Is there anything more to forces than being mathematical machinery?

[FactChecker]

I think @DrStupid's suggestion was taking a principle of superposition approach; i.e. if $\vec{F}_1 = \dot{p}_1$, $\vec{F}_2 = \dot{p}_2$ when acting individually then $\vec{F}_1 + \vec{F}_2 = m(\dot{v}_1 + \dot{v}_2) = \dot{p}_t$. Since the accelerations and rate of change of momenta are linear in force, it's valid to consider each action individually and then superpose them at the end.

Whilst of course the physical law is that the resultant force is $\frac{d\vec{p}}{dt}$.

Yes, I agree. I am afraid that I was getting obsessed with a detail and should leave it alone.

 

[etotheipi]

Yes, I agree. I am afraid that I was getting obsessed with a detail and should leave it alone.

No I think your objection was quite valid, since there wasn't an explicit mention of superposing solutions. I also agree that I find the definition of forces slightly unsatisfying, but it appears to be the case that we can't do much better .

 

[DrStupid]

I suppose you could think of it in terms of each force pumping in or siphoning out momentum, and just adding together all of the changes.

That's exactly how I see it. Force is an exchange of momentum between two systems. That implies both Newton II and Newton III. The energy equivalent would be the power of work and heat. According to the first law of thermodynamics the sum of work and heat transferred to a system is equal to the change of its internal energy. But that does not mean that there is no work or heat if the internal energy remains constant.

 

[DrStupid]

Your original statement "Force is defined as something that needs to be impressed upon a body to change its state of rest or uniform motion. " is one that, without explicitly stating it, does not require that there actually be a "change of state of rest or uniform motion", just that it "needs to be impressed" to make such a change.

I am quite sure that it is ment like that because Newton repeated it in the first law of motion. A force is required to change the state of motion but it is not sufficient. Several forces and the corresponding changes of momentum (and therefore also the resulting accelerations) can add to zero.

 

[etotheipi]

That's exactly how I see it. Force is an exchange of momentum between two systems. That implies both Newton II and Newton III. The energy equivalent would be the power of work and heat. According to the first law of thermodynamics the sum of work and heat transferred to a system is equal to the change of its internal energy. But that does not mean that there is no work or heat if the internal energy remains constant.

That's certainly a nice way of thinking of it!

 

[Dale]

The definition of forces says that they are proportioanl (or equal if you use proper units) to change of momentum.

That is pretty close to my preferred definition: “force is the rate of transfer of momentum”. Since momentum is a conserved quantity it’s makes sense to speak of transferring it. Both Newton’s 2nd and 3rd laws follow directly from that definition.

Edit: I see you said as much in a subsequent post!

 

[archaic]

That is pretty close to my preferred definition: “force is the rate of transfer of momentum”. Since momentum is a conserved quantity it’s makes sense to speak of transferring it. Both Newton’s 2nd and 3rd laws follow directly from that definition.

Edit: I see you said as much in a subsequent post!

Just for curiosity, how do you define mass? Something like volume times density would be tautological.

 

[jbriggs444]

Just for curiosity, how do you define mass? Something like volume times density would be tautological.

Historically, it had been defined in terms of another standard mass, using things like balance scales -- forces and torques as proxies for mass under the assumption of a locally uniform gravitational field. [One can also use spring compression as a proxy for force and make assumptions about temporally uniform gravitational fields and linear and unchanging spring properties].

The idea being that two bricks have twice as much mass as one. Or that a lump of platinum irridium over here that balances with a lump of platinum irridium over there must have the same mass.

The underlying assumptions about linearity, uniformity, and such have been well tested for millenia. Commerce and commercial advantage is a powerful motivator.

 

[Dale]

Just for curiosity, how do you define mass? Something like volume times density would be tautological.

I would define mass operationally. Something like “mass is the quantity measured by a balance scale”. This is similar to how time is defined.

 

[kuruman]

I would define mass operationally. Something like “mass is the quantity measured by a balance scale”. This is similar to how time is defined.

In that case wouldn't you have to define a uniform gravitational field or equivalent since the balance scale doesn't work very well in free space?

