In physics, mass (from Ancient Greek: μᾶζα) commonly refers to any of three properties of matter, which have been shown experimentally to be equivalent: inertial mass, active gravitational mass and passive gravitational mass. In everyday usage, mass is often taken to mean weight, but in scientific use, they refer to different properties.
The inertial mass of an object determines its acceleration in the presence of an applied force. According to Isaac Newton's second law of motion, if a body of mass m is subjected to a 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 m1 is placed at a distance r from a second body of mass m2, the first body experiences an attractive force F given by
where G is the universal constant of gravitation, equal to 6.67×10−11 kg−1 m3 s−2. This is sometimes referred to as gravitational mass (when a distinction is necessary, M is used to denote the active gravitational mass and m the passive gravitational mass). Repeated experiments since the seventeenth century have demonstrated that inertial and gravitational mass are equivalent; this is entailed in the equivalence principle of general relativity.
Special relativity provides a relationship between the mass of a body and its energy (E = mc2). Mass is a conserved quantity. From the viewpoint of any single observer, mass can neither be created or destroyed, and special relativity does not change this understanding. However, relativity adds the fact that all types of energy have an associated mass, and this mass is added to systems when energy is added, and the associated mass is subtracted from systems when the energy leaves. In nuclear reactions, for example, the system does not become less massive until the energy liberated by the reaction is allowed to leave whereby the "missing mass" is carried off with the energy, which itself has mass.
On the surface of the Earth, the weight W of an object is related to its mass m by
where g is the acceleration due to the Earth's gravity, 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.
Other units are accepted for use in SI:
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).
and the body's kinetic energy Ek to its velocity:
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
where c is the speed of light.
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 phenomena will no longer be considered a part of the abstract concept of mass.
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. The scale, by comparing weights, also compares masses. The balance scale is one of the oldest known devices for 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:
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:
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:
This example illustrates a fundamental principle of physical science: when values are related through simple fractions, there is a good possibility that the values stem from a common source.
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 1900’s.
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.
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.
When the French invented the metric system in the late 1700s, 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 manmade platinum-iridium bar known as the international prototype kilogram.
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.
|The Keplerian Planets|
|Semi-major axis||Sidereal orbital period||Mass of Sun|
|Mercury||0.387 099 AU||0.240 842 sidereal year|
|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:
|The Galilean moons|
|Semi-major axis||Sidereal orbital period||Mass of Jupiter|
|Io||0.002 819 AU||0.004 843 sidereal year|
|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|
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 the Galilean moons in honor of their discoverer) were the first celestial bodies observed to orbit something other than the Earth or Sun. Galileo continued to observe these moons over the next eighteen months, and by the middle of 1611 he had obtained remarkably accurate estimates for their periods. Later, the semi-major axis of each moon was also estimated, thus allowing the gravitational mass of Jupiter to be determined from the orbits of its moons. The gravitational mass of Jupiter was found to be approximately a thousandth of the gravitational mass of the Sun.
Sometime prior to 1638, Galileo turned his attention to the phenomenon of objects falling under the influence of Earth’s gravity, and he was actively 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. Galileo used a number of scientific experiments to characterize free fall motion. 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 to demonstrate that their time of descent was independent of their mass. 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:
Galileo found that for an object in free fall, the distance that the object has fallen is always proportional to the square of the elapsed time:
Galileo Galilei died in Arcetri, Italy (near Florence), on 8 January 1642. 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, the relationship between Galileo’s gravitational field and Kepler’s gravitational mass wasn’t comprehended during Galileo’s life time.
|Earth's Moon||Mass of Earth|
|Semi-major axis||Sidereal orbital period|
|0.002 569 AU||0.074 802 sidereal year||
|Earth's Gravity||Earth's Radius|
|0.00980665 km⁄sec2||6 375 km|
Robert Hooke published his concept of gravitational forces in 1674, stating that: “all Coelestial Bodies whatsoever, have an attraction or gravitating power towards their own Centers" [and] "they do also attract all the other Coelestial Bodies that are within the sphere of their activity”. He further states that gravitational attraction increases “by how much the nearer the body wrought upon is to their own center.” In a correspondence of 1679-1680 between Robert Hooke and Isaac Newton, Hooke conjectures that gravitational forces might decrease according to the square 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 of 1684, Isaac Newton sent a document to Edmund Halley, now lost but presumed to have been titled De motu corporum in gyrum (Latin: "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 1685-6, 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 of 1686-7 .
