Mass is both a property of a physical body and a measure of its
resistance to acceleration (a change in its state of motion) when a
net force is applied. It also determines the strength of its mutual
gravitational attraction to other bodies. The basic
SI unit of mass is
the kilogram (kg). In physics, mass is not the same as weight, even
though mass is often determined by measuring the object's weight using
a spring scale, rather than balance scale 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.
In Newtonian physics, mass can be generalized as the amount of matter
in an object. However, at very high speeds, special relativity states
that the kinetic energy of its motion becomes a significant additional
source of mass. Thus, any stationary body having mass has an
equivalent amount of energy, and all forms of energy resist
acceleration by a force and have gravitational attraction. In modern
physics, matter is not a fundamental concept because its definition
has proven elusive.
2 Units of mass
3 Definitions of mass
Weight vs. mass
3.2 Inertial vs. gravitational mass
3.3 Origin of mass
4 Pre-Newtonian concepts
Weight as an amount
4.2 Planetary motion
4.3 Galilean free fall
5 Newtonian mass
5.1 Newton's cannonball
5.2 Universal gravitational mass
5.3 Inertial mass
6 Atomic mass
Mass in relativity
7.2 General relativity
Mass in quantum physics
8.1 Tachyonic particles and imaginary (complex) mass
8.2 Exotic matter and negative mass
9 See also
12 External links
There are several distinct phenomena which 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 F = ma).
Active gravitational mass measures the gravitational force exerted by
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 the same
properties of physical bodies at the qualitative and quantitative
level respectively, by other words, the mass quantitatively describes
the inertia. According to
Newton's second law
Newton's second law of motion, 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 or is affected by a gravitational field. If a first body of
mass mA is placed at a distance r (center of mass to center of mass)
from a second body of mass mB, each body is subject to an attractive
force Fg = GmAmB/r2, where G =
6989667000000000000♠6.67×10−11 N kg−2 m2 is the "universal
gravitational constant". This is sometimes referred to as
gravitational mass.[note 1] Repeated experiments since the 17th
century have demonstrated that inertial and gravitational mass are
identical; since 1915, this observation has been entailed a priori in
the equivalence principle of general relativity.
Units of mass
Further information: Orders of magnitude (mass)
The kilogram is one of the seven
SI base units
SI base units and one of three which
is defined ad hoc (i.e. without reference to another base unit).
International System of Units
International System of Units (SI) unit of mass is the
kilogram (kg). The kilogram is 1000 grams (g), first defined in
1795 as one cubic decimeter of water at the melting point of ice. Then
in 1889, the kilogram was redefined as the mass of the international
prototype kilogram, and as such is independent of the meter, or the
properties of water. However, the mass of the international prototype
and its identical national copies have been found to be drifting over
time. It is expected that the re-definition of the kilogram and
several other units will change on May 20, 2019, following a final
vote by the
CGPM in November 2018. The new definition will use only
invariant quantities of nature: the speed of light, the caesium
hyperfine frequency, and the Planck constant.
Other units are accepted for use in SI:
the tonne (t) (or "metric ton") is equal to 1000 kg.
the electronvolt (eV) is a unit of energy, but because of the
mass–energy equivalence it can easily be converted to a unit of
mass, and is often used like one. In this context, the mass has units
of eV/c2 (where c is the speed of light). The electronvolt and its
multiples, such as the MeV (megaelectronvolt), are commonly used in
the atomic mass unit (u) is 1/12 of the mass of a carbon-12 atom,
approximately 6973166000000000000♠1.66×10−27 kg.[note 2] The
atomic mass unit is convenient for expressing the masses of atoms and
Outside the SI system, other units of mass include:
the slug (sl) is an Imperial unit of mass (about 14.6 kg).
the pound (lb) is a unit of both mass and force, used mainly in the
United States (about 0.45 kg or 4.5 N). In scientific
contexts where pound (force) and pound (mass) need to be
SI units are usually used instead.
Planck mass (mP) is the maximum mass of point particles (about
6992218000000000000♠2.18×10−8 kg). It is used in particle
the solar mass (M☉) is defined as the mass of the Sun. It is
primarily used in astronomy to compare large masses such as stars or
galaxies (≈7030199000000000000♠1.99×1030 kg).
the mass of a very small particle may be identified by its inverse
Compton wavelength (1 cm−1 ≈
the mass of a very large star or black hole may be identified with its
Schwarzschild radius (1 cm ≈
Definitions of mass
The relation between properties of mass and their associated physical
constants. Every massive object is believed to exhibit all five
properties. However, due to extremely large or extremely small
constants, it is generally impossible to verify more than two or three
properties for any object.
