Revisiting Newton's Third Law: Understanding Momentum and Additivity

[rahuldandekar]

Now a part of a project or anything, but a product of a question a friend of mine asked about the third law. Please correct any mistakes, and tell me what you think of this write-up I wrote.

http://anonicker.googlepages.com/Newton%27slaws"

 

[Marco_84]

Now a part of a project or anything, but a product of a question a friend of mine asked about the third law. Please correct any mistakes, and tell me what you think of this write-up I wrote.

http://anonicker.googlepages.com/Newton%27slaws"

i don't agree on the definition of mass you gave, that's a mole... we don't even know where mass come from.
I would say that mass is an intrinsic properties of matter, in CM is the constant of proportionality between F and A.
And the definition of system is too hurry and not detailed even in CM.

regards
marco

 

[rahuldandekar]

Actually, I don't mean system as in thermodynamics, I just mean the object or objects under consideration is the system.

If mass is the constant of proportionality between F and a, we lose a real definition of F. It isn't the case that it is defined only up to a proportionality constant, because that constant will be different for different objects. My point is, we Newton's second law defines force, and the third law can be interpreted as defining mass.

I am not sure about my approach regarding the third law, but conservation of momentum can be proved from the second law, since for a closed system, F_ext = 0, so the internal forces have to balance themselves. Now comes the difficult part: how can we prove these objects exert equal and opposite forces in pairs so as to satisfy that condition?

 

[rahuldandekar]

Mass is an intrinsic property, agreed. It is not known how mass arises, agreed. But that doesn't mean we cannot give a definition of mass based on how it manifests itself (in collisions).

 

[John Creighto]

Now a part of a project or anything, but a product of a question a friend of mine asked about the third law. Please correct any mistakes, and tell me what you think of this write-up I wrote.

http://anonicker.googlepages.com/Newton%27slaws"

I Chapter II of Mathematical Physics by Donald H. Menzel, the derivation of conservation of energy is derived via Newton's second law. Conservation of momentum is derived via Newton's third law.

I partly find it a shame that I did not learn this in undergraduate physics but I think the focus of undergraduate physics is more to teach student's to understand and visualize physics then it is to to show the beautiful mathematical foundation which it is founded upon.

 

[rahuldandekar]

How is the conservation of energy derived from the second law? Please elaborate. :)

I think the most you can get from the second law is the work-energy theorem. If you mean the conservation of PE + KE, then that does not follow from the second law, but from the specific law for the force. Namely, the force should be curl-less. Otherwise, you cannot gaurantee the conservation of PE + KE. For that, you need to specify a condition for F, which is not implicit in Newton's laws.

 

[John Creighto]

Right multiply Newtons second law equation by the velocity.

Rewrite as:

$$\frac{1}{2}m_i \frac{d}{dt} \left(\frac{dx_i}{dt} \right)^2=X_i\frac{dx_i}{dt}$$

Where $$X_i$$ is the force in the X direction.

Integrate both sides, Call the constant of integration E, and then sum over all particles and coordinates to obtain:

$$\sum_i\frac{1}{2}m_i \left[\left( \frac{dx_i}{dt} \right)^2+\left( \frac{dy_i}{dt} \right)^2+\left( \frac{dz_i}{dt} \right)^2 \right] -\sum_i \left[\int X_i dx_i + \int Y_i dy_i + \int Z_i dz_i \right] =E$$

Where $$X_i$$, $$Y_i$$, $$Z_i$$, is the force on the ith particle in the x direction y direction and z direction respectively.

I can write it out in more detail tomorrow if requested.

 

[rahuldandekar]

I do understand how you get the expression for E, what I don't understand is how it follows that E is conserved. In fact, if F = (Xi,Yi,Zi) is not curl-less, the second term (line integral of force) is not the same for all paths. This means we cannot define a potential energy.

What the above theorem says is just what the work-energy theorem says, that the change in KE is equal to the line integral of the force. That does follow from Newton's laws. But unless you can define a potential energy, you cannot say energy is conserved.
Or atleast that's how I see it. :)

 

[John Creighto]

I do understand how you get the expression for E, what I don't understand is how it follows that E is conserved. In fact, if F = (Xi,Yi,Zi) is not curl-less, the second term (line integral of force) is not the same for all paths. This means we cannot define a potential energy.

What the above theorem says is just what the work-energy theorem says, that the change in KE is equal to the line integral of the force. That does follow from Newton's laws. But unless you can define a potential energy, you cannot say energy is conserved.
Or atleast that's how I see it. :)

Look at it carefully. It says that if you do work on an object you loose kinetic energy, in such a way that the constant of integration (the total system energy) is conserved. I understand that it is counter intuitive that something as simple as Newtons second of law could result in conservation of energy but it does.

 

[rahuldandekar]

Good point you make there. Give me a day to think it over. :)

 

[rahuldandekar]

My point is, you cannot define a unique potential for a non-conservative (ie, curl-less) force, and yet in those situations KE is not conserved. (eg, friction, where you cannot define a unique potential). The work energy theorem is still valid though, but it is not a statement of energy conservation in such a case, because we cannot unique define the line-integral of the force.

I'll try another viewpoint. Consider a frictional force acting on an object being moved from X to Y through path 1. The line-integral of the force is the loss of KE of the object, that is true.

But there is no "energy" conservation as such, since you cannot form a quantity which, evaluated at X, is the same as that evaluated at Y. (The KE, it decreases.) So, can you define the "energy" for such a force, and then say that E_X = E_Y? You can do this only if (a) you have an expression for F, and (b) the force is conservative.

What I think is that conservation of momentum follows from Newton's 2nd law (for an isolated system... ie, one which is not interacting with anything. Conservation of energy and the specific expressions for E (eg, -GMm/r) follow from the laws for the particular type of interaction.

I'd appreciate your feedback. :)

 

[John Creighto]

I think your thinking along the right lines since that later in the book it shows conservation of energy follows from Newton's second law if you have a conservative field. However, perhaps if we use Newton's third law we can show the net work is zero in the absence of an external field.

 

[rahuldandekar]

I the absence of an external field, 1/2 m*v^2 is conserved, since the line integral of the force is zero. Maybe it follows from the conservation of momentum, if we assume mass is constant.

As far as I see it, Newton's third law is most useful for defining what mass actually is. I don't like to be unconventional in mindset, though ;) ... so if you could tell me if mass can be defined in another way, or how conservation of momentum does not follow from the II law, I'd be very happy to revert back to the conventional view. :)

 

[John Creighto]

I the absence of an external field, 1/2 m*v^2 is conserved, since the line integral of the force is zero. Maybe it follows from the conservation of momentum, if we assume mass is constant.

As far as I see it, Newton's third law is most useful for defining what mass actually is. I don't like to be unconventional in mindset, though ;) ... so if you could tell me if mass can be defined in another way, or how conservation of momentum does not follow from the II law, I'd be very happy to revert back to the conventional view. :)

You need both the second and third law to get conservation of momentum.

 

[rahuldandekar]

If I define momentum to be an additive property, then I can make do with the second law. For example, consider a system of two bodies with no external forces acting. Then the net rate of change of momentum is zero.

Thus, dp₁/dt + dp₂/dt = 0, giving ma₁ = -ma₂.

There is no other source for ma1 other than the interaction with body 2, so F21 = ma1, and similarly, F₁₂=ma₂. Thus F₁₂ = -F₂₁

Of course, there's the thing that here I assumed momentum is additive, which is not stated anywhere in Newton's Laws, and I just put it in the definition of momentum. However, I think it is the only consistent way in which we can get all the content of Newton's laws and get an operational definition of mass.