![]() I do not want to discuss it (on PSE, there are many related Q&A). If we maintain such a definition of mass, the further specification "in an object at rest relative to the observer" is understandable as a signal that who wrote the sentence is using the old-fashioned concept of relativistic mass. ![]() Last but not least, measuring a mass according to such a definition would require determining the exact isotopic composition of the sample. Moreover, we know that the mass of a nucleus is only approximately given by the sum of the masses of nucleons. Such a definition is meaningful but highly unsatisfactory because it is not directly related to dynamical properties. However, if by "substance" or "matter" one means the nucleons (protons and neutrons), it is possible to define the mass as a quantity approximately proportional to the number of nucleons. As you see, this is not directly the mass. In the International System (SI) of measurements, the mole is the unit for the quantity amount of substance or quantity of matter and is a measure of how many elementary entities of a given substance are in an object or sample. ![]() First, quantity of matter nowadays has a technical meaning. Today, such a definition is still present in textbooks and, in my opinion, could be used at the introductory level, but it would require some words of caution about its meaning and limits. Later, this definition was criticized and substituted with others in the nineteenth century. The definition of mass as a measurement of the quantity of matter goes back to Newton's Principia, which was enough for a long time. It underwent a long sequence of modifications as classical mechanics evolved, but also to cope with the conceptual changes introduced by Special Relativity firstly and Quantum Mechanics after. ![]() The concept of mass is firmly bound to the theory we use for the dynamics. The cited definition mixes in an uncontrolled way different concepts. So, staying at small velocities with a body, its constant $m$ plays the same role as in classical physics in the law used to define it. It turns out that there is a positive constant $m$, called the mass of the body, associated to our body and a second positive constant $m'$ associated to the other body interacting with the former, such that the vector Let us consider first the case of two-body interactions, assuming that the two bodies do not interact with anything except possibly each other. We consider a body and we make it interact with other bodies, describing what happens with respect to an inertial reference frame. Let us refer to bodies whose interactions are local (essentially contact interactions with some generalisation). What it is really necessary is just the law of conservation if total momentum. I will explicitly avoid to introduce the notion of force because, in my view, it makes the discussion even more complicated and it is by no means necessary (the notion of force is quite subtle and its use would open a number of related issues really unnecessary). Mach when replacing velocities with accelerations. There, in classical mechanics, the mass can be defined as follows, with an argument which can be traced back to E. The issue is a bit tangled because the notion of mass in relativity historically relies upon the notion of mass in classical physics. ![]()
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