Schwarzschild radius (rs) represents the ability of mass to cause
curvature in space and time.
The standard gravitational parameter (μ) represents the ability of a
massive body to exert Newtonian gravitational forces on other bodies.
Inertial mass (m) represents the Newtonian response of mass to forces.
Rest energy (E0) represents the ability of mass to be converted into
other forms of energy.
Compton wavelength (λ) represents the quantum response of mass to
In physical science, 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, 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 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.
Active gravitational mass[note 3] is a measure of the strength of an
object's gravitational flux (gravitational flux is equal to the
surface integral 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 acceleration. For example,
an object in free fall near the
Moon 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
Passive gravitational mass is a measure of the strength of an object's
interaction with a gravitational field. 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. This equivalence is exemplified in a large number of
physical processes including pair production, nuclear fusion, and the
gravitational bending of light.
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
Curvature of spacetime is a relativistic manifestation of the
existence of mass. Such curvature 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 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 satellite.
Quantum mass manifests itself as a difference between an object's
quantum frequency and its wave number. The quantum mass of an
electron, the Compton wavelength, can be determined through various
forms of spectroscopy and is closely related to the Rydberg constant,
the Bohr radius, and the classical electron radius. The quantum mass
of larger objects can be directly measured using a
Watt balance. In
relativistic quantum mechanics, mass is one of the irreducible
representation labels of the Poincaré group.
Weight vs. mass
Mass versus weight
In everyday usage, mass and "weight" 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) refers to an intrinsic property of an object, whereas
"weight" (measured in newtons) measures an object's resistance to
deviating from its natural course of free fall, 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 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 or the Moon. 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 W = mg, where g =
7000980665000000000♠9.80665 m/s2 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. 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 W = –ma, 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).
Macroscopically, mass is associated with matter, although matter is
not, ultimately, as clearly defined a concept as mass. On the
subatomic scale, not only fermions, the particles often associated
with matter, but also some bosons, the particles that act as force
carriers, have rest mass. Another problem for easy definition is that
much of the rest mass of ordinary matter derives from the binding
energy (potential energy) holding their quarks together and other
forms of energy rather than the sum of the rest masses of the
individual particle constituents. For example, only 1% of the rest
mass of matter is accounted for by the rest mass of its elementary
quarks and electrons. From a fundamental physics perspective, mass is
the number describing under which the representation of the little
group of the
Poincaré group a particle transforms. In the Standard
Model of particle physics, this symmetry is described as arising as a
consequence of a coupling of particles with rest mass to a postulated
additional field, known as the Higgs field.
The total mass of the observable universe is estimated at
1053 kg, corresponding to the rest mass of between 1079 and
1080 protons.
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
Newton's third law
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 developed his general theory of relativity starting
with the assumption of the intentionality of correspondence between
inertial and passive gravitational mass, and that no experiment will
ever detect a difference between them, in essence the equivalence
This particular equivalence often referred to as the "Galilean
equivalence principle" or the "weak equivalence principle" has the
most important consequence for freely falling objects. Suppose an
object with inertial and gravitational masses m and M, respectively.
If the only force acting on the object comes from a gravitational
field g, combining
Newton's second law
Newton's second law and the gravitational law
yields the acceleration
displaystyle a= frac M m g.
This says that the ratio of gravitational to inertial mass of any
object is equal to some constant K if and only if 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
obtained by dropping objects from the Leaning Tower of Pisa. This is
most likely apocryphal: he is more likely to have performed his
experiments with balls rolling down nearly frictionless inclined
planes 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 pendulum,
in 1889. As of 2008[update], 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
The universality of free-fall only applies to systems in which gravity
is the only acting force. All other forces, especially friction and
air resistance, 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, 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
David Scott did on the surface of the
Moon during Apollo
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. 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 of mass
Mass generation mechanism
In theoretical physics, a mass generation mechanism is a theory which
attempts to explain the origin of mass from the most fundamental laws
of physics. 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,
known as the Standard Model.
