$$E^2 = p^2c^2 + m^2c^4$$Where $m$ is the rest mass of particle. This is invariant, but not conserved.

Just as an attempt at a little provocation, doesn't this equation for energy in SR also involve a frame-dependent term?

 

[Juanda]

Thanks also to A.T. and Juanda. Is there any textbook on Newtonial physics that presents this simple but impactful idea: 'in Newtonian physics a particle is not allowed to change its mass' ?

I wish I could. Here in this forum, there are people way more capable of explaining the theoretical background and providing renowned sources. In fact, I feel you have a deeper background than myself already from all the books you mentioned.
I'm just a ME but I happen to have battled against this particular concept in the past because I couldn't make sense of it during my time in university. If you allow $\frac{dm}{dt}v$ to contribute to the force then you run into situations that are not possible as the one I described with the rocket expelling fuel symmetrically.
This of course applies to classical mechanics. How things work when Quantum Physics and Relativity are involved is beyond me so I'm gonna sit this one out and read your conversation that looks interesting.

 

[PeroK]

Just as an attempt at a little provocation, doesn't this equation for energy in SR also involve a frame-dependent term?

We can rearrange the equation to get:
$$m^2c^4 = E^2 - p^2c^2$$The invariance of $m$ then depends on the invariance of the RHS, which we recognise as the magnitude squared of the four-vector $(E, \vec p c)$ in Minkowski space. Proving this quantity is invariant under the Lorentz Transformation is a good exercise for the student.

 

[DaTario]

I wish I could. Here in this forum, there are people way more capable of explaining the theoretical background and providing renowned sources. In fact, I feel you have a deeper background than myself already from all the books you mentioned.
I'm just a ME but I happen to have battled against this particular concept in the past because I couldn't make sense of it during my time in university. If you allow $\frac{dm}{dt}v$ to contribute to the force then you run into situations that are not possible as the one I described with the rocket expelling fuel symmetrically.
This of course applies to classical mechanics. How things work when Quantum Physics and Relativity are involved is beyond me so I'm gonna sit this one out and read your conversation that looks interesting.

I have never discussed this topic in this depth. But regarding the symmetric expelling, I think may be we are neglecting something important as, for instance, the vectorial nature of this equation. The spatial distribution of the mass may have some consequences when taking into account its time derivative (as a kind of continuity equation, which will relate the time derivative of the mass with some mass current,...)

 

[DaTario]

We can rearrange the equation to get:
$$m^2c^4 = E^2 - p^2c^2$$The invariance of $m$ then depends on the invariance of the RHS, which we recognise as the magnitude squared of the four-vector $(E, \vec p c)$ in Minkowski space. Proving this quantity is invariant under the Lorentz Transformation is a good exercise for the student.

Ok, but it seems that we can use this argument also in the Newtonian equation: $$F = \frac{dm}{dt}v + m\frac {dv}{dt} $$ so that it becomes $$m\frac {dv}{dt} = F - \frac{dm}{dt}v.$$

 

[PeroK]

Ok, but it seems that we can use this argument also in the Newtonian equation: $$F = \frac{dm}{dt}v + m\frac {dv}{dt} $$ so that it becomes $$m\frac {dv}{dt} = F - \frac{dm}{dt}v.$$

Sorry, that's doesn't work at all. Only $v$ is frame dependent. Or, alternatively, you have to make $F$ frame dependent.

 

[DaTario]

why the term $\frac{dm}{dt}$ does not depend on the reference frame, since it is somehow associated with a continuity equation that brings with it mathematical elements such as divergence of a stream of matter?

I am deliberately taking risks in this argument