Contextual Relationships Between Momentum, Energy, and Force.

[Cameron Blake]

I apologize in advance for the length of this post, if you wish to reduce reading skip to paragraph 5. Or if you are super lazy, the final paragraph.

I've had long running difficulties with these classical concepts. Not to say I can't apply or use them. I've taken three different mechanics courses (due to bureaucratic reasons) and gave excellent performances each time. However, I still maintain a feeling of unease as to what I'm actually dealing with.

All the instruction I've ever had gave defininitonal based explanations: ƩF = ma, momentum = mv, and energy is work and work is energy (this is also a point of confusion). 'These things are what they are because that's how they are defined.' I don't want definitions, I want derivations. Not in a strict mathematical sense but conceptually. I understand these are rather nebulous demands, so I will demonstrate my current train of thought.

Noticing that Newton defined Force in terms momentum, I like to use momentum as the base for all the other concepts. I'll start with the assumption: Within a closed system, the amount of mass and the amount of velocity associated with that mass never change. Closed system in this case defined as a selection of particular masses that are imagined as the only masses in existence (no 'outside masses act on the 'inside' masses in any way). I love this starting point because it's simple and intuitive. Perhaps you could argue against the phrase "amount of velocity" but I feel it is sufficient.

Because of the use of "associated" with our assumption, velocity is added only as many times as there is mass that holds that velocity and vice-versa. So a quantity of velocity is added m times, mathematically stated as m*v. We have now arrived at the textbook definition of total momentum: m_total*v_{associated total} = constant. Good job guys.

If our closed system were to be breached and momentum altered without adding mass, (classically speaking) velocity is the only quantity subject to change. The derivative of velocity is acceleration, therefore the change in momentum would be m*a and is given the name "Force". This brings into question statements of 'forces' being applied but canceling out.

If force is the change in momentum, can a force exist if there is no change? Momentum can be separated into known components because that is our starting point; our initial condition's are the masses and their velocities. It is from there we derive notions of momentum. Our reasoning doesn't allow for reverse engineering, starting from momentum and then deriving masses and their velocities. Now apply that idea to force. Given simply just force (or momentum) allows for infinite possibilities. So the statement "a force is applied" is false, we can only say "a force is noted". Force is a derived term. In a closed system of a ball and the earth, if the ball were to be at rest on the Earth (ignoring internal changes) textbooks would say there are two forces: gravity down and normal force upwards, canceling out. Why can we say that? Under that logic, couldn't there be an infinite combination of forces? Maybe one leftwards and one identical in quantity rightwards. My current reasoning allows for forces to exist only when there is a change in momentum. Text books, too, only define net force, anything other doesn't seem to technically exist. What I want from you, Internet, is a logical thought process that satisfies current conventions.

Furthermore, energy mathematically seems to be the integral of momentum with respect to time m*v → (m*v²/2). But energy is also constant in a closed system which doesn't hold when momentum's constant is integrated with respect to time. Maybe I should have started with energy as the base concept and momentum and force as the derived concepts, I don't know. Also work seems to be the change of energy, via force (W = ∫f*dx). Work is derived form force so force represents a type of change in energy and momentum but I'm not to sure on how to explicitly state this.

These are my problems with classical mechanics. I suppose the heart of my difficulties lies in what are fundamentally derived concepts and what are initial statements, assumptions, or definitions. I struggle with momentum, energy, and force because they all seem so tauntingly related but I can't quite glue them together myself.

I am grateful for any enlightenment

 

[256bits]

You seem to be making it all more difficult for youself.

In a closed system of a ball and the earth, if the ball were to be at rest on the Earth (ignoring internal changes) textbooks would say there are two forces: gravity down and normal force upwards, canceling out. Why can we say that? Under that logic, couldn't there be an infinite combination of forces? Maybe one leftwards and one identical in quantity rightwards

When dealing with a problem, we are interesting with forces that cause change in motion of an object or the stresses that are induced in an object due to a force(s).

In your example of the ball, the textbook is attempting to explain to its readers certain concepts such as the gravitational force and the normal force. Certainly the ball is subject to atmospheric pressure, wind currents, pressure from light, perhaps from the magnetic field of the earth, electrostaic force if charged, etc etc. The textbook will assume those are negligable or balanced so as to give the student an easier grasp of the conceptual idea being discussed.

And in real life problems one has to also make similar distinctions or no problem will ever be able to be solved

 

[Cameron Blake]

256bits,

I suppose simplification for educational purposes makes since, however I consider the textbook a specific case. What I am asking is the question of "can a force exists if there is no acceleration?" If force is only defined from quantities of mass and acceleration I don't understand how one can "exert" a force without producing some sort of acceleration.

I'm imagining a controlled space-like void and the only known quantities are masses and there positions throughout all of time. There is momentum at and energy at all times, but "net" force only during acceleration. There could be many cancelling forces at all times, but it would be impossible to say. The only reason the textbook can make such a claim is because of other background information. My only information is masses and there positions throughout all of time.

