TheInfoList

Mass is the
quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measu ...
of ''
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic particl ...
'' in a
physical body A bubble of exhaled gas in water In common usage and classical mechanics, a physical object or physical body (or simply an object or body) is a collection of matter within a defined contiguous boundary in three-dimensional space Three-dimen ...
. It is also a measure of the body's ''
inertia Inertia is the resistance of any physical object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an ob ...

'', the resistance to
acceleration In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

(change of
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

) when a
net force In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximate ...

is applied. An object's mass also determines the
strength Physical strength *Physical strength, as in people or animals *Hysterical strength, extreme strength occurring when people are in life-and-death situations *Superhuman strength, great physical strength far above human capability *A common attrib ...

of its
gravitation Gravity (), or gravitation, is a natural phenomenon Types of natural phenomena include: Weather, fog, thunder, tornadoes; biological processes, decomposition, germination seedlings, three days after germination. Germination is th ...

al attraction to other bodies. The
SI base unit The SI base units are the standard defined by the (SI) for the seven of what is now known as the : they are notably a basic set from which all other can be . The units and their physical quantities are the for , the for , the for , the f ...

of mass is the
kilogram The kilogram (also kilogramme) is the base unit of mass Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change ...
(kg). In
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

, mass is not the same as
weight In science Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity, awareness, or understanding of someone or something, such as facts ( descriptive knowledge), skills (procedural knowledge ...

, even though mass is often determined by measuring the object's weight using a
spring scale A spring scale or spring balance or newton meter is a type of mechanical force gauge or weighing scale. It consists of a Spring (device), spring fixed at one end with a hook to attach an object at the other. It works by Hooke's Law, that spring is ...
, rather than
balance scale A scale or balance is a device to measure weight or mass. These are also known as mass scales, weight scales, mass balances, and weight balances. The traditional scale consists of two plates or bowls suspended at equal distances from a Lever, ...
comparing it directly with known masses. An object on the Moon would weigh less than it does on Earth because of the lower gravity, but it would still have the same mass. This is because weight is a force, while mass is the property that (along with gravity) determines the strength of this force.

# Phenomena

There are several distinct phenomena that can be used to measure mass. Although some theorists have speculated that some of these phenomena could be independent of each other, current experiments have found no difference in results regardless of how it is measured: * ''Inertial mass'' measures an object's resistance to being accelerated by a force (represented by the relationship ). * ''Active gravitational mass'' determines the strength of the gravitational field generated by an object. * ''Passive gravitational mass'' measures the gravitational force exerted on an object in a known gravitational field. The mass of an object determines its acceleration in the presence of an applied force. The inertia and the inertial mass describe this property of physical bodies at the qualitative and quantitative level respectively. According to
Newton's second law of motion Newton's laws of motion are three law Law is a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its ...
, if a body of fixed mass ''m'' is subjected to a single force ''F'', its acceleration ''a'' is given by ''F''/''m''. A body's mass also determines the degree to which it generates and is affected by a
gravitational field In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

. If a first body of mass ''m''A is placed at a distance ''r'' (center of mass to center of mass) from a second body of mass ''m''B, each body is subject to an attractive force , where is the "universal ". This is sometimes referred to as gravitational mass.When a distinction is necessary, the active and passive gravitational masses may be distinguished. Repeated experiments since the 17th century have demonstrated that inertial and gravitational mass are identical; since 1915, this observation has been incorporated ''
a priori ''A priori'' and ''a posteriori'' ('from the earlier' and 'from the later', respectively) are Latin phrases used in philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaph ...
'' in the
equivalence principle In the theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational ...
of
general relativity General relativity, also known as the general theory of relativity, is the of published by in 1915 and is the current description of gravitation in . General generalizes and refines , providing a unified description of gravity as a geome ...
.

# Units of mass

The
International System of Units International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' (Kevin Michael album), 2011 * International (New Order album), '' ...
(SI) unit of mass is the
kilogram The kilogram (also kilogramme) is the base unit of mass Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change ...
(kg). The kilogram is 1000 grams (g), and was first defined in 1795 as the mass of one cubic decimetre of water at the
melting point The melting point (or, rarely, liquefaction point) of a substance is the at which it changes from to . At the melting point the solid and liquid phase exist in . The melting point of a substance depends on and is usually specified at a such ...

of ice. However, because precise measurement of a cubic decimetre of water at the specified temperature and pressure was difficult, in 1889 the kilogram was redefined as the mass of a metal object, and thus became independent of the metre and the properties of water, this being a copper prototype of the
grave A grave is a location where a dead body A cadaver or corpse is a dead human body that is used by medical students A medical school is a tertiary educational institution, or part of such an institution, that teaches medicine, and awards ...
in 1793, the platinum
Kilogramme des Archives The grave, abbreviated ''gv'', is the unit of mass used in the first metric system which was implemented in France in 1793. In 1795, the grave was renamed as the kilogram The kilogram (also kilogramme) is the base unit of mass Mass is th ...
in 1799, and the platinum-iridium
International Prototype of the Kilogram The International Prototype of the Kilogram (referred to by metrologists as the IPK or Le Grand K; sometimes called the '' ur-kilogram,'' or ''urkilogram,'' particularly by German-language authors writing in English) is an object that was used to ...
(IPK) in 1889. However, the mass of the IPK and its national copies have been found to drift over time. The re-definition of the kilogram and several other units came into effect on 20 May 2019, following a final vote by the
CGPM The General Conference on Weights and Measures (GCWM; french: Conférence Générale des Poids et Mesures, CGPM) is the supreme authority of the International Bureau of Weights and Measures The International Bureau of Weights and Measures (f ...
in November 2018. The new definition uses only invariant quantities of nature: the
speed of light The speed of light in vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "Void (astronomy), void". An approximation to such vacuum is a region with a gaseous pressure m ...
, the caesium hyperfine frequency, the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature an ...
and the
elementary charge The elementary charge, usually denoted by or sometimes e is the electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive' ...
. Non-SI units accepted for use with SI units include: * the
tonne The tonne ( or ; symbol: t) is a metric unit of mass equal to 1,000 kilogram The kilogram (also kilogramme) is the base unit of mass in the International System of Units (SI), the current metric system, having the unit symbol kg. I ...
(t) (or "metric ton"), equal to 1000 kg * the
electronvolt In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular suc ...
(eV), a unit of
energy In , energy is the that must be to a or to perform on the body, or to it. Energy is a ; the law of states that energy can be in form, but not created or destroyed. The unit of measurement in the (SI) of energy is the , which is the ...

