# Definition of mass in Newtonian mechanics

I believe that for any physical quantity, to know what significance it has in any situation or that it's different from other quantities, we have to first start by assigning a value to it, or some means of comparison.

For example, empirical temperature started with a lot of experiments on thermal equilibrium and so on, until we knew what it was (at least it's effect on thermal equilibrium state) and then we made devices that can measure it and so we could assign a value to compare between temperatures.

I want to know what exactly is the definition of mass in Newtonian mechanics, I'm not looking for what do we know about mass now or how do we interpret it..etc, I want to know how was mass measured and assigned a numerical value and what are the criteria for saying that two bodies have the same "mass number", do they behave the same in specific experiments or what?

• Well, there are a number of ways we can work out mass, given it appears in many formulas. For example, one could have two masses $m_1$ and $m_2$, and from the angular frequency of a mass-spring system, providing the spring constant is known, one can work out the masses. – JamalS Jun 22 '17 at 15:55
• – Emilio Pisanty Jun 22 '17 at 15:55
• @JamalS , your answer is like saying; to calculate Temperature just get it from the ideal gas law, I know that. Maybe this is what it will turn out to be in the end, but I want to know how we got there. I'm talking about the definition of mass. I think Emilio Pisanty's comment is really close. I'll have a look – Khalid T. Salem Jun 22 '17 at 16:08

Consider a pair of bodies $b_1$ and $b_2$ in an inertial reference frame. If the bodies $b_1$ and $b_2$ are far from the other objects of the universe and to each other, they have constant velocity. As soon as they become sufficiently close to each other accelerations take place in view of the interactions between them. However physical evidence shows that, inedpendently form the nature of the interaction, there are two strictly positive constants $m_1,m_2$ such that $$m_1 \vec{v}_1 + m_2 \vec{v}_2 = \vec{constant} \quad \mbox{in time}\tag{1}$$ even if $\vec{v}_i$ change in time.

If you replace $b_2$ for $b'_2$, you see that $m_1$ does not change, it is a property of $b_1$ only.

Furthermore, changing inertial reference frame masses do not change.

Another classical property of the mass is that if the two (or more) bodies impact and give rise to a third body $b_3$ it turns out that $m_3 = m_1+m_2$. The same happens if a body breaks down into two (or more) bodies.

(1) can ideally be exploited to measure the mass of bodies. Assume per definition that a fixed body has unit mass $1$. To measure the mass $m$ of $b$, just measure the velocities in two different instants when they are different in view of the interaction of the bodies, $$1\vec{V}(t) + m \vec{v}(t) = 1\vec{V}(t') + m \vec{v}(t')$$ and thus $$1(\vec{V}(t) -\vec{V}(t')) = m (\vec{v}(t')-\vec{v}(t))$$ this identity determines $m$ univocally.

• "If the bodies b1 and b2 are far from the other objects of the universe and to each other, they have constant velocity. As soon as they become sufficiently close to each other accelerations take place in view of the interactions between them." I don't understand the "interactions" part, could you illustrate more what is meant by interactions? – Khalid T. Salem Jun 23 '17 at 11:30
• It is an elementary physical fact one experiences everyday, when two bodies are sufficiently close to each other their motion changes. I just meat this practical fact... – Valter Moretti Jun 23 '17 at 12:41

In the first page of his Principia, Newton defined mass as "the amount of matter which is determined by its volume and density". Of course this is a tautology. We can precisely define inertial mass in classical mechanics in exactly the same way we define temperature in thermodynamics. In this case, the analogous of the zeroth law of thermodynamics is the third law of mechanics, as stated by Mach (see section 2.4 and 2.5).

Let us consider a set of particles and an inertial frame. If we let any two of these particles to mechanically pairwise interact, isolated from the rest, then it is an empirical fact that they accelerate with opposite accelerations $\vec a_i$ and $\vec a_j$ whose magnitudes have the constant ratio $|\vec a_i|/|\vec a_j|$. This is the third law of mechanics. Moreover, if we measure that $|\vec a_A|=|\vec a_B|$ and $|\vec a_B|=|\vec a_C|$, then we also measure $|\vec a_A|=|\vec a_C|$.

Those empirical facts allow us to split the original set of particles into subsets where all its belonging particles pairwise interact in the same way. Each subset forms an equivalence class and we attribute a label, $m$, to the subset. This label is called inertial mass.

By arbitrarily choosing the particle $i=0$ as a reference particle and observing its interaction with the others, we obtain that the inertial mass of every particle is determined from the inertial mass of the reference particle, $$m=\frac{|\vec a_0|}{|\vec a|}m_0.$$

The other mass to be defined in classical mechanics is gravitational mass. This is to be consider as a gravitational charge. It is defined through Newton's law of Universal Gravitation simply as the charge $m_g$ satisfying the relation $$F=\frac{Gm_{g,1}m_{g,2}}{r^2}.$$ It turns out however that the inertial and gravitational mass are numerically the same and that is the basis for the Equivalence Principle and the general theory of relativity.

• So the number or label m is quite arbitrary, just like temperature, what does it have to do with kilograms? How do we make this connection? And why would newton say this about mass? I mean, after all, he must've known that it's a label, so that must mean that there was a connection between that and what is measured empirically and represents the amount of matter. – Khalid T. Salem Jun 22 '17 at 18:19
• The kilogram is, by definition, the mass of the reference particle described above. – Diracology Jun 22 '17 at 18:39
• "Let us consider a set of particles and an inertial frame. If we let any two of these particles to mechanically pairwise interact, isolated from the rest, then it is an empirical fact that they accelerate with opposite accelerations." I don't quite understand what is mean by "interact"? What do we do to let the particles "interact"? – Khalid T. Salem Jun 23 '17 at 11:22
• It means we let two of them close to each other and infinitely far from the rest. – Diracology Jun 23 '17 at 12:05

Definition of mass in classical mechanics

Classical mechanics is a mathematical model describing the kinematics of observables.

.Classical mechanics is concerned with the set of physical laws describing the motion of bodies under the influence of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology. It is also known as Newtonian mechanics,

As all mathematical models classical mechanics depends on a series of vocabulary/definitions which describe objects and defines their behavior in space as a function of time. All these are logical extension from everyday observations organized in a logical sequence.

Then come the laws,for classical mechanics Newton's laws of motion. When modeling mathematically the behavior of nature, the axiomatic setup of mathematics needs extra axioms so that a subset of all the possible solutions can be defined that pertain to physical observables. These extra axioms are sometimes called laws, and sometimes called postulates ( in quantum mechanics).

The laws of Newton pick up the subset of solutions of the relevant differential equations that have to do with physical phenomena:

First law: In an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.

Second law: In an inertial reference frame, the vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration a of the object: F = ma. (It is assumed here that the mass m is constant - see below.)

Third law: When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

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

Forces are used to assign a value to mass. In a fixed location, the gravitational force plays this role, identifying mass with weight.

and what are the criteria for saying that two bodies have the same "mass number",

Two objects have the same mass if they behave the same way in the proportionality measurements with the same force. Simple example the gravitational force.

Mass is the resistance to change in rectilinear constant velocity motion.

• This is based on Newton's 2nd law, and it's an interpretation of the mass number. – Khalid T. Salem Jun 22 '17 at 16:09