Weight as an amount
Main article: weight
Depiction of early balance scales in the
Papyrus of Hunefer
Papyrus of Hunefer (dated to
the 19th dynasty, ca. 1285 BC). The scene shows
Anubis weighing the
heart of Hunefer.
The concept of amount 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 to the number of objects in
displaystyle 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:
displaystyle frac W_ n n = frac W_ m m
, or equivalently
displaystyle frac W_ n W_ m = frac n m .
An early use of this relationship is a balance scale, 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 seed (carat or
siliqua) as a measurement standard. If an object's weight was
equivalent to 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:
displaystyle frac mathrm ounce mathrm pound = frac W_
144 W_ 1728 = frac 144 1728 = frac 1 12 .
See also: Kepler's laws of planetary motion
In 1600 AD,
Johannes Kepler sought employment with Tycho Brahe, 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. Kepler discovered
that the square of the orbital period of each planet is directly
proportional to the cube of the semi-major axis of its orbit, or
equivalently, that the ratio of these two values is constant for all
planets in the Solar System.[note 4]
On 25 August 1609,
Galileo Galilei 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
Galilean moons in honor of their discoverer) were the first
celestial bodies observed to orbit something other than the
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
Galileo Galilei (1636)
Distance traveled by a freely falling ball is proportional to the
square of the elapsed time
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 stated that Galileo had
dropped balls of the same material, but different masses, from the
Leaning Tower of Pisa
Leaning Tower of Pisa to demonstrate that their time of descent was
independent of their mass.[note 5] 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, 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 angles 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
Distance propto text
Time ^ 2
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
Isaac Newton 1689
Mass of Earth
Sidereal orbital period
0.002 569 AU
0.074 802 sidereal year
displaystyle 1.2pi ^ 2 cdot 10^ -5 frac text AU ^ 3 text y
^ 2 =3.986cdot 10^ 14 frac text m ^ 3 text s ^ 2
9.806 65 m/s2
6 375 km
Robert Hooke had published his concept of gravitational forces in
1674, stating that all celestial bodies 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 their own center. In
Isaac Newton 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, 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, 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
De motu corporum in gyrum (Latin for "On the motion of
bodies in an orbit"). Halley presented Newton's findings to the
Royal Society of London, with a promise that a fuller presentation
would follow. Newton later recorded his ideas in a three book set,
Philosophiæ Naturalis Principia Mathematica
Philosophiæ Naturalis Principia Mathematica (Latin:
"Mathematical Principles of Natural Philosophy"). The first was
received by the
Royal Society on 28 April 1685–6; the second on 2
March 1686–7; and the third on 6 April 1686–7. The Royal Society
published Newton’s entire collection at their own expense in May
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:
displaystyle mathbf g =-mu frac hat mathbf R mathbf R
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) 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.
A cannon on top of a very high mountain shoots a cannonball
horizontally. If the speed is low, the cannonball quickly falls back
Earth (A,B). At intermediate speeds, it will revolve around Earth
along an elliptical orbit (C,D). At a sufficiently high speed, it will
Earth altogether (E).
Main article: Newton's cannonball
Newton's cannonball was a thought experiment 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.":513 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."[citation
Universal gravitational mass
An apple experiences gravitational fields directed towards every part
of the Earth; however, the sum total of these many fields produces a
single gravitational field directed towards the Earth's center
In contrast to earlier theories (e.g. celestial spheres) 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,:397 and inversely proportional to the
square of the distance to the body's center.:221[note 6]
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 it is
a simple matter of abstraction to realize that any traditional mass
unit can theoretically be used to measure gravitational mass.
Vertical section drawing of Cavendish's torsion balance instrument
including the building in which it was housed. The large balls were
hung from a frame so they could be rotated into position next to the
small balls by a pulley from outside. Figure 1 of Cavendish's paper.
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. 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
Given two objects A and B, of masses MA and MB, separated by a
displacement RAB, Newton's law of gravitation states that each object
exerts a gravitational force on the other, of magnitude
displaystyle mathbf F _ text AB =-GM_ text A M_ text B frac
hat mathbf R _ text AB mathbf R _ text AB ^ 2
where G is the universal gravitational constant. 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
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 beneath the weighing pan, as per Hooke's
law, and the scales are 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 balance measures relative
weight, giving the relative gravitation mass of each object.
