Mass

/div>

Jump to: navigation, search

This article is about the scientific concept. For the substance of which all physical objects consist, see Matter. For other uses, see Mass (disambiguation).

Classical mechanics
$\mathbf{F} = m \mathbf{a}$ Newton's Second Law
History of classical mechanics · Timeline of classical mechanics
Branches Statics · Dynamics / Kinetics · Kinematics · Applied mechanics · Celestial mechanics · Continuum mechanics · Statistical mechanics
Formulations Newtonian mechanics (Vectorial mechanics) Analytical mechanics: Lagrangian mechanics Hamiltonian mechanics
Fundamental concepts Space · Time · Velocity · Speed · Mass · Acceleration · Gravity · Force · Impulse · Torque / Moment / Couple · Momentum · Angular momentum · Inertia · Moment of inertia · Reference frame · Energy · Kinetic energy · Potential energy · Mechanical work · Virtual work · D'Alembert's principle
Core topics Rigid body · Rigid body dynamics · Euler's equations (rigid body dynamics) · Motion · Newton's laws of motion · Newton's law of universal gravitation · Euler's laws of motion · Equations of motion · Inertial frame of reference · Non-inertial reference frame · Rotating reference frame · Fictitious force · Linear motion · Mechanics of planar particle motion · Displacement (vector) · Relative velocity · Friction · Simple harmonic motion · Harmonic oscillator · Vibration · Damping · Damping ratio · Rotational motion · Circular motion · Uniform circular motion · Non-uniform circular motion · Centripetal force · Centrifugal force · Centrifugal force (rotating reference frame) · Reactive centrifugal force · Coriolis force · Pendulum · Rotational speed · Angular acceleration · Angular velocity · Angular frequency · Angular displacement
Scientists Galileo Galilei · Isaac Newton · Jeremiah Horrocks · Leonhard Euler · Jean le Rond d'Alembert · Alexis Clairaut · Joseph Louis Lagrange · Pierre-Simon Laplace · William Rowan Hamilton · Siméon-Denis Poisson
v d e

In physics, mass (from Greek μᾶζα "barley cake, lump (of dough)"), more specifically inertial mass, can be defined as a quantitative measure of an object's resistance to the change of its speed. In addition to this, gravitational mass can be described as a measure of magnitude of the gravitational force which is

exerted by an object (active gravitational mass), or
experienced by an object (passive gravitational force)

when interacting with a second object. The SI unit of mass is the kilogram (kg).

In everyday usage, mass is often referred to as weight, the units of which are often taken to be kilograms (for instance, a person may state that their weight is 75 kg). In scientific use, however, the term weight refers to a different, yet related, property of matter. Weight is the gravitational force acting on a given body, while mass is an intrinsic property of this body. On the surface of the Earth, the weight W of an object is related to its mass m by W = mg, where g is the Earth's gravitational field strength, equal to about 9.81 m s⁻². An object's weight depends on its environment, while its mass does not: an object with a mass of 50 kilograms weighs 491 Newtons on the surface of the Earth; on the surface of the Moon, the same object still has a mass of 50 kilograms but weighs only 81.5 Newtons.

The inertial mass of an object determines its acceleration in the presence of an applied force. According to Newton's second law of motion, if a body of fixed mass M is subjected to a force F, its acceleration α 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 M_A is placed at a distance r from a second body of mass M_B, each body experiences an attractive force F_G whose magnitude is F_G= G M_AM_B r⁻², where G is the universal constant of gravitation, equal to 6.67×10⁻¹¹ N m²kg^-2. This is sometimes referred to as gravitational mass^[1] Repeated experiments since the 17th century have demonstrated that inertial and gravitational mass are equivalent; since 1915, this observation has been entailed a priori in the equivalence principle of general relativity.

Special relativity shows that rest mass (or invariant mass) and rest energy are essentially equivalent, via the well-known relationship (E=mc²). This same equation also connects relativistic mass and "relativistic energy" (total system energy). These are concepts that are related to their "rest" counterparts, but they do not have the same value, in systems where there is a net momentum. In order to deduce any of these four quantities from any of the others, in any system which has a net momentum, an equation that takes momentum into account is needed. Mass (so long as the type and definition of mass is agreed upon) is a conserved quantity over time. From the viewpoint of any single unaccelerated observer, mass can neither be created or destroyed, and special relativity does not change this understanding (though different observers may not agree on how much mass is present, all agree that the amount does not change over time).