 

[FactChecker]

I am quite sure that it is ment like that because Newton repeated it in the first law of motion. A force is required to change the state of motion but it is not sufficient. Several forces and the corresponding changes of momentum (and therefore also the resulting accelerations) can add to zero.

Yes. The way you stated it there makes sense and takes care of my concern.

 

[DrStupid]

Both Newton’s 2nd and 3rd laws follow directly from that definition.

Only if you do not insist on a limitation of Newton III to two-body forces because that does not follow directly from the interpretation of force as transfer of momentum .

 

[DrStupid]

Just for curiosity, how do you define mass?

In classical mechanics it is indirectly given by the definition of momentum, Newton II + III, the principle of relativity, isotropy and the transformation. (But be careful: If you use this definition in relativity you get something that is like a red rag to a bull for physicists.) I do not know if it also works without isotropy. It is at least required in the one-dimensional case.

Something like volume times density would be tautological.

Is is a tautology today because we use this definition the other way around to define density. I do not think that was the case when Newton defined mass this way centuries ago. It provides at least a general impression what mass is and what it not is (e.g. amount of substance, which is sometimes confused with Newtons term "quantity of matter").

 

[Dale]

In that case wouldn't you have to define a uniform gravitational field or equivalent since the balance scale doesn't work very well in free space?

I would leave details like that for the manufacturer of the balance scale to explain in the owner’s manual. I would not make it part of the definition.

 

[Dale]

Only if you do not insist on a limitation of Newton III to two-body forces because that does not follow directly from the interpretation of force as transfer of momentum .

I still strongly disagree with your take on that topic but have less than 0 interest in renewing a discussion on it.

 

[jbriggs444]

I would leave details like that for the manufacturer of the balance scale to explain in the owner’s manual. I would not make it part of the definition.

Lack of gravity is not a problem encountered in typical use but artificial gravity (e.g. rotation) can readily be employed if needed. The Acme balance scale, orbital version, has the requisite information in appendix B.

 

[sophiecentaur]

It doesn't really tell you what the forces are though.

I don't think this is anything more than a way of communicating a shared experience with other people - same as any other piece of language. You can draw up a list of effects that can be attributed to the result of a "force'. The introduction of Maths into it makes no fundamental difference; Maths just improves the precision of communication between people and allows more reliable 'reasoning'.
You and I (I'm assuming) had similar educations so we can communicate about a process that involves what we both call "a force" by using Newton's Laws of Motion. If you try to build up knowledge about the World without using Science, you end up with statements like "Nature abhors a vacuum" and we know that seriously falls short when you want to build a rocket or a radio set.
Otoh, if you want to get all philosophical then you need to go elsewhere than PF for your answers.

 

[atyy]

I do apologise, that wasn't the intent of my OP! If the best answer is that force is a quantity defined as $m\mathbf{a}$, I would consider that a perfectly good answer.

That is only part of the answer. Consider $F = ma$. This is not that physically meaningful until we say what $F$ is. $F$ is a push or a pull - an interaction - between two (or more) bodies, with the strength and direction of the push or pull being dependent on the properties of both bodies.

In the case, of gravitation, the relevant properties of the bodies are their masses: $F = Gm_{1}m_{2}/r^{2}$

In the case of the electrical force (in electrostatic situations), the relevant properties of the bodies are their charges: $F = Kq_{1}q_{2}/r^{2}$

Combining the electrostatic force law with Newton' second law gives $Kq_{1}q_{2}/r^{2} = ma$, which shows that the properties of bodies that determine how they interact (ie. the LHS of $F=ma$) do not necessarily involve the inertial mass (in the RHS of $F=ma$).

In the case of the gravitational force, the property on the LHS is called the gravitational mass. Amazingly, in the unique case of the gravitational force, a body's gravitational mass (on the LHS) is always proportional to its inertial mass (on the RHS). This coincidence is one form of something called the Principle of Equivalence, which inspired Einstein's approach to General Relativity, a relativistic theory of gravity. Reformulated versions of the Principle of Equivalence hold in General Relativity.

 

[atyy]

Really then, it appears that introducing the concept of forces is just a means to an end (i.e. you formulate your laws of physics using the construction of 'forces' and can start calculating results), since we could cut them out and just say that the interaction between particles causes motion (though admittedly it would be rather more complicated to calculate anything...). This might well be a stupid question, but is there a physical underlying motivation for introducing forces, or are they just a mathematical abstraction of the more fundamental interactions?

