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Everything I am talking about here takes place in the Newtonian mechanics, but it is also interesting to discuss everything in the context of special theory of relativity, quantum-mechanics or even QFT.

We say that the mass is something that measures the resistance to acceleration. In practice, in order to get the body to accelerate, we need to apply force. But want to define the force, we say that the 1N of force measures the needed force in order to accelerate the 1kg body with the acceleration $1 \frac{m}{s^2}$. Thus we completed the loop - in order to define the acceleration, we need the force and in order to define the force we need the acceleration.

Does that mean that we need to define the mass and force simultaneously? If you think about it, the concept of mass, historically speaking, was very intuitive and known in the ancient times and they could also measure it (using balance scale for example). Force on the other hand, even though very intuitive did not have any way to be measured in the ancient times. Only after the Newton's second law $F = ma$ we finally get the formula that permits us to measure the force (and maybe even define it quantitatively? ). Going back to my question, I want to find out if we can define the mass and the force before introducing the Newton's second law?

Андреј
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  • Definitely seems like a nice question, but I would bet it is not that easy. Is it possible to do this in analogous situations? Can you define Resistance, Voltage and Current without referring to $V = RI$? – JGBM Jan 11 '21 at 23:05
  • Your question has already been asked. For example here: https://physics.stackexchange.com/questions/246906/why-is-the-definition-of-inertial-mass-circular?rq=1 If you feel that the answers there are not satisfactory, you could wait to have enough reputation and put a bounty for a better answer. – GiorgioP-DoomsdayClockIsAt-90 Jan 11 '21 at 23:51
  • "[I]n order to define the acceleration, we need the force and in order to define the force we need the acceleration"-- In order to define acceleration, we don't need to define force. Acceleration is purely kinematic. You take a ruler and a watch, and measure $d^2 x/dt^2$. I suppose you meant mass, if so, you kindly consider editing it. –  Jan 12 '21 at 00:14

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I believe that force is also known from ancient times, linked to the elastic displacement of stuff (bow and arrow for example are very old).

So, it is possible to measure the force (with a spring) and the acceleration of an object. And change the mass by changing the volume while keeping the same material.

What happens is that after the second law was verified many times, net force became defined as the product of mass and acceleration.

And if a spring shows some experimental deviation (when $F = ma \neq kx$), after careful measurements, we say that it is not perfectly linear in the range.

Claudio Saspinski
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Qualitatively speaking, Mass is just an amount of matter contained in an object you don't need force to define that and force is an interaction or an external agent which if is unbalanced gives rise to the motion. We have other types of forces too which doesn't depend upon masses at all ( Lorentz Force) but in classical mechanics (Newtonian Mechanics) force happen to be linked mass. In advanced mechanics we don't even give emphasize on concept of force at all.

Durgesh Gaikwad
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  • I guess we can @Johny – Durgesh Gaikwad Jan 11 '21 at 23:34
  • The Lorentz force can be handled in Newtonian mechanics. Not sure what argument you are trying to make. – kaylimekay Jan 12 '21 at 00:19
  • -1: "In advanced mechanics we don't even give emphasize on concept of force at all" -- If you mean that Lagrangian and Hamiltonian mechanics don't use the concept of force explicitly, that is completely irrelevant. That doesn't help answer OP's question one way or the other. Regardless, what do you mean when you say that Lorentz force doesn't depend on mass but forces in Newtonian mechanics are linked to mass? :/ [1/2] –  Jan 13 '21 at 11:42
  • Finally, mass is not a measure of how much matter is contained in an object. OP is right that it is the measure of the inertia of an object. In fact, most of the mass of any object that you see doesn't come from matter at all. It comes from the inertia associated with the energy of strong-interactions in the nucleus. Also, it's not true that you don't need to define force to define mass, whether you need to define force to define mass is complicated and ultimately depends on the formulation of Newtonian mechanics you prefer (of course, nothing is circular in any formulation). [2/2] –  Jan 13 '21 at 11:46