, used to express mass in units of eV/''c''2 through
mass–energy equivalence In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
* the
dalton Dalton may refer to: Science * Dalton (crater), a lunar crater * Dalton (program), chemistry software * Dalton (unit) (Da), the atomic mass unit Entertainment * Dalton (Buffyverse), minor character from ''Buffy the Vampire Slayer'' television s ...
(Da), equal to 1/12 of the mass of a free
carbon-12 Carbon-12 (12C) is the more abundant of the two stable A stable is a building in which livestock Livestock is commonly defined as domesticated Domestication is a sustained multi-generational relationship in which one group of organisms ...

atom, approximately .The dalton is convenient for expressing the masses of atoms and molecules. Outside the SI system, other units of mass include: * the
slug Slug, or land slug, is a common name Common may refer to: Places * Common, a townland in County Tyrone, Northern Ireland * Boston Common Boston Common (also known as the Common) is a central public park in downtown Boston, Massachusetts. ...
(sl), an
Imperial unit The imperial system of units, imperial system or imperial units (also known as British Imperial or Exchequer Standards of 1826) is the system of units A system of measurement is a collection of units of measurement A unit of measuremen ...
of mass (about 14.6 kg) * the
pound Pound or Pounds may refer to: Units * Pound (currency) A pound is any of various units of currency A currency, "in circulation", from la, currens, -entis, literally meaning "running" or "traversing" in the most specific sense is money Im ...
(lb), a unit of mass (about 0.45 kg), which is used alongside the similarly named
pound (force) The pound of force or pound-force (symbol: lbf, sometimes lbf,) is a unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete p ...
(about 4.5 N), a unit of forceThese are used mainly in the United States except in scientific contexts where SI units are usually used instead. * the Planck mass (about ), a quantity derived from fundamental constants * the
solar mass The solar mass () is a standard unit of mass in astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial obje ...
(), defined as the mass of the
Sun The Sun is the star A star is an astronomical object consisting of a luminous spheroid of plasma (physics), plasma held together by its own gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many othe ...

, primarily used in astronomy to compare large masses such as stars or galaxies (≈ ) * the mass of a particle, as identified with its inverse
Compton wavelength The Compton wavelength is a quantum mechanical Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the f ...
() * the mass of a star or , as identified with its
Schwarzschild radiusThe Schwarzschild radius (sometimes historically referred to as the gravitational radius) is a physical parameter that shows up in the Schwarzschild solution to Einstein's field equations, corresponding to the radius In classical geometry, a rad ...
().

# Definitions

In
physical science Physical science is a branch of natural science that studies abiotic component, non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together called the "physical sciences". D ...
, one may distinguish conceptually between at least seven different aspects of ''mass'', or seven physical notions that involve the concept of ''mass''. Every experiment to date has shown these seven values to be
proportional Proportionality, proportion or proportional may refer to: Mathematics * Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant * Ratio, of one quantity to another, especially of a part compared ...
, and in some cases equal, and this proportionality gives rise to the abstract concept of mass. There are a number of ways mass can be measured or operationally defined: * Inertial mass is a measure of an object's resistance to acceleration when a
force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...

is applied. It is determined by applying a force to an object and measuring the acceleration that results from that force. An object with small inertial mass will accelerate more than an object with large inertial mass when acted upon by the same force. One says the body of greater mass has greater
inertia Inertia is the resistance of any physical object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an ob ...

. * Active gravitational massThe distinction between "active" and "passive" gravitational mass does not exist in the Newtonian view of gravity as found in classical mechanics, and can safely be ignored for many purposes. In most practical applications, Newtonian gravity is assumed because it is usually sufficiently accurate, and is simpler than General Relativity; for example, NASA uses primarily Newtonian gravity to design space missions, although "accuracies are routinely enhanced by accounting for tiny relativistic effects". The distinction between "active" and "passive" is very abstract, and applies to post-graduate level applications of General Relativity to certain problems in cosmology, and is otherwise not used. There is, nevertheless, an important conceptual distinction in Newtonian physics between "inertial mass" and "gravitational mass", although these quantities are identical; the conceptual distinction between these two fundamental definitions of mass is maintained for teaching purposes because they involve two distinct methods of measurement. It was long considered anomalous that the two distinct measurements of mass (inertial and gravitational) gave an identical result. The property, observed by Galileo, that objects of different mass fall with the same rate of acceleration (ignoring air resistance), shows that inertial and gravitational mass are the same. is a measure of the strength of an object's gravitational flux (gravitational flux is equal to the
surface integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of gravitational field over an enclosing surface). Gravitational field can be measured by allowing a small "test object" to fall freely and measuring its
free-fall In Newtonian physics, free fall is any motion of a body where gravity Gravity (), or gravitation, is a list of natural phenomena, natural phenomenon by which all things with mass or energy—including planets, stars, galaxy, galaxies, ...

acceleration. For example, an object in free-fall near the
Moon The Moon is Earth's only natural satellite. At about one-quarter the diameter of Earth (comparable to the width of Australia (continent), Australia), it is the largest natural satellite in the Solar System relative to the size of its plane ...

is subject to a smaller gravitational field, and hence accelerates more slowly, than the same object would if it were in free-fall near the Earth. The gravitational field near the Moon is weaker because the Moon has less active gravitational mass. * Passive gravitational mass is a measure of the strength of an object's interaction with a
gravitational field In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

. Passive gravitational mass is determined by dividing an object's weight by its free-fall acceleration. Two objects within the same gravitational field will experience the same acceleration; however, the object with a smaller passive gravitational mass will experience a smaller force (less weight) than the object with a larger passive gravitational mass. * Energy also has mass according to the principle of
mass–energy equivalence In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
. This equivalence is exemplified in a large number of physical processes including
pair production Pair production is the creation of a subatomic particle and its antiparticle from a neutral boson In quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical ...

,
nuclear fusion 400 px, The nuclear binding energy curve. The formation of nuclei with masses up to iron-56 releases energy, as illustrated above. Nuclear fusion is a nuclear reaction, reaction in which two or more atomic nuclei are combined to form one or m ...

