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In Section 2.5, he gives an operational definition of mass without ever resorting to the concept of force (thus avoiding circularity). The definition is as follows. Take an airtrack with a cart holding whatever object you want to measure, tie a rubber band to it, and pull on the rubber band in such a way so that the band is always elongated at ...
Also, the mass as inertia definition is more philosophical than physical since does not let us determine the mass of a particle or a system of particles, so it is bogus. Regarding the mass as amount of matter definition, it is a tautology, since when you look for the definition of matter it says that is "anything that has mass and volume".
The above gives a complete definition of mass. Seen this way, mass is the propensity of a body to resist acceleration. In practice measurements of the kind I have been discussing are often done using a balance and comparing weights, which amounts to comparing gravitational attractions between planet Earth and two different objects.
Although the second definition seems rigorous, it is even more confusing because it presumes definition of force which is defined as force = mass * acceleration. A little contemplation will reveal that such definition of mass as force/acceleration is nothing but a circular definition.
So mass is postulated to be the proportionality constant between the measured acceleration of an object and the force . The force is defined as dp/dt, the change in momentum of an object). This is the classical definition of mass, assumed constant for each specific object.. I want to know how was mass measured and assigned a numerical value
Let us define the inertial mass, gravitational mass and rest mass of a particle. Inertial mass: To every particle in nature we can associate a real number with it so that the value of the number gives the measure of inertia (the amount of resistance of the particle to accelerate for a definite force applied on it) of the particle.
What is Newton's original definition of mass. The following is from the English translation of Newton's Principia by Andrew Motte. "The quantity of matter is the measure of the same, arising from its density and bulk conjointly." "Thus air of double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity.
With that out of the way, then we can give a definition of mass, valid in the SR limit, whereby the spatial extent of the system of particles is not so big as to require GR corrections, as $$\tag1m_0=+c^{-2}\sqrt{(\sum E)^2-(\sum\vec pc)^2}$$ You will find that this definition immediately settles a lot of the deeper confusions.
The inertial mass defined using Newton's laws is the same as the gravitational mass defined by the force a body exerts in a gravitational field. So if you take a 1kg mass at the Earth's surface, the weight of 9.81 Newtons it exerts is exactly the same as the force you'd need to accelerate the 1kg mass at 9.81m/s$^2$.
Based on this definition you would measure the mass by applying a known force to an object and measuring its acceleration. Mass is rarely measured this way (e.g. in a mass spectrometer). More commonly we just put objects on a scale and measure the gravitational force (=weight) which is proportional to the mass. This is an indirect measurement.