The mass (or inertial mass, from Greek: μᾶζα, máza, barley cake, lump of dough) is a physical quantity that represents the amount of matter proper to material bodies that determines their dynamic behavior when subjected to the influence of external forces. In classical mechanics the term mass can refer to three different scalar physical quantities, distinct from each other:

  • inertial mass is proportional to the inertia of a body, which is the resistance to changing its state of motion when a force is applied;
  • passive gravitational mass is proportional to the force of a body’s interaction with the force of gravity;
  • active gravitational mass, on the other hand, is proportional to the intensity of the gravitational field produced by a body.

Inertial mass and gravitational mass have been experimentally proven to be equivalent, although conceptually they are distinct. The first experiments aimed to establish this equivalence were those of Galileo Galilei.

Throughout the history of physics, in particular classical physics, mass has been considered an intrinsic property of matter, representable by a scalar value and which is conserved in time and space, remaining constant in any isolated system. In addition, the term mass has been used to denote two potentially distinct quantities: the interaction of matter with the gravitational field and the relationship between the force applied to a body and the acceleration induced on it. However, the equivalence of the two masses has been verified in numerous experiments (put in place by Galileo Galilei first).

The mass does not correspond with the amount of substance, the physical quantity for which it has been introduced in the SI a fundamental quantity, the mole (symbol mol). The mass of a body is commonly determined by measuring its inertia which is opposed to a change in its state of motion or the gravitational attraction to other bodies by comparison with a sample (see balance).

Unlike space and time, for which operational definitions can be given in terms of natural phenomena, defining the concept of mass requires explicit reference to the physical theory that describes its meaning and properties. Pre-physical intuitive concepts of quantity of matter (not to be confused with quantity of substance, measured in moles) are too vague for an operational definition, and refer to common properties, inertia and weight, which are considered quite distinct from the first theory to introduce mass in quantitative terms, Newtonian dynamics.

The concept of mass becomes more complex at the level of particle physics where the presence of elementary particles with mass (electrons, quarks, and so on) and without mass (photons, gluons) still has no explanation in fundamental terms. In other words, it is not clear why some particles have mass and others do not. The main theories that try to give an interpretation to mass are: the Higgs mechanism, string theory and loop quantum gravity; of these, as of July 4, 2012 thanks to the LHC particle accelerator, only the Higgs theory has had the first experimental evidence.

In the current International System of Units (SI), mass has been chosen as a fundamental physical quantity, i.e., not expressible solely in terms of other fundamental quantities. Its unit of measurement is the kilogram, indicated by the symbol: kg.

In the CGS system, the unit of mass is the gram. In the United Kingdom and the United States the pound (approximately 454 g) and the stone (14 pounds) are commonly used. Other units of measurement are commonly used in specific fields of physics.

In atomic physics and physics of matter are commonly used the Hartree units, based on the mass of the electron or the atomic mass unit, roughly equivalent to the mass of a proton. In chemistry is frequently used the mole that, even if it is not a mass unit, it is linked by a simple proportionality factor. In nuclear and sub-nuclear physics is common the use of atomic mass unit. However, especially in the field of high energies, it is used to express the mass (at rest or invariant) by its equivalent energy E = mc2. The energy is in turn expressed in: eV.

Conservation of mass

In classical mechanics there is the fundamental law of conservation of mass, in various formulations. In general, given a control volume, fixed, the variation of mass contained in it is equal to the outgoing flow of mass through the frontier of the system, that is through the closed surface that delimits the volume, changed of sign: in simple words, the variation of mass of a system is equal to the incoming mass minus the outgoing mass; this implies, for example, that the mass can be neither created nor destroyed, but only moved from one place to another. In chemistry, Antoine Lavoisier established in the 18th century that in a chemical reaction the mass of the reactants is equal to the mass of the products.

The principle of conservation of mass is valid with very good approximation in everyday experience, but it ceases to be valid in nuclear reactions and, in general, in phenomena involving relativistic energies: in this case it is incorporated in the principle of conservation of energy.

Inertial mass

Newtonian definition

The inertial mass \(m_i\) of a body is defined in the Principia as the amount of matter by tying it to the principle of proportionality as the constant of proportionality between the applied force \(\vec{F}\) and the acceleration undergone \(\vec{a}\):


The inertial mass can actually be obtained operationally by measuring the acceleration of the body subjected to a known force, being an index of the resistance of a body to accelerate when subjected to a force, i.e., of the inertia of the body. The problem to use this property as a definition is that it needs the previous concept of force; to avoid the vicious circle generated by Newton who did not specify the instrument to measure it, often the force is then defined by linking it to the elongation of a spring following Hooke’s law, definition clearly unsatisfactory as particular and not general. In addition, this definition has given rise to several problems, particularly related to the reference system in which the measurement is made: the concept of inertia, like that of force, was in fact historically criticized by many thinkers, including Berkeley, Ernst Mach, Percy Williams Bridgman and Max Jammer.