, and the gravitational . Pair production and nuclear fusion are processes in which measurable amounts of mass are converted to energy or vice versa. In the gravitational bending of light, photons of pure energy are shown to exhibit a behavior similar to passive gravitational mass. * Curvature of
spacetime In , spacetime is any which fuses the and the one of into a single . can be used to visualize effects, such as why different observers perceive differently where and when events occur. Until the 20th century, it was assumed that the three ...
is a relativistic manifestation of the existence of mass. Such
curvature In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

is extremely weak and difficult to measure. For this reason, curvature was not discovered until after it was predicted by Einstein's theory of general relativity. Extremely precise
atomic clocks An atomic clock is a clock whose timekeeping mechanism is based on the interaction of electromagnetic radiation with the Excited state, excited states of certain Atom, atoms. Specifically, either a Hyperfine structure, hyperfine transition in t ...

on the surface of the Earth, for example, are found to measure less time (run slower) when compared to similar clocks in space. This difference in elapsed time is a form of curvature called gravitational time dilation. Other forms of curvature have been measured using the
Gravity Probe B Gravity Probe B (GP-B) was a satellite-based experiment to test two unverified predictions of general relativity: the geodetic effect and frame-dragging. This was to be accomplished by measuring, very precisely, tiny changes in the direction of s ...

satellite. * Quantum mass manifests itself as a difference between an object's quantum
frequency Frequency is the number of occurrences of a repeating event per unit of time A unit of time is any particular time Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparen ...

and its
wave number In the physical sciences, the wavenumber (also wave number or repetency) is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per ...
. The quantum mass of a particle is proportional to the inverse
Compton wavelength The Compton wavelength is a quantum mechanical Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the f ...
and can be determined through various forms of
spectroscopy Spectroscopy is the study of the interaction Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way ...

. In relativistic quantum mechanics, mass is one of the irreducible representation labels of the Poincaré group.

## Weight vs. mass

In everyday usage, mass and "
weight In science Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity, awareness, or understanding of someone or something, such as facts ( descriptive knowledge), skills (procedural knowledge ...

" are often used interchangeably. For instance, a person's weight may be stated as 75 kg. In a constant gravitational field, the weight of an object is proportional to its mass, and it is unproblematic to use the same unit for both concepts. But because of slight differences in the strength of the Earth's gravitational field at different places, the distinction becomes important for measurements with a precision better than a few percent, and for places far from the surface of the Earth, such as in space or on other planets. Conceptually, "mass" (measured in
kilograms The kilogram (also kilogramme) is the SI base unit, base unit of mass in the International System of Units (SI), the metric system, having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is oft ...
) refers to an intrinsic property of an object, whereas "weight" (measured in
newtons The newton (symbol: N) is the International System of Units International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' ( ...
) measures an object's resistance to deviating from its natural course of
free fall #REDIRECT Free fall In Newtonian physics, free fall is any motion of a body where gravity Gravity (), or gravitation, is a list of natural phenomena, natural phenomenon by which all things with mass or energy—including planets, star ...

, which can be influenced by the nearby gravitational field. No matter how strong the gravitational field, objects in free fall are weightless, though they still have mass. The force known as "weight" is proportional to mass and
acceleration In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

in all situations where the mass is accelerated away from free fall. For example, when a body is at rest in a gravitational field (rather than in free fall), it must be accelerated by a force from a scale or the surface of a planetary body such as the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Earth, covered wi ...

or the
Moon The Moon is Earth's only natural satellite. At about one-quarter the diameter of Earth (comparable to the width of Australia (continent), Australia), it is the largest natural satellite in the Solar System relative to the size of its plane ...

. This force keeps the object from going into free fall. Weight is the opposing force in such circumstances and is thus determined by the acceleration of free fall. On the surface of the Earth, for example, an object with a mass of 50 kilograms weighs 491 newtons, which means that 491 newtons is being applied to keep the object from going into free fall. By contrast, on the surface of the Moon, the same object still has a mass of 50 kilograms but weighs only 81.5 newtons, because only 81.5 newtons is required to keep this object from going into a free fall on the moon. Restated in mathematical terms, on the surface of the Earth, the weight ''W'' of an object is related to its mass ''m'' by , where is the acceleration due to Earth's gravitational field, (expressed as the acceleration experienced by a free-falling object). For other situations, such as when objects are subjected to mechanical accelerations from forces other than the resistance of a planetary surface, the weight force is proportional to the mass of an object multiplied by the total acceleration away from free fall, which is called the
proper acceleration In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at r ...
. Through such mechanisms, objects in elevators, vehicles, centrifuges, and the like, may experience weight forces many times those caused by resistance to the effects of gravity on objects, resulting from planetary surfaces. In such cases, the generalized equation for weight ''W'' of an object is related to its mass ''m'' by the equation , where ''a'' is the proper acceleration of the object caused by all influences other than gravity. (Again, if gravity is the only influence, such as occurs when an object falls freely, its weight will be zero).

## Inertial vs. gravitational mass

Although inertial mass, passive gravitational mass and active gravitational mass are conceptually distinct, no experiment has ever unambiguously demonstrated any difference between them. In classical mechanics, Newton's third law implies that active and passive gravitational mass must always be identical (or at least proportional), but the classical theory offers no compelling reason why the gravitational mass has to equal the inertial mass. That it does is merely an empirical fact.
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born , widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the , but he also made important contributions to the develo ...

developed his
general theory of relativity General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern ph ...
starting with the assumption that the inertial and passive gravitational masses are the same. This is known as the
equivalence principle In the theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational ...
. The particular equivalence often referred to as the "Galilean equivalence principle" or the "
weak equivalence principle In the theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational ...
" has the most important consequence for freely falling objects. Suppose an object has inertial and gravitational masses ''m'' and ''M'', respectively. If the only force acting on the object comes from a gravitational field ''g'', the force on the object is: : $F = M g.$ Given this force, the acceleration of the object can be determined by Newton's second law: : $F = m a.$ Putting these together, the gravitational acceleration is given by: : $a=\fracg.$ This says that the ratio of gravitational to inertial mass of any object is equal to some constant ''K''
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...
all objects fall at the same rate in a given gravitational field. This phenomenon is referred to as the "universality of free-fall". In addition, the constant ''K'' can be taken as 1 by defining our units appropriately. The first experiments demonstrating the universality of free-fall were—according to scientific 'folklore'—conducted by
Galileo Galileo di Vincenzo Bonaiuti de' Galilei ( , ; 15 February 1564 – 8 January 1642), commonly referred to as Galileo, was an astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific qu ...

obtained by dropping objects from the
Leaning Tower of Pisa The Leaning Tower of Pisa ( it, torre pendente di Pisa) or simply the Tower of Pisa (''torre di Pisa'' ) is the ''campanile A bell tower is a tower that contains one or more bells, or that is designed to hold bells even if it has none. Such a ...

. This is most likely apocryphal: he is more likely to have performed his experiments with balls rolling down nearly frictionless
inclined plane An inclined plane, also known as a ramp, is a flat supporting surface tilted at an angle, with one end higher than the other, used as an aid for raising or lowering a load. The inclined plane is one of the six classical simple machines defin ...

s to slow the motion and increase the timing accuracy. Increasingly precise experiments have been performed, such as those performed by Loránd Eötvös, using the
torsion balance :'' Torsion coefficient links here.'' A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy In physical sciences, mechanical energy is the sum of potent ...

pendulum, in 1889. , no deviation from universality, and thus from Galilean equivalence, has ever been found, at least to the precision 10−12. More precise experimental efforts are still being carried out. The universality of free-fall only applies to systems in which gravity is the only acting force. All other forces, especially
friction Friction is the force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related en ...

and
air resistance In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding f ...
, must be absent or at least negligible. For example, if a hammer and a feather are dropped from the same height through the air on Earth, the feather will take much longer to reach the ground; the feather is not really in ''free''-fall because the force of air resistance upwards against the feather is comparable to the downward force of gravity. On the other hand, if the experiment is performed in a
vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Gree ...