 

[jbriggs444]

If your workshop consists of balls with known masses and observable trajectories, that is sufficient to come up with theories of force. Scientists have reasoned from far less direct evidence.

You could observe an inverse square pattern in the interaction of two bodies. You might observe that this is quantized (e.g. the Millikan oil-drop experiment). You could observe that these interactions add like vectors. You could decide to model the n body problem as n * (n-1) / 2 independently interacting peers and observe that the results match experiment.

If you called these interactions "forces" then such a model would claim that "forces" exist even when the net acceleration is zero.

Whether that characterization is physically correct is a philosophical question. The scientific test is whether the predictions match experiment.

 

[Cameron Blake]

Okay jbriggs444, what I gather from you is that we can say separable forces exist only if we can

A.) make a distinction at one point in time that only one force is acting upon an object (which I would think is technically, by-definition, impossible)
B.) if we are able to make claim as to the causes behind that force (i.e. mass or distance from than mass).
C.) if theses causes are present at all times, then if the object is at rest, there must be a force or combination of forces to counter-act the 'known' force.

Perhaps this process would argue that the definition of 'single' forces depend on more than just acceleration, but the total history of an object's position (requiring some other sort of reasoning) and even then, there will always be uncertainty.

Also, the model I was thinking of doesn't necessarily have to follow traditional labels of forces like gravity, just the concepts of a force. They could be totally nonsensical forces.

 

[AlephZero]

Furthermore, energy mathematically seems to be the integral of momentum with respect to time m*v → (m*v²/2).

Either you made a mistake there, or you have some basic misunderstanding about calculus.

You seem to be trying to integrate momentum with respect to velocity (not time) - and it's not very obvious what that means.

Also, the model I was thinking of doesn't necessarily have to follow traditional labels of forces like gravity, just the concepts of a force. They could be totally nonsensical forces.

The main purpose of the "concepts of mechnanics" (and everything else in physics) is to be useful in understanding how the universe works - so a philosophical discussion about "totally nonsensical forces" probably won't lead anywhere useful.

 

[Cameron Blake]

AlephZero,

Sorry, I misspoke about the integration. I agree that it's 'meaning' is not obvious, it was just attempt I made to connect momentum and energy without creating two separate definitions. My objective of this argument (whether or not I made it clear enough) is to create a logical template that allows for only one assumption, which followed through leads to concepts of momentum, energy, work, and force. One definition and three derivations. I just feel like I'm missing connections between these ideas.

The point about "nonsensical forces" is just to make the template more general, trying to avoid using ends to reach ends. I want to start blind to everything but masses and there positions through time. I want a set of logic that could work in any situation or universe, if that's clear enough.

Something like how did all of these concepts get reasoned in the first place? Were they made as separate definitions? This is similar to the chicken and the egg question.

Or am I just totally thinking about everything wrong?

 

[WannabeNewton]

Perhaps this process would argue that the definition of 'single' forces depend on more than just acceleration, but the total history of an object's position (requiring some other sort of reasoning) and even then, there will always be uncertainty.

This is incorrect. $F = m\ddot{x}$ is a local equation.

I'm having trouble seeing what the purpose of the thread is. Metaphysical rhetoric aside, what is the crux of your question as far as actual physics goes?

 

[Dale]

All the instruction I've ever had gave defininitonal based explanations: ƩF = ma, momentum = mv, and energy is work and work is energy (this is also a point of confusion). 'These things are what they are because that's how they are defined.' I don't want definitions, I want derivations. Not in a strict mathematical sense but conceptually...

These are my problems with classical mechanics. I suppose the heart of my difficulties lies in what are fundamentally derived concepts and what are initial statements, assumptions, or definitions.

I think that what you are asking for is fundamentally impossible. You simply cannot derive a definition. Furthermore, definitions are more basic than derivations, since you can define term without derivations, but you cannot do any derivations without definitions. You MUST start with definitions, not the other way around. Your instruction is correct in that sense.

 

[Cameron Blake]

I think that what you are asking for is fundamentally impossible. You simply cannot derive a definition. Furthermore, definitions are more basic than derivations, since you can define term without derivations, but you cannot do any derivations without definitions. You MUST start with definitions, not the other way around. Your instruction is correct in that sense.

I agree. What I'm trying to say is that, as I see it now, there are three definitions: energy/work, momentum, and force. Because these all deal with just mass and it's position through time, differing only by how they are combined together, I feel like they can be related together better than I know now. I want one definition/assumption that leads to the other two concepts, not three separate definitions/assumptions. The cheesy line "two sides of the same coin" comes to mind.

Or it could very well be that these are all entirely separate ideas that are only connected in the sense that they deal with similar quantities. Where would I find the "initial" claims to these concepts? I already know of Newton's Principia.

This is incorrect. $F = m\ddot{x}$ is a local equation.

What I'm arguing is that ƩF =m*a is for net force only, therefore any claims to knowledge of the "specific" forces a play can either only be based on other information or is an incorrect statement.