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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".
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 ...
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.
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
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.
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.
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.
In general, this definition of mass is independent from the choice of the Lagrangian. However, the momentum and/or momentum density cannot be defined arbitrarily. It has to be defined in such a way that
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$^2$.
Centre of mass & gravity coincides until they have unifrom gravitational field. The time uniform gravitational field is lost we rather consider centre of mass than centre of gravity. However, they both're interchangeable.