, in which there is no air resistance, the hammer and the feather should hit the ground at exactly the same time (assuming the acceleration of both objects towards each other, and of the ground towards both objects, for its own part, is negligible). This can easily be done in a high school laboratory by dropping the objects in transparent tubes that have the air removed with a vacuum pump. It is even more dramatic when done in an environment that naturally has a vacuum, as
David Scott David Randolph Scott (born June 6, 1932) is an American retired test pilot A test pilot is an aircraft pilot with additional training to fly and evaluate experimental, newly produced and modified aircraft with specific maneuvers, known ...

did on the surface of the
Moon The Moon is Earth's only natural satellite. At about one-quarter the diameter of Earth (comparable to the width of Australia (continent), Australia), it is the largest natural satellite in the Solar System relative to the size of its plane ...

during
Apollo 15 Apollo 15 was the ninth crewed mission in the United States' Apollo program The Apollo program, also known as Project Apollo, was the third United States human spaceflight program carried out by the NASA, National Aeronautics and Spac ...

. A stronger version of the equivalence principle, known as the ''Einstein equivalence principle'' or the ''strong equivalence principle'', lies at the heart of the
general theory of relativity General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern ph ...
. Einstein's equivalence principle states that within sufficiently small regions of space-time, it is impossible to distinguish between a uniform acceleration and a uniform gravitational field. Thus, the theory postulates that the force acting on a massive object caused by a gravitational field is a result of the object's tendency to move in a straight line (in other words its inertia) and should therefore be a function of its inertial mass and the strength of the gravitational field.

## Origin

In
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, natural phenomena. This is in contrast to experimental phy ...
, a mass generation mechanism is a theory which attempts to explain the origin of mass from the most fundamental laws of
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

. To date, a number of different models have been proposed which advocate different views of the origin of mass. The problem is complicated by the fact that the notion of mass is strongly related to the gravitational interaction but a theory of the latter has not been yet reconciled with the currently popular model of
particle physics Particle physics (also known as high energy physics) is a branch of that studies the nature of the particles that constitute and . Although the word ' can refer to various types of very small objects (e.g. , gas particles, or even household d ...
, known as the
Standard Model The Standard Model of particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsi ...

.

# Pre-Newtonian concepts

## Weight as an amount

The concept of
amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact of existence. The term has a history of use reaching ba ...
is very old and predates recorded history. Humans, at some early era, realized that the weight of a collection of similar objects was
directly proportional In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
to the number of objects in the collection: : $W_n \propto n,$ where ''W'' is the weight of the collection of similar objects and ''n'' is the number of objects in the collection. Proportionality, by definition, implies that two values have a constant
ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

: : $\frac = \frac$, or equivalently $\frac = \frac.$ An early use of this relationship is a
balance scale A scale or balance is a device to measure weight or mass. These are also known as mass scales, weight scales, mass balances, and weight balances. The traditional scale consists of two plates or bowls suspended at equal distances from a Lever, ...
, which balances the force of one object's weight against the force of another object's weight. The two sides of a balance scale are close enough that the objects experience similar gravitational fields. Hence, if they have similar masses then their weights will also be similar. This allows the scale, by comparing weights, to also compare masses. Consequently, historical weight standards were often defined in terms of amounts. The Romans, for example, used the
carob The carob (''Ceratonia siliqua'') is a Flowering plant, flowering evergreen tree or shrub in the legume family, Fabaceae. It is widely cultivated for its edible Seed pod, pods, and as an ornamental tree in gardens and landscapes. The carob tree i ...
seed ( carat or
siliqua siliqua, c. 363 Image:Siliqua Constantine III-RIC 1355.jpg, 300px, Constantine III (usurper) The siliqua (plural ''siliquae'') is the modern name given (without any ancient evidence to confirm the designation) to small, thin, Roman silver coins ...

) as a measurement standard. If an object's weight was equivalent t
1728 carob seeds
then the object was said to weigh one Roman pound. If, on the other hand, the object's weight was equivalent to 144 carob seeds then the object was said to weigh one Roman ounce (uncia). The Roman pound and ounce were both defined in terms of different sized collections of the same common mass standard, the carob seed. The ratio of a Roman ounce (144 carob seeds) to a Roman pound (1728 carob seeds) was: : $\frac = \frac = \frac = \frac.$

## Planetary motion

In 1600 AD,
Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe as ...

sought employment with
Tycho Brahe Tycho Brahe ( ; born Tyge Ottesen Brahe; 14 December 154624 October 1601) was a Danish , known for his accurate and comprehensive astronomical observations. He was born in , which became part of Sweden in the next century. Tycho was well known ...

, who had some of the most precise astronomical data available. Using Brahe's precise observations of the planet Mars, Kepler spent the next five years developing his own method for characterizing planetary motion. In 1609, Johannes Kepler published his three laws of planetary motion, explaining how the planets orbit the Sun. In Kepler's final planetary model, he described planetary orbits as following elliptical paths with the Sun at a focal point of the
ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

. Kepler discovered that the
square In Euclidean geometry, a square is a regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * Regular (Badfinger ...
of the
orbital period The orbital period is the time a given astronomical object takes to complete one orbit around another object, and applies in astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of th ...
of each planet is directly
proportional Proportionality, proportion or proportional may refer to: Mathematics * Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant * Ratio, of one quantity to another, especially of a part compared ...
to the
cube In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
of the
semi-major axis In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...
of its orbit, or equivalently, that the
ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

of these two values is constant for all planets in the
Solar System The Solar SystemCapitalization Capitalization ( North American English) or capitalisation ( British English) is writing a word with its first letter as a capital letter (uppercase letter) and the remaining letters in lower case, in writin ...

