Definition of mass, force, Newton's law and Foucault's experiment

Question

In a "modern way of teaching Newtonian mechanics", there are several notions to define before stating Newton's laws, which appear in the laws.

(i) Position and time in reference frames. I'll consider that it is possible to measure it so that acceleration in a given reference frame is computable.
(ii) Mass should be defined. How can we define such a notion? How can we measure it?
(iii) Force is to be defined too and, of course, a way to measure it.

My understanding of Newtonian's law is the following (I will not talk about Newton's third law):

(1) An inertial reference of frame is, by definition, a frame in which if there are no net forces, that is, if the sum of all forces acting on an object is equal to zero, then $a=0$.

(2) In such inertial frames, $ma=F$.

(3) The action-reaction law.

These statements are somewhat "circular", this is no news as many other questions on this SE can attest this fact.

It seems that the second Law (2) is a definition of both force and mass: you measure a and you observe that there is a proportionality constant $m$ depending on the object such that $m_1a_1=m_2a_2$ when the two objects are in the same conditions. With this notion of mass, you identify $F$ as the thing equal to $ma$.

But in order to do so, you need to work in inertial frame of references ... which requires the notion of force to be defined! And the force is also by definition ma in intertial frames ... this seems circular. How do we make it uncircular?

Is there a rigorous construction of mass, force, and Newton's first/second law?

Let me give a troubling example, may somebody explain to me why I am wrong: Foucault's experiment is considered as a proof that the frame of reference given by the earth in which we are sitting in, is not an inertial frame. To do so, one cooks up an inventory of forces, let us say that all these forces add up to $F$. Then one observes the acceleration of the pendulum $a$, and observe that $ma\neq F$. To restore the equality one has to add a "fictitious" force, $F_f$, so that $ma=F+F_f$. My objection is then: why do we say that this frame of reference is not inertial, rather than admitting that we forget a force in our inventory?

Just to be clear, I am not a first-year undergraduate student who does not understand anything to Newtonian physics. I am pretty convinced that the earth is not an inertial frame of reference and pretty aware that in the appropriate frame of reference in which Earth is rotating, the "fictitious force" is just incorporated in the acceleration. But ... Newton's law (2) states that "acceleration is force".

If you think that this question needs modification or clarification, do not hesitate to tell me. I should mention that I did not find any other questions on the Physics Stack Exchange properly addressing my question.

Answer 1

The issue that you raise is addressed in the book 'Gravitation', by Misner, Thorne, and Wheeler.

The following paragraphs go into the question of how to do physics at all.

Paragraph 12.3

Point of principle: how can one write down the laws of gravity and properties of spacetime in Galilean coordinates first (par. 12.1), and only afterwards (here) com to grip with the nature of the coordinate system and its nonuniqueness? Answer: (a quotation from par. 3.1, slightly modified): "Here and elsewhere in science, as emphasized not least by Henri Poincaré, that view is out of date which used to say 'Define your terms before you proceed.' All the laws and theories of physics, including Newton's laws of gravity, have this deep and subtle character, that they both define the concepts they use (here Galilean coordinates) and make statements about these concepts."

The discussion in section 3.1 of the book goes as follows:

All the laws and theories of physics, including the Lorentz force law, have this deep and subtle character, that they both define the concepts they use (here B and E) and make statements about these concepts. Contrariwise, the absence of some body of theory, law, and principle deprives one of the means properly to define or even use concepts.

Any forward step in human knowledge is creative in this sense: that theory, concept, law, and method of measurement - forever inseparable - are born into the world in union.

How I understand the above paragraphs:

The defninitions that we are using in the course of doing physics are operational definitions.

Take the example of mass.
We know we can do setups with air tracks, to get near frictionless motion.

You start with a bulk quantity of something uniform. (To make sure personally you can take a lump of clay and knead it until you are sure it's all uniformly mixed.)

In collision experiments two units of volume of this uniform density material will behave differently than one unit of volume, in accordance with that 1:2 ratio.

Operationally you find that such a thing as mass is consistent, so much so that you can make it a fundamental unit.

The justification for the concept of mass is that with the proper operational definition it becomes a powerful tool to do physics.

Moving to the general case:
When ideas are applied in technology is when the rubber meets the road.

The engineer/physicist designs a new machine, and if the theory that underlies the design is good the machine will operate as designed.

Conversely, if there is no rubber-meets-the-road then, yeah, it might all be circular reasoning.

Inertial frame of reference

In theory of motion we have the equivalence class of inertial coordinate systems. The equivalence class of inertial coordinate systems expresses the physical properties of inertia.

There is a parallel between inertia and inductance. Inductance: a coil with self-inductance will oppose change of current strength. (When there is a change of voltage the current tends to change. The change of current strength induces a magnetic field that opposes the change of current strength.)

Inertia offers zero resistence to velocity; inertia opposes change of velocity.

In order to formulate a theory of motion we must grant the existence of inertia. We find that if we do grant the existence of inertia we can formulate a powerful theory of motion.

Now a specific example: the operational definition of the inertial coordinate system that is co-moving with the solar system.

If you use an inertial coordinate system to describe the motions of the planets then all the planets move according to a single law: Newton's law of universal gravity. If you use any other coordinate system, one that is rotating with respect to the inertial coordinate system, then you would have to describe the motion of each individual planet with a bespoke law.

Ultimately there is only one way to identify the inertial coordinate system. Newton's laws of motion and the law of universal gravity obtain if and only if you are using an inertial coordinate system.

(Sure, mathematically you can use a rotating coordinate system. When you use a rotating coordinate system then the equation of motion contains terms with the angular velocity of the rotating coordinate system with respect to the inertial coordinate system. That is: you can use a rotating coordinate system if and only if you retain the inertial coordinate system as the underlying reference.)

So let me end with repeating the quote from Misner, Thorne and Wheeler

All the laws and theories of physics, including Newton's laws of gravity, have this deep and subtle character, that they both define the concepts they use and make statements about these concepts.