Machian definition

The concept of inertial mass was revolutionized by Mach’s work. He was able to eliminate metaphysical elements that persisted in classical mechanics, reformulating the definition of mass in an operationally precise way and without logical contradictions. From this redefinition, general relativity started, even if Einstein himself was not able to include Mach principle within general relativity.

The Machian definition is based on the principle of action-reaction, leaving the principle of proportionality to define the force later. Consider an isolated system consisting of two interacting (point-like) bodies. Whatever is the force acting between the two bodies, it is observed experimentally that the accelerations undergone by the two bodies are always proportional and in constant ratio between them:


What is particularly relevant is that the ratio \(\mu_{12}\) between the two instantaneous accelerations is not only constant over time, but does not depend on the initial state of the system: it is therefore associated with an intrinsic physical property of the two bodies under consideration. By changing one of the two bodies, the constant of proportionality also varies. Let’s suppose then to use three bodies, and carry out separately three experiments with the three possible pairs (it is always assumed the absence of external forces). In this way we can measure the constants \(\mu_{12},\mu_{23},\mu_{31}\). Note that by definition:


Comparing the values of the observed constants, one will invariably find that they satisfy the relation \(\mu_{12}\cdot \mu_{23}\cdot \mu_{31}=1\). Thus the product \(\mu_{12}\cdot \mu_{31}\) does not depend on the nature of body 1, since it is equal to the inverse of \(\mu_{23}\), namely \(\mu_{32}\), which is independent of it due to the independence of \(\mu_{23}\). From this it follows that any coefficient \(\mu_{ij}\) must be able to be expressed as a product of two constants, each dependent on only one of the two bodies.

\[\mu_{ab}=\nu_{b}\cdot m_{a}=\dfrac{1}{\nu_a m_b}=\dfrac{1}{\mu_{ba}} \quad \Rightarrow\begin{cases} \nu_a=\dfrac{1}{m_a} \\ \nu_b=\dfrac{1}{m_b} \end{cases}\]

\[\mu_{ij}=\dfrac{m_i}{m_j}\quad\Rightarrow\quad m_i\vec{a}_i=-m_j\vec{a}_j\]

at any instant of time, for any pair of bodies. The quantity \(m\) that results so defined (unless a constant factor, which corresponds to the choice of the unit of measurement) is called inertial mass of the body: it is therefore possible to measure the mass of a body by measuring the accelerations due to interactions between this and another body of known mass, without needing to know what are the forces acting between the two points (provided that the system formed by the two bodies can be considered isolated, ie not subject to external forces). The link between the masses is given by:


Gravitational mass

If a body, such as a basket ball, is left free in the air, it is attracted downwards by a force, in first approximation constant, called weight force. By means of a plate scale, it can be seen that different bodies, in general, are attracted differently by the weight force, i.e. they weigh differently. The plate balance can be used to give an operational definition of gravitational mass: a unit mass is assigned to a sample object and the other objects have a mass equal to the number of samples needed to balance the plates.

Passive gravitational mass is a physical quantity proportional to the interaction of each body with the gravitational field. Within the same gravitational field, a body with small gravitational mass experiences a smaller force than a body with large gravitational mass: the gravitational mass is proportional to the weight, but while the latter varies with the gravitational field, the mass remains constant. By definition, the weight force \(F_w\) is expressed as the product of the gravitational mass \(m_g\) by a vector \(\vec{g}\), called gravity acceleration, which depends on the place where the measurement is made and whose units depend on that of the gravitational mass.

The active gravitational mass of a body is proportional to the intensity of the gravitational field generated by it. The greater is the active gravitational mass of a body, the more intense is the gravitational field generated by it, and therefore the force exerted by the field on another body; to make an example, the gravitational field generated by the Moon is less (at the same distance from the center of the two celestial bodies) than that generated by the Earth because its mass is less. Measurements of active gravitational masses can be performed, for example, with torsion balances like the one used by Henry Cavendish in the determination of the constant of universal gravitation.

The gravitational mass is to all intents and purposes the charge of the gravitational field, exactly in the same sense in which the electric charge is the charge of the electric field: it simultaneously generates and suffers the effects of the gravitational field. Objects with zero gravitational mass (e.g. photons) would not suffer the effects of the field: in fact a result of general relativity is that any body follows a trajectory due to gravitational field.

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