.This constant ratio was later shown to be a direct measure of the Sun's active gravitational mass; it has units of distance cubed per time squared, and is known as the
standard gravitational parameter In celestial mechanics, the standard gravitational parameter ''μ'' of a celestial body is the product of the gravitational constant ''G'' and the mass ''M'' of the body. :\mu=GM \ For several objects in the Solar System The Solar Syste ...
: : $\mu=4\pi^2\frac\propto\text$
On 25 August 1609,
Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei (; 15 February 1564 – 8 January 1642) was an Italian astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the ...

demonstrated his first telescope to a group of Venetian merchants, and in early January 1610, Galileo observed four dim objects near Jupiter, which he mistook for stars. However, after a few days of observation, Galileo realized that these "stars" were in fact orbiting Jupiter. These four objects (later named the
Galilean moons 's four Galilean moons, in a composite image depicting part of Jupiter and their relative sizes (positions are illustrative, not actual). From top to bottom: Io (moon), Io, Europa (moon), Europa, Ganymede (moon), Ganymede, Callisto (moon), Callisto. ...
in honor of their discoverer) were the first celestial bodies observed to orbit something other than the Earth or Sun. Galileo continued to observe these moons over the next eighteen months, and by the middle of 1611, he had obtained remarkably accurate estimates for their periods.

## Galilean free fall

Sometime prior to 1638, Galileo turned his attention to the phenomenon of objects in free fall, attempting to characterize these motions. Galileo was not the first to investigate Earth's gravitational field, nor was he the first to accurately describe its fundamental characteristics. However, Galileo's reliance on scientific experimentation to establish physical principles would have a profound effect on future generations of scientists. It is unclear if these were just hypothetical experiments used to illustrate a concept, or if they were real experiments performed by Galileo, but the results obtained from these experiments were both realistic and compelling. A biography by Galileo's pupil
Vincenzo Viviani Vincenzo Viviani (April 5, 1622 – September 22, 1703) was an Italian Italian may refer to: * Anything of, from, or related to the country and nation of Italy ** Italians, an ethnic group or simply a citizen of the Italian Republic ** Italian lang ...
stated that Galileo had dropped
ball A ball is a round object (usually spherical, but can sometimes be ovoid An oval (from Latin ''ovum'', "egg") is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas ( projective ...

s of the same material, but different masses, from the
Leaning Tower of Pisa The Leaning Tower of Pisa ( it, torre pendente di Pisa) or simply the Tower of Pisa (''torre di Pisa'' ) is the ''campanile A bell tower is a tower that contains one or more bells, or that is designed to hold bells even if it has none. Such a ...

to demonstrate that their time of descent was independent of their mass.At the time when Viviani asserts that the experiment took place, Galileo had not yet formulated the final version of his law of free fall. He had, however, formulated an earlier version that predicted that bodies ''of the same material'' falling through the same medium would fall at the same speed. See In support of this conclusion, Galileo had advanced the following theoretical argument: He asked if two bodies of different masses and different rates of fall are tied by a string, does the combined system fall faster because it is now more massive, or does the lighter body in its slower fall hold back the heavier body? The only convincing resolution to this question is that all bodies must fall at the same rate. A later experiment was described in Galileo's ''Two New Sciences'' published in 1638. One of Galileo's fictional characters, Salviati, describes an experiment using a bronze ball and a wooden ramp. The wooden ramp was "12 cubits long, half a cubit wide and three finger-breadths thick" with a straight, smooth, polished groove. The groove was lined with "
parchment Parchment is a writing material Writing material refers to the materials that provide the surfaces on which humans use writing instruments A writing implement or writing instrument is an object used to produce writing Writing is a mediu ...

, also smooth and polished as possible". And into this groove was placed "a hard, smooth and very round bronze ball". The ramp was inclined at various
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...

s to slow the acceleration enough so that the elapsed time could be measured. The ball was allowed to roll a known distance down the ramp, and the time taken for the ball to move the known distance was measured. The time was measured using a water clock described as follows: :"a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results." Galileo found that for an object in free fall, the distance that the object has fallen is always proportional to the square of the elapsed time: : $\propto$ Galileo had shown that objects in free fall under the influence of the Earth's gravitational field have a constant acceleration, and Galileo's contemporary, Johannes Kepler, had shown that the planets follow elliptical paths under the influence of the Sun's gravitational mass. However, Galileo's free fall motions and Kepler's planetary motions remained distinct during Galileo's lifetime.

# Newtonian mass

Robert Hooke Robert Hooke FRS FRS may also refer to: Government and politics * Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States * Family Resources ...
had published his concept of gravitational forces in 1674, stating that all
celestial bodies In astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses ...
have an attraction or gravitating power towards their own centers, and also attract all the other celestial bodies that are within the sphere of their activity. He further stated that gravitational attraction increases by how much nearer the body wrought upon is to its own center. In correspondence with
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...

from 1679 and 1680, Hooke conjectured that gravitational forces might decrease according to the double of the distance between the two bodies. Hooke urged Newton, who was a pioneer in the development of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

, to work through the mathematical details of Keplerian orbits to determine if Hooke's hypothesis was correct. Newton's own investigations verified that Hooke was correct, but due to personal differences between the two men, Newton chose not to reveal this to Hooke. Isaac Newton kept quiet about his discoveries until 1684, at which time he told a friend,
Edmond Halley Edmond (or Edmund) Halley (; – ) was an English astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such ...

, that he had solved the problem of gravitational orbits, but had misplaced the solution in his office. After being encouraged by Halley, Newton decided to develop his ideas about gravity and publish all of his findings. In November 1684, Isaac Newton sent a document to Edmund Halley, now lost but presumed to have been titled ''
De motu corporum in gyrum :''For other works by a similar name see De Motu (disambiguation)''. ''De motu corporum in gyrum'' ('On the motion of bodies in an orbit') is the presumed title of a manuscript by Isaac Newton Sir Isaac Newton (25 December 1642 – ...
'' (Latin for "On the motion of bodies in an orbit"). Halley presented Newton's findings to the
Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. Founded on 28 November 1660, it was granted a royal charter by Charles II of ...
of London, with a promise that a fuller presentation would follow. Newton later recorded his ideas in a three-book set, entitled ''
Philosophiæ Naturalis Principia Mathematica (from Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it bec ...
'' (Latin: ''Mathematical Principles of Natural Philosophy''). The first was received by the Royal Society on 28 April 1685–86; the second on 2 March 1686–87; and the third on 6 April 1686–87. The Royal Society published Newton's entire collection at their own expense in May 1686–87. Isaac Newton had bridged the gap between Kepler's gravitational mass and Galileo's gravitational acceleration, resulting in the discovery of the following relationship which governed both of these: : $\mathbf=-\mu\frac$ where g is the apparent acceleration of a body as it passes through a region of space where gravitational fields exist, ''μ'' is the gravitational mass (
standard gravitational parameter In celestial mechanics, the standard gravitational parameter ''μ'' of a celestial body is the product of the gravitational constant ''G'' and the mass ''M'' of the body. :\mu=GM \ For several objects in the Solar System The Solar Syste ...
) of the body causing gravitational fields, and R is the radial coordinate (the distance between the centers of the two bodies). By finding the exact relationship between a body's gravitational mass and its gravitational field, Newton provided a second method for measuring gravitational mass. The mass of the Earth can be determined using Kepler's method (from the orbit of Earth's Moon), or it can be determined by measuring the gravitational acceleration on the Earth's surface, and multiplying that by the square of the Earth's radius. The mass of the Earth is approximately three-millionths of the mass of the Sun. To date, no other accurate method for measuring gravitational mass has been discovered.