Macroscopically, mass is associated with matter. But on the sub-atomic scale, not only fermions, the particles associated with matter, but also some bosons, the particles that act as force carriers, have rest mass. In the Standard Model of particle physics, mass is described as arising as a consequence of a coupling of the field of which the massive particles are quanta to a postulated additional field, known as the Higgs field.

The total mass of the observable universe is estimated at between 10⁵² kg and 10⁵³ kg, corresponding to the rest mass of between 10⁷⁹ and 10⁸⁰ protons.

[ Units of mass

Further information: Orders of magnitude (mass)

The kilogram is one of the seven SI base units; among these, it is one of three (besides the second and the Kelvin) which is defined ad hoc, without reference to another base unit, since 1889 by means of the international prototype kilogram.^[2]

Balance scales allow to directly compare gravitational mass within a gravitational field. Such devices have been in use since at least the Middle Bronze Age (shown is a balance for weighing tobacco dating to the mid-19th century).

In the International System of Units (SI), mass is measured in kilograms (kg). The gram (g) is ¹⁄₁₀₀₀ of a kilogram. The gram was first introduced in 1795, with a definition based on the density of water (so that at the temperature of melting ice, one cubic centimeter of water would have a mass of one gram; while the meter at the time was defined as the 10,000,000th part of the distance from the Earth's equator to the North Pole). Since 1889, the kilogram has been defined as the mass of the international prototype kilogram, and as such is independent of the meter, or the properties of water. In October 2011, the 24th General Conference on Weights and Measures resolved to "take note of the intention" to redefine the kilogram in terms of the Planck constant, scheduled for 2014.

Other units are accepted for use in SI:

The tonne (t) is equal to 1000 kg.
The electronvolt (eV) is primarily a unit of energy, but because of the mass-energy equivalence it can also function as a unit of mass. In this context it is denoted eV/c², or simply as eV. The electronvolt is common in particle physics.
The atomic mass unit (u) is defined so that a single carbon-12 atom has a mass of 12 u; 1 u is approximately 1.66×10⁻²⁷ kg.^{[note 1]} The atomic mass unit is convenient for expressing the masses of atoms and molecules.

Outside the SI system, a variety of different mass units are used, depending on context, such as the slug (sl), the pound (lb), the Planck mass (m_P), and the solar mass

In normal situations, the weight of an object is proportional to its mass, which usually makes it unproblematic to use the same unit for both concepts. However, the distinction between mass and weight becomes important for measurements with a precision better than a few percent (because of slight differences in the strength of the Earth's gravitational field at different places), and for places far from the surface of the Earth, such as in space or on other planets.

A mass can sometimes be expressed in terms of length. The mass of a very small particle may be identified with 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).

[ Summary of mass concepts and formalisms

In classical mechanics, mass has a central role in determining the behavior of bodies. Newton's second law relates the force F exerted in a body of mass M to the body's acceleration α:

$\mathbf{F}=M\boldsymbol{\alpha}\$ .

Additionally, mass relates a body's momentum p to its velocity v:

$\mathbf{p}=M\mathbf{v}\$ ,

and the body's kinetic energy K to its velocity:

$\mathbf{K}=\tfrac{1}{2}Mv^2\$ .

In special relativity, relativistic mass is a formalism which accounts for relativistic effects by having the mass increase with velocity.

$M=\gamma m_0\!$

$E=Mc^2\!$

Since energy is dependent on reference frame (upon the observer) it is convenient to formulate the equations of physics in a way such that mass values are invariant (do not change) between observers, and so the equations are independent of the observer. For a single particle, this quantity is the rest mass; for a system of bound or unbound particles, this quantity is the invariant mass. The invariant mass M of a body is related to its energy E and the magnitude of its momentum p by

$Mc^2=\sqrt{E^2-(pc)^2},\!$

where c is the speed of light.

[ Summary of mass related phenomena

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.

The Schwarzschild radius (r_s) 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.
The Compton wavelength (λ) represents the quantum response of mass to local geometry.

In physical science, one may distinguish conceptually between at least seven attributes of mass, or seven physical phenomena that can be explained using the concept of mass:^[3]

The amount of matter in certain types of samples can be exactly determined through electrodeposition^{[clarification needed]} or other precise processes. The mass of an exact sample is determined in part by the number and type of atoms or molecules it contains, and in part by the energy involved in binding it together (which contributes a negative "missing mass," or mass deficit).

