Newton's laws in variable mass systems

[D H]

(Specifically, F = \dot mV_e for any rocket motor, in which \dot m is the fuel mass flow rate, and V_e is the exhaust velocity)

That is correct only if you toss the definition of force as F=dp/dt and use m·dv/dt=F_ext+u·dm/dt in its place, where u is relative velocity of expelled material. Defining F_reaction≡u·dm/dt let's one simply use F=ma, even for a system with non-constant mass. This form is admittedly very useful as the basis for the equations of motion of a rocket. It is however absolutely useless for computing things like work precisely because it throws out the connection with the conservation laws.

It is this \dot mV_e term that this entire silly imbroglio is all about. Is it a force or is it something that happens to have units of force? The answer lies in how one interprets Newton's second law. Is force defined by F=dp/dt or F=ma? Different textbooks do use different definitions, and the two definitions definitely are not the same in the case of a variable mass system.

 

[Dickfore]

It is a force. It comes from the ejected particles who act on the reamining part of the rocket according to 3rd Newton's Law.

 

[D H]

Fine. Is this force the same in all frames?

Answering yes puts you in the F=ma camp.
Answering no puts you in the F=dp/dt camp.

 

[Dickfore]

Fine. Is this force the same in all frames?

Answering yes puts you in the F=ma camp.
Answering no puts you in the F=dp/dt camp.

Forces are Galilean invariants. This one is too. To see this, notice that both \mathbf{u} and \dot{m} are Galilean invariants. There are no camps. Your transformation formula for the force was incorrect.

 

[D H]

Then tell that to Marion, and Goldstein (or the authors who have taken over for them) and others as well.

And please do find the flaw in that derivation. HINT: it is not my derivation.

 

[Dickfore]

The "third camp", the one you said retreated from the discussion, was right. I guess they saw that they were dealing with obtuse opponents, so there was no point for them to propagate the discussion.

You (and I) derived something which is not Second Newton's Law. It simply gives the acceleration of the center of mass of a system with variable mass in terms of the external forces (real ones) acting on the particles that happen to consist the system at that instant and the momentum flux (according to the CM frame) that goes out and into the system. The equation happens to be of the form:

<br /> m(t) \, \mathbf{a}(t) = \sum{\mathbf{F}_{\mathrm{ext}}(t)} - \mathbf{\Pi}_{\mathrm{out}}(t) + \mathbf{\Pi}_{\mathrm{in}}(t) <br />

However, this is NOT Second Newton's Law! It is an equation of motion. I don't even think Newton wrote down an equation of motion for a system with variable mass. I think this equation is derived by Meshchersky in the beginning of the XX century.

Furthermore, this equation tells us about the motion of the CM of the variable mass system. As an extreme example, a variable mass systme is a two-body system where one of the particles exits the (geometric) boundaries. It is obvious that the above equation is of little use in that case. The case where this equation is useful is where it makes sense to approximate the momentum flux $\mathbf{\Pi}$ as a continuous function of time and the object you are considering has a much bigger mass than the mass it exchanges. Also, it is convenient if the geometric boundaries of the object are actually rigid walls and the motion of the object is then that of a rigid body (with a variable mass fuel compartment). This is an abstraction of a 'rocket'.

As you mentioned very well, this equation is not sufficient to predict the motion of such a 'rocket'. One still needs the equation of motion for the rotation angles in relation to the external torques. As you noted, the problem is quite complicated in realistic situations.

 

[arildno]

DH's position is just silly.
It is unsurprising that he employs the logical fallacy ad authoritam

To take a case where it is utterly nonsensical to describe the momentum flux as a force, we can look at a fluid moving at constant (horizontal) velocity U, and choosing as our control volume that at t=0 starts out as a line segment of length 1, broadening into a rectangle, where one side remains stationary at the initial position of the line segment, the other vertical side moving with (horizontal) velocity V.
Letting d be the density of the fluid, the momentum containe in our control volume is simply U*V*d*t, with a rate of change U*V*d

In this case, the rate of change of momentum within our control volume cannot be ascribed as the effect of an acting force.
Forces act solely upon material particles, not upon arbitrarily chosen spatial regions*. The rate of change of moementum is solely due to flux of momentum, i.e, a quantity that has the same units as force, but is still wholly distinct from force.


Note:
Some use "momentum flux" to designate what I'd call "momentum flux density", i.e, the rate of momentum transfer per unit area(by means of momentum-carrying particles leaving the control volume).


*Remember that in the classical world time&space are dynamically inactive quantities, merely the empty box within which dynamics and the play of forces occur.

 

[D H]

My view might be silly, but it is the one most widely supported in literature.

Oh, and it is also useful.

 

[Dale]

Can a rocket, as opposed to the rocket + exhaust, be modeled with a Lagrangian?

 

[D H]

If it is going through a non-conservative medium (e.g., the atmosphere), no. If it has a control system that turns thrusters on and off, throttles them, gimbals them, or activates some other effector to keep the vehicle under control, no.

In short, the answer is no.

 

[Dickfore]

DH's position is just silly.
It is unsurprising that he employs the logical fallacy ad authoritam

To take a case where it is utterly nonsensical to describe the momentum flux as a force, we can look at a fluid moving at constant (horizontal) velocity U, and choosing as our control volume that at t=0 starts out as a line segment of length 1, broadening into a rectangle, where one side remains stationary at the initial position of the line segment, the other vertical side moving with (horizontal) velocity V.
Letting d be the density of the fluid, the momentum containe in our control volume is simply U*V*d*t, with a rate of change U*V*d

In this case, the rate of change of momentum within our control volume cannot be ascribed as the effect of an acting force.
Forces act solely upon material particles, not upon arbitrarily chosen spatial regions*. The rate of change of moementum is solely due to flux of momentum, i.e, a quantity that has the same units as force, but is still wholly distinct from force.