## Newton's cannonball

Newton's cannonball was a
thought experiment A thought experiment is a hypothetical situation in which a hypothesis A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can t ...
used to bridge the gap between Galileo's gravitational acceleration and Kepler's elliptical orbits. It appeared in Newton's 1728 book ''A Treatise of the System of the World''. According to Galileo's concept of gravitation, a dropped stone falls with constant acceleration down towards the Earth. However, Newton explains that when a stone is thrown horizontally (meaning sideways or perpendicular to Earth's gravity) it follows a curved path. "For a stone projected is by the pressure of its own weight forced out of the rectilinear path, which by the projection alone it should have pursued, and made to describe a curve line in the air; and through that crooked way is at last brought down to the ground. And the greater the velocity is with which it is projected, the farther it goes before it falls to the Earth." Newton further reasons that if an object were "projected in an horizontal direction from the top of a high mountain" with sufficient velocity, "it would reach at last quite beyond the circumference of the Earth, and return to the mountain from which it was projected."

## Universal gravitational mass

In contrast to earlier theories (e.g.
celestial spheres The celestial spheres, or celestial orbs, were the fundamental entities of the cosmological Cosmology (from Greek κόσμος, ''kosmos'' "world" and -λογία, ''-logia'' "study of") is a branch of astronomy Astronomy (from ...
) which stated that the heavens were made of entirely different material, Newton's theory of mass was groundbreaking partly because it introduced universal gravitational mass: every object has gravitational mass, and therefore, every object generates a gravitational field. Newton further assumed that the strength of each object's gravitational field would decrease according to the square of the distance to that object. If a large collection of small objects were formed into a giant spherical body such as the Earth or Sun, Newton calculated the collection would create a gravitational field proportional to the total mass of the body, and inversely proportional to the square of the distance to the body's center.These two properties are very useful, as they allow spherical collections of objects to be treated exactly like large individual objects. For example, according to Newton's theory of universal gravitation, each carob seed produces a gravitational field. Therefore, if one were to gather an immense number of carob seeds and form them into an enormous sphere, then the gravitational field of the sphere would be proportional to the number of carob seeds in the sphere. Hence, it should be theoretically possible to determine the exact number of carob seeds that would be required to produce a gravitational field similar to that of the Earth or Sun. In fact, by
unit conversion Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in ...
it is a simple matter of abstraction to realize that any traditional mass unit can theoretically be used to measure gravitational mass. Measuring gravitational mass in terms of traditional mass units is simple in principle, but extremely difficult in practice. According to Newton's theory, all objects produce gravitational fields and it is theoretically possible to collect an immense number of small objects and form them into an enormous gravitating sphere. However, from a practical standpoint, the gravitational fields of small objects are extremely weak and difficult to measure. Newton's books on universal gravitation were published in the 1680s, but the first successful measurement of the Earth's mass in terms of traditional mass units, the Cavendish experiment, did not occur until 1797, over a hundred years later. Henry Cavendish found that the Earth's density was 5.448 ± 0.033 times that of water. As of 2009, the Earth's mass in kilograms is only known to around five digits of accuracy, whereas its gravitational mass is known to over nine significant figures. Given two objects A and B, of masses ''M''A and ''M''B, separated by a Displacement (vector), displacement RAB, Newton's law of gravitation states that each object exerts a gravitational force on the other, of magnitude : $\mathbf_=-GM_M_\frac\$, where ''G'' is the universal . The above statement may be reformulated in the following way: if ''g'' is the magnitude at a given location in a gravitational field, then the gravitational force on an object with gravitational mass ''M'' is : $F=Mg$. This is the basis by which masses are determined by weighing. In simple spring scales, for example, the force ''F'' is proportional to the displacement of the spring (device), spring beneath the weighing pan, as per Hooke's law, and the scales are calibration, calibrated to take ''g'' into account, allowing the mass ''M'' to be read off. Assuming the gravitational field is equivalent on both sides of the balance, a Beam balance, balance measures relative weight, giving the relative gravitation mass of each object.

## Inertial mass

''Inertial mass'' is the mass of an object measured by its resistance to acceleration. This definition has been championed by Ernst MachOri Belkind, "Physical Systems: Conceptual Pathways between Flat Space-time and Matter" (2012) Springer (''Chapter 5.3'') and has since been developed into the notion of Operationalization, operationalism by Percy W. Bridgman. The simple classical mechanics definition of mass differs slightly from the definition in the theory of special relativity, but the essential meaning is the same. In classical mechanics, according to Newton's second law, we say that a body has a mass ''m'' if, at any instant of time, it obeys the equation of motion : $\mathbf=m \mathbf,$ where F is the resultant
force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...

acting on the body and a is the
acceleration In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

of the body's centre of mass.In its original form, Newton's second law is valid only for bodies of constant mass. For the moment, we will put aside the question of what "force acting on the body" actually means. This equation illustrates how mass relates to the
inertia Inertia is the resistance of any physical object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an ob ...

of a body. Consider two objects with different masses. If we apply an identical force to each, the object with a bigger mass will experience a smaller acceleration, and the object with a smaller mass will experience a bigger acceleration. We might say that the larger mass exerts a greater "resistance" to changing its state of motion in response to the force. However, this notion of applying "identical" forces to different objects brings us back to the fact that we have not really defined what a force is. We can sidestep this difficulty with the help of Newton's third law, which states that if one object exerts a force on a second object, it will experience an equal and opposite force. To be precise, suppose we have two objects of constant inertial masses ''m''1 and ''m''2. We isolate the two objects from all other physical influences, so that the only forces present are the force exerted on ''m''1 by ''m''2, which we denote F12, and the force exerted on ''m''2 by ''m''1, which we denote F21. Newton's second law states that : $\begin \mathbf & =m_1\mathbf_1,\\ \mathbf & =m_2\mathbf_2, \end$ where a1 and a2 are the accelerations of ''m''1 and ''m''2, respectively. Suppose that these accelerations are non-zero, so that the forces between the two objects are non-zero. This occurs, for example, if the two objects are in the process of colliding with one another. Newton's third law then states that : $\mathbf_=-\mathbf_;$ and thus : $m_1=m_2\frac\!.$ If is non-zero, the fraction is well-defined, which allows us to measure the inertial mass of ''m''1. In this case, ''m''2 is our "reference" object, and we can define its mass ''m'' as (say) 1 kilogram. Then we can measure the mass of any other object in the universe by colliding it with the reference object and measuring the accelerations. Additionally, mass relates a body's momentum p to its linear
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

