Mass

**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. An object's mass also determines the strength of its 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.

## Units of mass

Further information: Orders of magnitude (mass)

The standard 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. However, because precise measurement of a cubic decimeter of water at the proper temperature and pressure was difficult, in 1889 the kilogram was redefined as the mass of the international prototype of the kilogram of cast iron, and thus became independent of the meter and the properties of water.

However, the mass of the international prototype and its supposedly identical national copies have been found to be drifting over time. The re-definition of the kilogram and several other units occurred on 20 May 2019, following a final vote by the CGPM in November 2018. The new definition uses 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/*c*2 (where *c* is the speed of light). The electronvolt and its multiples, such as the MeV (megaelectronvolt), are commonly used in particle physics.
* the atomic mass unit (u) is 1/12 of the mass of a carbon-12 atom, approximately 1.66×10−27 kg. The atomic mass unit is convenient for expressing the masses of atoms and molecules.

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 distinguished, SI units are usually used instead.
* the Planck mass (*m*P) is the maximum mass of point particles (about 2.18×10−8 kg). It is used in particle physics.
* 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 (≈1.99×1030 kg).
* the mass of a very small particle may be identified by its inverse Compton wavelength (1 cm−1 ≈ 3.52×10−41 kg).
* the mass of a very large star or black hole may be identified with its Schwarzschild radius (1 cm ≈ 6.73×1024 kg).

## Definitions

Template:Properties of mass

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

### Weight vs. mass

Main articles: 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* = 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).

### Origin

Main articles: 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.

## Newtonian mass

π
2

⋅

10

−
5

AU
3

y
2

=
3.986
⋅

10

14

m
3

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 correspondence with 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 been titled *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, entitled *Philosophiæ Naturalis Principia Mathematica* (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:

g

=
−
μ

R

^

R

2

{\displaystyle \mathbf{g}=-\mu\frac{\hat{\mathbf{R}}}{|\mathbf{R}|^2}}

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.

### Newton's cannonball

Main articles: 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." 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) 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.

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.

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 figures.

Given two objects A and B, of masses *M*A and *M*B, separated by a displacement **R**AB, Newton's law of gravitation states that each object exerts a gravitational force on the other, of magnitude

F

AB

=
−
G

M

A

M

B

R

^

AB

R

AB

2

{\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

# F
M
g

{\displaystyle 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 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.

### Inertial mass

*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 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

F

=
m

a

,

{\displaystyle \mathbf{F}=m \mathbf{a},}

where **F** is the resultant force acting on the body and **a** is the acceleration of the body's centre of 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 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 **F**12, and the force exerted on *m*2 by *m*1, which we denote **F**21. Newton's second law states that

F

12

=

m
1

a

1

,

F

21

=

m
2

a

2

,

{\displaystyle
\begin{align}
\mathbf{F\_{12}} & =m\_1\mathbf{a}\_1,\\
\mathbf{F\_{21}} & =m\_2\mathbf{a}\_2,
\end{align}}

where **a**1 and **a**2 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

F

12

=
−

F

21

;

{\displaystyle \mathbf{F}\_{12}=-\mathbf{F}\_{21};}
and thus

m
1

=

m
2

a

2

a

1

.

If |**a**1| 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 **v**:

p

=
m

v

{\displaystyle \mathbf{p}=m\mathbf{v}},
and the body's kinetic energy *K* to its velocity:

# K

1
2

m

v

2

{\displaystyle K=\dfrac{1}{2}m|\mathbf{v}|^2}.

## Atomic mass

Main articles: 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 of the 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 (meaning, one dalton, that is, one unified atomic mass unit) 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 it.

## 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