Massmeter, a device for measuring the inertial mass of an astronaut in
weightlessness. The mass is calculated via the oscillation period for
a spring with the astronaut attached (Tsiolkovsky State Museum of the
History of Cosmonautics)
Inertial mass is the mass of an object measured by its resistance to
acceleration. This definition has been championed by Ernst
Mach and has since been developed into the notion of
operationalism by Percy W. Bridgman. The simple classical
mechanics definition of mass is slightly different than the definition
in the theory of special relativity, but the essential meaning is the
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
displaystyle mathbf F =mmathbf a ,
where F is the resultant force acting on the body and a is the
acceleration of the body's centre of mass.[note 7] For the moment, we
will put aside the question of what "force acting on the body"
This equation illustrates how mass relates to the inertia 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
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 m1
and m2. We isolate the two objects from all other physical influences,
so that the only forces present are the force exerted on m1 by m2,
which we denote F12, and the force exerted on m2 by m1, which we
Newton's second law
Newton's second law states that
displaystyle begin aligned mathbf F_ 12 &=m_ 1 mathbf a
_ 1 ,\mathbf F_ 21 &=m_ 2 mathbf a _ 2 ,end aligned
where a1 and a2 are the accelerations of m1 and m2, 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
displaystyle mathbf F _ 12 =-mathbf F _ 21 ;
displaystyle m_ 1 =m_ 2 frac mathbf a _ 2 mathbf a _ 1
If a1 is non-zero, the fraction is well-defined, which allows us to
measure the inertial mass of m1. In this case, m2 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
displaystyle mathbf p =mmathbf v
and the body's kinetic energy K to its velocity:
displaystyle K= dfrac 1 2 mmathbf v ^ 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 m2 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 m1 (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 m1
/m2 = a2 / a1. 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.
Main article: Atomic mass unit
Typically, the mass of objects is measured in relation to that of the
kilogram, which is defined as the mass of the international prototype
kilogram (IPK), a platinum alloy cylinder stored in an
environmentally-monitored safe secured in a vault at the International
Bureau of Weights and Measures in France. However, the IPK is not
convenient for measuring the masses of atoms and particles of similar
scale, as it contains trillions of trillions of atoms, and has most
certainly lost or gained a little mass over time despite the best
efforts to prevent this. It is much easier to precisely compare an
atom's mass to that of another atom, thus scientists developed the
atomic mass unit (or Dalton). By definition, 1 u is exactly one
twelfth of the mass of a carbon-12 atom, and by extension a carbon-12
atom has a mass of exactly 12 u. This definition, however, might
be changed by the proposed redefinition of SI base units, which will
leave the Dalton very close to one, but no longer exactly equal to
Mass in relativity
Mass in special relativity
In special relativity, there are two kinds of mass: rest mass
(invariant mass),[note 8] and relativistic mass (which increases with
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 c2. The two are related by
the following equation:
displaystyle m_ mathrm relative =gamma (m_ mathrm rest )!
is the Lorentz factor:
displaystyle gamma = frac 1 sqrt 1-v^ 2 /c^ 2
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:
displaystyle (m_ mathrm rest )c^ 2 = sqrt E_ mathrm total
^ 2 -(mathbf p c)^ 2 .!
So long as the system is 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.[note 9] 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 E = mc2, yielding rest
energy and "relativistic energy" (total system energy) respectively:
displaystyle E_ mathrm rest =(m_ mathrm rest )c^ 2 !
displaystyle E_ mathrm total =(m_ mathrm relative )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 proportional to the energy, it has gradually
fallen into disuse among physicists. There is disagreement over
whether the concept remains useful 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 (c2 = 9×1016 m2/s2).
Mass in general relativity
In general relativity, the equivalence principle is any of several
related concepts dealing with the equivalence of gravitational and
inertial mass. At the core of this assertion is 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 pseudo-force
experienced by an observer in a non-inertial (i.e. accelerated) frame
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 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.
Mass in quantum physics
In classical mechanics, the inert mass of a particle appears in the
Euler–Lagrange equation as a parameter m:
displaystyle frac mathrm d mathrm d t left(, frac
partial L partial dot x _ i ,right) = m, ddot x _ i
After quantization, replacing the position vector x with a wave
function, the parameter m appears in the kinetic energy operator:
displaystyle ihbar frac partial partial t Psi (mathbf r
,,t)=left(- frac hbar ^ 2 2m nabla ^ 2 +V(mathbf r )right)Psi
(mathbf r ,,t)
In the ostensibly covariant (relativistically invariant) Dirac
equation, and in natural units, this becomes:
displaystyle (-igamma ^ mu partial _ mu +m)psi =0,
where the "mass" parameter m is now simply a constant associated with
the quantum described by the wave function ψ.