v: : $\mathbf=m\mathbf$, and the body's kinetic energy ''K'' to its velocity: : $K=\dfracm, \mathbf, ^2$. The primary difficulty with Mach's definition of mass is that it fails to take into account the potential energy (or binding energy) needed to bring two masses sufficiently close to one another to perform the measurement of mass. This is most vividly demonstrated by comparing the mass of the proton in the nucleus of deuterium, to the mass of the proton in free space (which is greater by about 0.239%—this is due to the binding energy of deuterium). Thus, for example, if the reference weight ''m''2 is taken to be the mass of the neutron in free space, and the relative accelerations for the proton and neutron in deuterium are computed, then the above formula over-estimates the mass ''m''1 (by 0.239%) for the proton in deuterium. At best, Mach's formula can only be used to obtain ratios of masses, that is, as ''m''1 / ''m''2 =  / . An additional difficulty was pointed out by Henri Poincaré, which is that the measurement of instantaneous acceleration is impossible: unlike the measurement of time or distance, there is no way to measure acceleration with a single measurement; one must make multiple measurements (of position, time, etc.) and perform a computation to obtain the acceleration. Poincaré termed this to be an "insurmountable flaw" in the Mach definition of mass.

# Atomic masses

Typically, the mass of objects is measured in terms of the kilogram, which since 2019 is defined in terms of fundamental constants of nature. The mass of an atom or other particle can be compared more precisely and more conveniently to that of another atom, and thus scientists developed the
dalton Dalton may refer to: Science * Dalton (crater), a lunar crater * Dalton (program), chemistry software * Dalton (unit) (Da), the atomic mass unit Entertainment * Dalton (Buffyverse), minor character from ''Buffy the Vampire Slayer'' television s ...
(also known as the unified atomic mass unit). By definition, 1 Da (one
dalton Dalton may refer to: Science * Dalton (crater), a lunar crater * Dalton (program), chemistry software * Dalton (unit) (Da), the atomic mass unit Entertainment * Dalton (Buffyverse), minor character from ''Buffy the Vampire Slayer'' television s ...
) is exactly one-twelfth of the mass of a
carbon-12 Carbon-12 (12C) is the more abundant of the two stable A stable is a building in which livestock Livestock is commonly defined as domesticated Domestication is a sustained multi-generational relationship in which one group of organisms ...

atom, and thus, a carbon-12 atom has a mass of exactly 12 Da.

# In relativity

## Special relativity

In some frameworks of special relativity, physicists have used different definitions of the term. In these frameworks, two kinds of mass are defined: rest mass (invariant mass),It is possible to make a slight distinction between "rest mass" and "invariant mass". For a system of two or more particles, none of the particles are required be at rest with respect to the observer for the system as a whole to be at rest with respect to the observer. To avoid this confusion, some sources will use "rest mass" only for individual particles, and "invariant mass" for systems. and relativistic mass (which increases with velocity). Rest mass is the Newtonian mass as measured by an observer moving along with the object. ''Relativistic mass'' is the total quantity of energy in a body or system divided by speed of light, ''c''2. The two are related by the following equation: : $m_\mathrm=\gamma \left(m_\mathrm\right)\!$ where $\gamma$ is the Lorentz factor: : $\gamma = \frac$ The invariant mass of systems is the same for observers in all inertial frames, while the relativistic mass depends on the observer's frame of reference. In order to formulate the equations of physics such that mass values do not change between observers, it is convenient to use rest mass. The rest mass of a body is also related to its energy ''E'' and the magnitude of its momentum p by the relativistic energy-momentum equation: : $(m_\mathrm)c^2=\sqrt.\!$ So long as the system is Closed system, closed with respect to mass and energy, both kinds of mass are conserved in any given frame of reference. The conservation of mass holds even as some types of particles are converted to others. Matter particles (such as atoms) may be converted to non-matter particles (such as photons of light), but this does not affect the total amount of mass or energy. Although things like heat may not be matter, all types of energy still continue to exhibit mass.For example, a nuclear bomb in an idealized super-strong box, sitting on a scale, would in theory show no change in mass when detonated (although the inside of the box would become much hotter). In such a system, the mass of the box would change only if energy were allowed to escape from the box as light or heat. However, in that case, the removed energy would take its associated mass with it. Letting heat or radiation out of such a system is simply a way to remove mass. Thus, mass, like energy, cannot be destroyed, but only moved from one place to another. Thus, mass and energy do not change into one another in relativity; rather, both are names for the same thing, and neither mass nor energy ''appear'' without the other. Both rest and relativistic mass can be expressed as an energy by applying the well-known relationship Mass–energy equivalence, ''E'' = ''mc''2, yielding rest energy and "relativistic energy" (total system energy) respectively: : $E_\mathrm=(m_\mathrm)c^2\!$ : $E_\mathrm=(m_\mathrm)c^2\!$ The "relativistic" mass and energy concepts are related to their "rest" counterparts, but they do not have the same value as their rest counterparts in systems where there is a net momentum. Because the relativistic mass is Mass–energy equivalence, proportional to the energy, it has gradually fallen into disuse among physicists. There is disagreement over whether the concept remains useful Pedagogy, pedagogically. In bound systems, the binding energy must often be subtracted from the mass of the unbound system, because binding energy commonly leaves the system at the time it is bound. The mass of the system changes in this process merely because the system was not closed during the binding process, so the energy escaped. For example, the binding energy of atomic nuclei is often lost in the form of gamma rays when the nuclei are formed, leaving nuclides which have less mass than the free particles (nucleons) of which they are composed. Mass–energy equivalence also holds in macroscopic systems. For example, if one takes exactly one kilogram of ice, and applies heat, the mass of the resulting melt-water will be more than a kilogram: it will include the mass from the thermal energy (latent heat) used to melt the ice; this follows from the conservation of energy. This number is small but not negligible: about 3.7 nanograms. It is given by the latent heat of melting ice (334 kJ/kg) divided by the speed of light squared (''c''2 ≈ ).

## General relativity

In
general relativity General relativity, also known as the general theory of relativity, is the of published by in 1915 and is the current description of gravitation in . General generalizes and refines , providing a unified description of gravity as a geome ...
, the
equivalence principle In the theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational ...
is the equivalence of Gravitational mass, gravitational and Inertial Mass, inertial mass. At the core of this assertion is Albert Einstein, Albert Einstein's idea that the gravitational force as experienced locally while standing on a massive body (such as the Earth) is the same as the ''Fictitious force, pseudo-force'' experienced by an observer in a non-Inertial frame of reference, inertial (i.e. accelerated) frame of reference. However, it turns out that it is impossible to find an objective general definition for the concept of invariant mass in general relativity. At the core of the problem is the Nonlinear system, non-linearity of the Einstein field equations, making it impossible to write the gravitational field energy as part of the stress–energy tensor in a way that is invariant for all observers. For a given observer, this can be achieved by the stress–energy–momentum pseudotensor.

