Mass is a fundamental quantity in physics that characterizes a body's resistance to acceleration and its role in gravitation. In classical mechanics, it is commonly understood as a measure of the amount of matter in an object. In relativistic physics, mass is more precisely related to the energy of a system in its rest frame. Mass is distinct from weight: mass describes a property of a body, whereas weight is a force associated with gravity. Its SI unit is the kilogram, symbol kg. (arxiv.org)
Inertial and gravitational mass
Inertial mass measures resistance to changes in motion. For a body of constant mass moving at speeds much lower than the speed of light, Newton’s second law takes the form
[ \mathbf{F}=m\mathbf{a}, ]
where (\mathbf{F}) is the net force and (\mathbf{a}) is acceleration. Under the same net force, a body with greater mass accelerates less. Mass ratios can also be established by comparing changes in motion during interactions between bodies. (einstein-online.info)
In Isaac Newton’s gravitational theory, gravitational mass determines both how strongly a body produces gravitational attraction and how strongly it responds to another body’s gravitational field. These roles are called active and passive gravitational mass. Inertial and passive gravitational mass describe conceptually different properties, but their proportionality explains why freely falling bodies have the same acceleration in a given gravitational field when other forces are negligible. (einstein-online.info)
This universality of free fall is the weak equivalence principle. It provides a foundation for Albert Einstein’s general relativity, in which free-falling test bodies follow paths determined by the geometry of spacetime, rather than by a gravitational force in the Newtonian sense. (einstein-online.info)
Mass, weight, and density
Weight is measured in newtons, not kilograms. In a local gravitational field its magnitude is commonly written
[ W=mg, ]
where (g) is the local acceleration of free fall. Moving a body to a location with a different gravitational field changes its weight without, by itself, changing its mass. Everyday expressions such as “weighs five kilograms” ordinarily refer to mass, despite using the language of weight. (nist.gov)
Density describes mass per unit volume. For a uniform material,
[ \rho=\frac{m}{V}. ]
Mass and volume therefore express different properties: objects of equal volume need not have equal masses. For nonuniform matter, total mass is obtained by adding the masses of its constituent volume elements, represented mathematically by integrating the local density over the volume. (www1.grc.nasa.gov)
Units and measurement
The kilogram is an SI base unit. Since May 20, 2019, its definition has fixed the numerical value of the Planck constant at exactly
[ h=6.62607015\times10^{-34}\ \mathrm{J,s}. ]
Because the joule is expressed in kilograms, metres, and seconds, this fixes the kilogram through a fundamental constant together with the definitions of the metre and second. It replaced the definition based on the international prototype kilogram, a platinum–iridium artifact. (nist.gov)
A Kibble balance realizes the unit by linking mechanical measurements to electrical measurements. In its weighing mode, an electromagnetic force balances the weight of a test mass; a separate motion mode establishes the relation needed to connect that force to electrical quantities. Routine mass measurements are transferred through calibrated reference standards. High-accuracy comparisons require corrections for effects such as air buoyancy, which depends on the volumes of the compared objects and the density of the surrounding air. (nist.gov)
Relativistic mass and rest energy
In the theory of relativity, modern usage generally identifies mass with invariant mass, also called rest mass. It is independent of an observer’s inertial reference frame and satisfies
[ E^2=p^2c^2+m^2c^4, ]
where (E) is total energy, (p) is the magnitude of momentum, and (c) is the speed of light. In the rest frame, this becomes
[ E_0=mc^2. ]
The older expression “relativistic mass” denotes (E/c^2), a frame-dependent quantity. Distinguishing it from invariant mass avoids interpreting an increase in a moving body’s energy as an increase in its rest mass. (arxiv.org)
Mass–energy equivalence means that internal motion and interactions contribute to a composite system’s mass. A system’s mass therefore need not equal the sum of its constituents’ individual masses. When separated constituents form a bound system and release energy, the resulting system has less mass; the difference corresponds to its binding energy divided by (c^2). This relationship is important in nuclear physics and also applies, on a much smaller energy scale, to chemical binding. (arxiv.org)
Conservation and microscopic origins
In ordinary chemical and engineering calculations, conservation of mass is an extremely accurate approximation. Relativistically, changes in a system’s energy content entail corresponding mass changes. Thus, the sum of individual particle masses is not a universal conserved quantity, although the invariant mass of an isolated system remains fixed through conservation of its total energy and momentum. (www1.grc.nasa.gov)
The Standard Model distinguishes elementary-particle masses from the masses of composite particles. Interactions with the Higgs field account for the masses of charged leptons and quarks, as well as the W and Z bosons. The Higgs boson is an excitation of this field. (home.web.cern.ch)
Most of the mass of a proton does not arise from simply adding its constituent quark masses. It emerges from the dynamics and energy of quarks and gluons governed by the strong interaction. The mass of ordinary matter consequently reflects both elementary-particle properties and the energy associated with composite structures. (arxiv.org)