Inertial mass is a measure of an object's resistance to changing its state of motion 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 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 freely fall and measuring its free-fall acceleration. For example, an object in free-fall near the Moon will experience less gravitational field, and hence accelerate slower 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. 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 through which measurable amounts of mass and energy are converted into each other. 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. Curvature is extremely weak and difficult to measure. For this reason, curvature wasn’t 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) than 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 watt balance.

Inertial mass, gravitational mass, and the various other mass-related phenomena are conceptually distinct. However, every experiment to date has shown these values to be proportional, and this proportionality gives rise to the abstract concept of mass. If, in some future experiment, one of the mass-related phenomena is shown to not be proportional to the others, then that specific phenomenon will no longer be considered a part of the abstract concept of mass.

[ Weight and amount

Main article: weight

A depiction of early balance scales in the Papyrus of Hunefer (dated to the 19th dynasty, ca. 1285 BC). The scene shows Anubis weighing the heart of Hunefer.

Weight, by definition, is a measure of the force which must be applied to support an object (i.e. hold it at rest) in a gravitational field. The Earth’s gravitational field causes items near the Earth to have weight. Typically, gravitational fields change only slightly over short distances, and the Earth’s field is nearly uniform at all locations on the Earth’s surface; therefore, an object’s weight changes only slightly when it is moved from one location to another, and these small changes went unnoticed through much of history. This may have given early humans the impression that weight is an unchanging, fundamental property of objects in the material world.

In the Egyptian religious illustration to the right, Anubis is using a balance scale to weigh the heart of Hunefer. A balance scale 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, and gives it the distinction of being one of the oldest known devices capable of measuring mass.

The concept of amount is very old and predates recorded history, so any description of the early development of this concept is speculative in nature. However, one might reasonably assume that humans, at some early era, realized that the weight of a collection of similar objects was directly proportional 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:

$\frac{w_n}{n} = \frac{w_m}{m}$ , or equivalently $\frac{w_n}{w_m} = \frac{n}{m}$ .

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:

$\frac{ounce}{pound} = \frac{w_{144}}{w_{1728}} = \frac{144}{1728} = \frac{1}{12}$ .

This example illustrates a common occurrence in physical science: when values are related through simple fractions, there is a good possibility that the values stem from a common source.

Various atoms and molecules as depicted in John Dalton's A New System of Chemical Philosophy (1808).

The name atom comes from the Greek ἄτομος/átomos, α-τεμνω, which means uncuttable, something that cannot be divided further. The philosophical concept that matter might be composed of discrete units that cannot be further divided has been around for millennia. However, empirical proof and the universal acceptance of the existence of atoms didn’t occur until the early 20th century.

As the science of chemistry matured, experimental evidence for the existence of atoms came from the law of multiple proportions. When two or more elements combined to form a compound, their masses are always in a fixed and definite ratio. For example, the mass ratio of nitrogen to oxygen in nitric oxide is seven eights. Ammonia has a hydrogen to nitrogen mass ratio of three fourteenths. The fact that elemental masses combined in simple fractions implies that all elemental mass stems from a common source. In principle, the atomic mass situation is analogous to the above example of Roman mass units. The Roman pound and ounce were both defined in terms of different sized collections of carob seeds, and consequently, the two mass units were related to each other through a simple fraction. Comparatively, since all of the atomic masses are related to each other through simple fractions, then perhaps the atomic masses are just different sized collections of some common fundamental mass unit.

In 1805, the chemist John Dalton published his first table of relative atomic weights, listing six elements, hydrogen, oxygen, nitrogen, carbon, sulfur, and phosphorus, and assigning hydrogen an atomic weight of 1. And in 1815, the chemist William Prout concluded that the hydrogen atom was in fact the fundamental mass unit from which all other atomic masses were derived.