Standard Model of particle physics 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 mψ in the Lagrangian
displaystyle G_ psi overline psi phi psi
. This shifts the explanandum of the value for the mass of each
elementary particle to the value of the unknown couplings Gψ.
Tachyonic particles and imaginary (complex) mass
Tachyonic field and
Tachyon § Mass
A tachyonic field, or simply tachyon, is a quantum field with an
imaginary mass. Although tachyons (particles that move faster than
light) are a purely hypothetical concept not generally believed to
exist, fields with imaginary mass have come to play an
important role in modern physics and are discussed in
popular books on physics. 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 current
models of particle physics.
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
condensation of the
Higgs boson in particle physics, and
ferromagnetism in condensed matter physics.
Although the notion of a tachyonic 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 fields, the field operators at spacelike separated points
still 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 mass, with the
real part being their mass in the usual sense, and the imaginary part
being the 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. An 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
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:
displaystyle E^ 2 =p^ 2 c^ 2 +m^ 2 c^ 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:
displaystyle E= frac mc^ 2 sqrt 1- frac v^ 2 c^ 2 .
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", as the value under the radical is negative.
Because the total energy must be 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.
Exotic matter and negative mass
Main article: negative mass
The negative mass exists in the model to describe dark energy (phantom
energy) and radiation in negative-index metamaterial in a unified
way. In this way, the negative mass is associated with negative
momentum, negative pressure, negative kinetic energy and FTL
Mass versus weight
Effective mass (spring–mass system)
Effective mass (solid-state physics)
International System of Quantities
Proposed redefinition of SI base units
^ When a distinction is necessary, M is used to denote the active
gravitational mass and m the passive gravitational mass.
^ Since the
Avogadro constant NA is defined as the number of atoms in
12 g of carbon-12, it follows that 1 u is exactly
^ The 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".www2.jpl.nasa.gov/basics/bsf3-2.php 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.
^ 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:
displaystyle mu =4pi ^ 2 frac text distance ^ 3 text time
^ 2 propto text gravitational 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 which predicted
that bodies of the same material falling through the same medium would
fall at the same speed. See Drake, S. (1978). Galileo at Work.
University of Chicago Press. pp. 19–20.
^ These two properties are very useful, as they allow spherical
collections of objects to be treated exactly like large individual
^ In its original form,
Newton's second law
Newton's second law is valid only for bodies
of constant 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.
^ 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.
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SI base quantities
l, x, r, (etc.)
r = 10 m
m = 10 kg
t = 10 s
I , i
I = 10 A
T = 10 K
amount of substance
n = 10 mol
Iv = 10 cd
The quantity (not the unit) can have a specification: Tmax = 300 K
A quantity Q is expressed in the base quantities:
displaystyle Q=fleft( mathit l,m,t,I,T,n,I mathrm _ v right)
dim Q = La · Mb · Tc · Id · Θe ·
Nf · Jg (Superscripts a–g are algebraic exponents, usually a
positive, negative or zero integer.)
Quantity acceleration = l1 · t−2, dim acceleration =
L1 · T−2
possible units: m1 · s−2, km1 · Ms−2, etc.
History of the metric system
International System of Quantities
Systems of measurement
Classical mechanics SI units
distance: d, position: r, s, x, displacement
angle: θ, angular displacement: θ
solid angle: Ω
speed: v, velocity: v
kinematic viscosity: ν,
specific angular momentum: h
angular speed: ω, angular velocity: ω
angular acceleration: α
angular jerk: ζ
moment of inertia: I
momentum: p, impulse: J
kg m s−1, N s
action: 𝒮, actergy: ℵ
kg m2 s−1, J s
angular momentum: L, angular impulse: ΔL
kg m2 s−1
action: 𝒮, actergy: ℵ
kg m2 s−1, J s
force: F, weight: Fg
kg m s−2, N
energy: E, work: W
kg m2 s−2, J
torque: τ, moment: M
kg m2 s−2, N m
energy: E, work: W
kg m2 s−2, J
kg m s−3, N s−1
kg m2 s−3, W
kg m2 s−3, N m s−1
kg m2 s−3, W
BNF: cb11973906k (data)