The concept of force is the way the interactions are described in elementary Newtonian physics. In relativity, and especially relativistic quantum mechanics, the force concept of Newton is not so useful. But although we no longer use the mathematical formalism of Newton in those cases, we still call the interactions between bodies "forces".

And yes, even classically you can get rid of forces eg. the Hamiltonian and Lagrangian formalisms.

 

[sophiecentaur]

And yes, even classically you can get rid of forces eg. the Hamiltonian and Lagrangian formalisms.

Good point BUT I very seldom come across a situation where my body tells me I'm dealing with Potential Energy, rather than with Forces.

I've had many useful conversations with young teenagers about the 'nature' of Forces and Energy and they clearly aren't ready for any more than what our experiences of those two things involves. To my mind, that implies that we start off with lists of what effects can be put down to Force or Energy. I have never been asked (at that level) what they 'really are'.

Many people want to fill in that gap between the totally intuitive / subjective experience and the mathematical model by looking for a 'really is' model. But, without the Maths, the 'really is' is little more than a dressed up version of intuition.

 

[Adesh]

As I see it, the force found its existence due to the philosophy of causation. Its hard to think that motion can be caused without any cause. And we see that due to this philosophy we something called "Fictitious Forces", these fictitious forces do not exist (no one is there to cause them) but the motion is caused so we have to accept their existence.

 

[Dale]

As I see it, the force found its existence due to the philosophy of causation. Its hard to think that motion can be caused without any cause.

I think that is a very tenuous position to hold given Newton’s first law

 

[FactChecker]

As I see it, the force found its existence due to the philosophy of causation. Its hard to think that motion can be caused without any cause. And we see that due to this philosophy we something called "Fictitious Forces", these fictitious forces do not exist (no one is there to cause them) but the motion is caused so we have to accept their existence.

Where you say "motion", you really should be saying "acceleration". In a Newtonian inertial coordinate system, there is no unexplained acceleration. The "fictitious forces" are there to allow us to study acceleration in non-Newtonian coordinate systems. It is just a computational convenience with no philosophical implications.

 

[sophiecentaur]

I think that is a very tenuous position to hold given Newton’s first law

In the context of this discussion, I would say that N1 is no more than a self referential statement. It more or less says that All Things behave like this. No real causality is implied.

 

[DrStupid]

No real causality is implied.

According to N1 force is the cause and the change of motion the effect. How is this not a causality?

 

[etotheipi]

I thought that Newton's first law is just a statement that there exist inertial frames of reference in which objects remain at rest or in constant rectilinear motion if no forces act on them. I think that's along the lines of what @sophiecentaur was saying (correct me if I'm wrong!).

 

[DrStupid]

I thought that Newton's first law is just a statement that there exist inertial frames of reference in which objects remain at rest or in constant rectilinear motion if no forces act on them.

I usually refer to the original wording. That reads in the English translation:

Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.

That means that forces are the only cause for changes in the state of motion of bodies. Frames of references are not even mentioned.

 

[Adesh]

I usually refer to the original wording. That reads in the English translation:
That means that forces are the only cause for changes in the state of motion of bodies. Frames of references are not even mentioned.

I think those are really the original words.

 

[etotheipi]

That means that forces are the only cause for changes in the state of motion of bodies. Frames of references are not even mentioned.

Right, but the notion of an inertial reference frame is implied. If you sit in a car that accelerates, everything outside the window will accelerate backward without any real forces being applied.

I just treat the first law as the definition of an inertial frame of reference.

 

[vanhees71]

Nevertheless this holds only for inertial frames of reference. The 1st Law rather states that there are reference frames, where this 1st Law holds, the inertial frames. Then the 2nd Law says that the causes of changes of the state of motion (the deviation from rectilinear uniform motion) are forces, which are equal to $\dot{\vec{p}}$, where for a point particle $\vec{p}=m \vec{a}$ with $m$ a measure for inertia ("inertial mass") and $\vec{a}$ the acceleration of the particle relative to the inertial reference frame.