Note:
Some use "momentum flux" to designate what I'd call "momentum flux density", i.e, the rate of momentum transfer per unit area(by means of momentum-carrying particles leaving the control volume).


*Remember that in the classical world time&space are dynamically inactive quantities, merely the empty box within which dynamics and the play of forces occur.

I don't have the faintest idea what you are talking about.

 

[arildno]

My view might be silly, but it is the one most widely supported in literature.

Again, just meaningless appeal to authority

 

[stewartcs]

Again, just meaningless appeal to authority

That's actually not a meaningless appeal to authority.

If DH had used himself as the authority, then made a claim that his position was true because he said it was, then it would be a logical fallacy. However, since DH is referring to separate reputable sources (i.e. peer reviewed literature) it is not a logical fallacy.

If using reputable literature as an authoritative reference is a logical fallacy then no one on this site should be posting unless they have performed original research (implying they would be committing a logical fallacy).

However, with that being said, if DH is a "rocket scientist" then he may very well qualify as an expert on this subject and may indeed make authoritative claims without committing a logical fallacy.

CS

 

[Dickfore]

That's actually not a meaningless appeal to authority.

If DH had used himself as the authority, then made a claim that his position was true because he said it was, then it would be a logical fallacy. However, since DH is referring to separate reputable sources (i.e. peer reviewed literature) it is not a logical fallacy.

If using reputable literature as an authoritative reference is a logical fallacy then no one on this site should be posting unless they have performed original research (implying they would be committing a logical fallacy).

However, with that being said, if DH is a "rocket scientist" then he may very well qualify as an expert on this subject and may indeed make authoritative claims without committing a logical fallacy.

I have yet to see him provide a peer-reviewed Journal or a redacted textbook in which the equation:

<br /> \mathbf{F} = \mathbf{F}&#039; + \dot{m} \, \mathbf{V}<br />

is derived (see the beginning of post #14).

 

[D H]

I have yet to see him provide a peer-reviewed Journal or a redacted textbook in which the equation:

<br /> \mathbf{F} = \mathbf{F}&#039; + \dot{m} \, \mathbf{V}<br />

is derived (see the beginning of post #14).

You have been given a reference to exactly such a derivation twice in this thread. Here they are (both link to the same article):

Plastino & Muzzio, Celestial Mechanics and Dynamical Astronomy, 53:3 (1992) http://articles.adsabs.harvard.edu//full/1992CeMDA..53..227P/0000227.000.html.

This may be of use here:
http://articles.adsabs.harvard.edu//full/1992CeMDA..53..227P/0000227.000.html

 

[Dickfore]

You have been given a reference to exactly such a derivation twice in this thread. Here they are (both link to the same article):

What equation exactly in the first reference has the derivation?

 

[D H]

Equation (2). Technical papers don't need to spell out the blatantly obvious; they just need to say that some result is blatantly obvious (e.g., the stock phrase "the reader can readily see"). From the paper (emphasis mine):

If we consider the simple case of a variable mass, and we write Newton's second law as:

\vec F = m\frac{d\vec v}{dt} + \vec v\frac{dm}{dt}

we can easily see that it violates the relativity principle under Galilean transformations.​

Be honest: Do technical papers need to spell out the obvious?

 

[Dickfore]

Exactly! And, if you understood this as an argument in favor to your claim, you are illiterate when it comes to scientific literature.

 

[D H]

No. I placed this paper in camp 3, Newton's laws don't apply to variable mass systems, which in my mind is a stick one's head in the sand position. Aerospace engineering is plain old Newtonian mechanics. We've been flying for over a century now, and flying modern rockets for nearly that long. Descriptions of flying machinery is almost always done in a Newtonian (read: not Lagrangian) sense. So to say that Newtonian mechanics doesn't apply to such devices is just silly.

 

[Dickfore]

No. I placed this paper in camp 3, Newton's laws don't apply to variable mass systems, which in my mind is a stick one's head in the sand position. Aerospace engineering is plain old Newtonian mechanics. We've been flying for over a century now, and flying modern rockets for nearly that long. Descriptions of flying machinery is almost always done in a Newtonian (read: not Lagrangian) sense. So to say that Newtonian mechanics doesn't apply to such devices is just silly.

Your position has clearly failed. Do not change the thesis. It would be the most mature thing you can do if you admitted your error about the fallacious Transformation Law that you presented in post #14.

 

[D H]

Please. I have not changed my thesis and there is no flaw in post #14 given the definition F=dp/dt.

 

[D H]

There is some flawed mathematics in this thread. Post #61 implicitly assumes there is no momentum flux inside the control volume -- i.e., it assumes that all particles are moving at the same velocity.

 

[Dickfore]

lol, keep striking.

 

[D H]

The only swingin' and missin' is yours. This is a standard problem in graduate level aerospace engineering classes. A rocket is not a point mass. The center of mass of the fuel is not the center of mass of the vehicle as a whole. As fuel is consumed the overall vehicle's center of mass will move inside the vehicle. This motion of the center of mass has an effect on vehicle dynamics. Challenge to you: What are the equations of motion for such a rocket?

 

[Dickfore]

Everything in that post is well defined. You didn't read it.