# In quantum physics

In classical mechanics, the inert mass of a particle appears in the Euler–Lagrange equation as a parameter ''m'': : $\frac \ \left( \, \frac \, \right) \ = \ m \, \ddot_i$. After quantization, replacing the position vector ''x'' with a wave function, the parameter ''m'' appears in the kinetic energy operator: : $i\hbar\frac \Psi(\mathbf,\,t) = \left(-\frac\nabla^2 + V(\mathbf)\right)\Psi(\mathbf,\,t)$. In the ostensibly Covariance and contravariance of vectors#Informal usage, covariant (relativistically invariant) Dirac equation, and in natural units, this becomes: : $(-i\gamma^\mu\partial_\mu + m) \psi = 0\,$ where the "Invariant mass, mass" parameter ''m'' is now simply a constant associated with the quantum described by the wave function ψ. In the
Standard Model The Standard Model of particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsi ...

of
particle physics Particle physics (also known as high energy physics) is a branch of that studies the nature of the particles that constitute and . Although the word ' can refer to various types of very small objects (e.g. , gas particles, or even household d ...
as developed in the 1960s, this term arises from the coupling of the field ψ to an additional field Φ, the Higgs field. In the case of fermions, the Higgs mechanism results in the replacement of the term ''m''ψ in the Lagrangian with $G_ \overline \phi \psi$. This shifts the explanandum of the value for the mass of each elementary particle to the value of the unknown Coupling constant, coupling constant ''G''ψ.

## Tachyonic particles and imaginary (complex) mass

A tachyonic field, or simply tachyon, is a quantum field with an imaginary number, imaginary mass. Although tachyons (particles that move faster-than-light, faster than light) are a purely hypothetical concept not generally believed to exist,Lisa Randall, ''Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions'', p.286: "People initially thought of tachyons as particles travelling faster than the speed of light...But we now know that a tachyon indicates an instability in a theory that contains it. Regrettably for science fiction fans, tachyons are not real physical particles that appear in nature." field (physics), fields with imaginary mass have come to play an important Higgs boson, role in modern physics and are discussed in popular books on physics.Brian Greene, ''The Elegant Universe'', Vintage Books (2000) Under no circumstances do any excitations ever propagate faster than light in such theories—the presence or absence of a tachyonic mass has no effect whatsoever on the maximum velocity of signals (there is no violation of causality). While the ''field'' may have imaginary mass, any physical particles do not; the "imaginary mass" shows that the system becomes unstable, and sheds the instability by undergoing a type of phase transition called tachyon condensation (closely related to second order phase transitions) that results in symmetry breaking in Standard Model, current models of
particle physics Particle physics (also known as high energy physics) is a branch of that studies the nature of the particles that constitute and . Although the word ' can refer to various types of very small objects (e.g. , gas particles, or even household d ...
. The term "tachyon" was coined by Gerald Feinberg in a 1967 paper, but it was soon realized that Feinberg's model in fact did not allow for superluminal speeds. Instead, the imaginary mass creates an instability in the configuration:- any configuration in which one or more field excitations are tachyonic will spontaneously decay, and the resulting configuration contains no physical tachyons. This process is known as tachyon condensation. Well known examples include the Higgs mechanism, condensation of the Higgs boson in
particle physics Particle physics (also known as high energy physics) is a branch of that studies the nature of the particles that constitute and . Although the word ' can refer to various types of very small objects (e.g. , gas particles, or even household d ...
, and ferromagnetism in condensed matter physics. Although the notion of a tachyonic imaginary number, imaginary mass might seem troubling because there is no classical interpretation of an imaginary mass, the mass is not quantized. Rather, the scalar field is; even for tachyonic quantum field theory, quantum fields, the field operators at Minkowski space, spacelike separated points still Canonical commutation relation, commute (or anticommute), thus preserving causality. Therefore, information still does not propagate faster than light, and solutions grow exponentially, but not superluminally (there is no violation of causality). Tachyon condensation drives a physical system that has reached a local limit and might naively be expected to produce physical tachyons, to an alternate stable state where no physical tachyons exist. Once the tachyonic field reaches the minimum of the potential, its quanta are not tachyons any more but rather are ordinary particles with a positive mass-squared. This is a special case of the general rule, where unstable massive particles are formally described as having a complex number, complex mass, with the real part being their mass in the usual sense, and the imaginary part being the Particle decay#Decay rate, decay rate in natural units. However, in quantum field theory, a particle (a "one-particle state") is roughly defined as a state which is constant over time; i.e., an eigenvalue of the Hamiltonian (quantum mechanics), Hamiltonian. An Particle decay, unstable particle is a state which is only approximately constant over time; If it exists long enough to be measured, it can be formally described as having a complex mass, with the real part of the mass greater than its imaginary part. If both parts are of the same magnitude, this is interpreted as a resonance appearing in a scattering process rather than a particle, as it is considered not to exist long enough to be measured independently of the scattering process. In the case of a tachyon, the real part of the mass is zero, and hence no concept of a particle can be attributed to it. In a Lorentz invariant theory, the same formulas that apply to ordinary slower-than-light particles (sometimes called "bradyons" in discussions of tachyons) must also apply to tachyons. In particular the energy–momentum relation: :$E^2 = p^2c^2 + m^2c^4 \;$ (where p is the relativistic momentum of the bradyon and m is its rest mass) should still apply, along with the formula for the total energy of a particle: :$E = \frac.$ This equation shows that the total energy of a particle (bradyon or tachyon) contains a contribution from its rest mass (the "rest mass–energy") and a contribution from its motion, the kinetic energy. When ''v'' is larger than ''c'', the denominator in the equation for the energy is imaginary number, "imaginary", as the value under the square root, radical is negative. Because the total
energy In , energy is the that must be to a or to perform on the body, or to it. Energy is a ; the law of states that energy can be in form, but not created or destroyed. The unit of measurement in the (SI) of energy is the , which is the ...

must be real number, real, the numerator must ''also'' be imaginary: i.e. the rest mass m must be imaginary, as a pure imaginary number divided by another pure imaginary number is a real number.

# See also

* Mass versus weight * Effective mass (spring–mass system) * Effective mass (solid-state physics) * Extension (metaphysics) * International System of Quantities * 2019 redefinition of SI base units

# External links

* * * * * * * {{Authority control Mass, Physical quantities SI base quantities