Carbon atoms in graphite (image obtained with a Scanning tunneling microscope)

If Prout's hypothesis had proven accurate, then the abstract concept of mass, as we now know it, might never have evolved, since mass could always be defined in terms of amounts of the hydrogen atomic mass. Prout’s hypothesis; however, was found to be inaccurate in two major respects. First, further scientific advancements revealed the existence of smaller particles, such as electrons and quarks, whose masses are not related through simple fractions. And second, the elemental masses themselves were found to not be exact multiples of the hydrogen atom mass, but rather, they were near multiples. Einstein’s theory of relativity explained that when protons and neutrons come together to form an atomic nucleus, some of the mass of the nucleus is released in the form of binding energy. The more tightly bound the nucleus, the more energy is lost during formation and this binding energy loss causes the elemental masses to not be related through simple fractions.

Hydrogen, for example, with a single proton, has an atomic weight of 1.007825 u. The most abundant isotope of iron has 26 protons and 30 neutrons, so one might expect its atomic weight to be 56 times that of the hydrogen atom, but in fact, its atomic weight is only 55.9383 u, which is clearly not an integer multiple of 1.007825. Prout’s hypothesis was proven inaccurate in many respects, but the abstract concepts of atomic mass and amount continue to play an influential role in chemistry, and the atomic mass unit continues to be the unit of choice for very small mass measurements.

Weights as in a box with weights

When the French invented the metric system in the late 18th century, they used an amount to define their mass unit. The kilogram was originally defined to be equal in mass to the amount of pure water contained in a one-liter container. This definition, however, was inadequate for the precision requirements of modern technology, and the metric kilogram was redefined in terms of a man-made platinum-iridium bar known as the international prototype kilogram.

[ Gravitational mass

Active gravitational mass is a property of the mass of an object that produces a gravitational field in the space surrounding the object, and these gravitational fields govern large-scale structures in the Universe. Gravitational fields hold the galaxies together. They cause clouds of gas and dust to coalesce into stars and planets. They provide the necessary pressure for nuclear fusion to occur within stars. And they determine the orbits of various objects within the Solar System. Since gravitational effects are all around us, it is impossible to pin down the exact date when humans first discovered gravitational mass. However, it is possible to identify some of the significant steps towards our modern understanding of gravitational mass and its relationship to the other mass phenomena. Some terms associated with gravitational mass and its effects are the Gaussian gravitational constant, the standard gravitational parameter and the Schwarzschild radius.

[ Keplerian gravitational mass

Main article: Kepler's laws of planetary motion

Johannes Kepler 1610.

English name	The Keplerian planets
English name	Semi-major axis	Sidereal orbital period	Mass of Sun
Mercury	0.387 099 AU	0.240 842 sidereal year	$4 \pi^2 \frac{AU^3}{year^2}$
Venus	0.723 332 AU	0.615 187 sidereal year
Earth	1.000 000 AU	1.000 000 sidereal year
Mars	1.523 662 AU	1.880 816 sidereal year
Jupiter	5.203 363 AU	11.861 776 sidereal year
Saturn	9.537 070 AU	29.456 626 sidereal year

Johannes Kepler was the first to give an accurate description of the orbits of the planets, and by doing so; he was the first to describe gravitational mass. In 1600 AD, Kepler sought employment with Tycho Brahe and consequently gained access to astronomical data of a higher precision than any previously available. Using Brahe’s precise observations of the planet Mars, Kepler realized that the traditional astronomical methods were inaccurate in their predictions, and he spent the next five years developing his own method for characterizing planetary motion.

In Kepler’s final planetary model, he successfully described planetary orbits as following elliptical paths with the Sun at a focal point of the ellipse. The concept of active gravitational mass is an immediate consequence of Kepler's third law of planetary motion. 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. This constant ratio is 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:

$\mu = 4 \pi^2 \frac{Distance^3}{Time^2} \propto Gravitational \, Mass$

English name	The Galilean moons
English name	Semi-major axis	Sidereal orbital period	Mass of Jupiter
Io	0.002 819 AU	0.004 843 sidereal year	$0.0038 \pi^2 \frac{AU^3}{year^2}$
Europa	0.004 486 AU	0.009 722 sidereal year
Ganymede	0.007 155 AU	0.019 589 sidereal year
Callisto	0.012 585 AU	0.045 694 sidereal year

Main article: Galilean moons

In 1609, Johannes Kepler published his three rules known as Kepler's laws of planetary motion, explaining how the planets follow elliptical orbits under the influence of the Sun. On August 25 of that same year, Galileo Galilei demonstrated his first telescope to a group of Venetian merchants, and in early January of 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 thee CC-BY-SA.

mass