Isaac Newton had bridged the gap between Kepler’s gravitational mass and Galileo’s gravitational acceleration, and proved the following relationship:
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 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 an 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.” Newton’s thought experiment is illustrated in the image to the right. A cannon on top of a very high mountain shoots a cannon ball in a horizontal direction. If the speed is low, it simply falls back on Earth (paths A and B). However, if the speed is equal to or higher than some threshold (orbital velocity), but not high enough to leave Earth altogether (escape velocity), it will continue revolving around Earth along an elliptical orbit (C and D).
Newton's cannonball illustrated the relationship between the Earth’s gravitational mass and its gravitational field; however, a number of other ambiguities still remained. Robert Hooke had asserted in 1674 that: "all Coelestial Bodies whatsoever, have an attraction or gravitating power towards their own Centers", but Hooke had neither explained why this gravitating attraction was unique to celestial bodies, nor had he explained why the attraction was directed towards the center of a celestial body.
To answer these questions, Newton introduced the entirely new concept that gravitational mass is “universal”: meaning that 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. With these assumptions in mind, Newton calculated what the overall gravitational field would be if a large collection of small objects were formed into a giant spherical body. Newton found that a giant spherical body (like the Earth or Sun, with roughly uniform density at each given radius), would have a gravitational field which was proportional to the total mass of the body , and inversely proportional to the square of the distance to the body’s center .
Newton's concept of universal gravitational mass is illustrated in the image to the left. Every piece of the Earth has gravitational mass and every piece creates a gravitational field directed towards that piece. However, the overall effect of these many fields is equivalent to a single powerful field directed towards the center of the Earth. The apple behaves as if a single powerful gravitational field were accelerating it towards the Earth’s center.
Newton’s concept of universal gravitational mass puts gravitational mass on an equal footing with the traditional concepts of weight and amount. For example, the ancient Romans had used the carob seed as a weight standard. The Romans could place an object with an unknown weight on one side of a balance scale and place carob seeds on the other side of the scale, increasing the number of seeds until the scale was balanced. If an object’s weight was equivalent to 1728 carob seeds, then the object was said to weigh one Roman pound.
According to Newton’s theory of universal gravitation, each carob seed produces gravitational fields. 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. And since the Roman weight units were all defined in terms of carob seeds, then knowing the Earth’s, or Sun's “carob seed mass” would allow one to calculate the mass in Roman pounds, or Roman ounces, or any other Roman unit.
This possibility extends beyond Roman units and the carob seed. The British avoirdupois pound, for example, was originally defined to be equal to 7,000 barley grains. Therefore, if one could determine the Earth’s “barley grain mass” (the number of barley grains required to produce a gravitational field similar to that of the Earth), then this would allow one to calculate the Earth’s mass in avoirdupois pounds. Also, the original kilogram was defined to be equal in mass to a litre of pure water (the modern kilogram is defined by the manmade international prototype kilogram). Thus, the mass of the Earth in kilograms could theoretically be determined by ascertaining how many litres of pure water (or international prototype kilograms) would be required to produce gravitational fields similar to those of the Earth. In fact, 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. And if one were to collect an immense number of objects, the resulting sphere would probably be too large to construct on the surface of the Earth, and too expensive to construct in space. Newton’s books on universal gravitation were published in the 1680’s, but the first successful measurement of the Earth’s mass in terms of traditional mass units, the Cavendish experiment, didn’t 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 digits .
Although inertial mass, passive gravitational mass and active gravitational mass are conceptually distinct, no experiment has ever unambiguously demonstrated any difference between them. In classical mechanics, Newton's third law implies that active and passive gravitational mass must always be identical (or at least proportional), but the classical theory offers no compelling reason why the gravitational mass has to equal the inertial mass. That it does is merely an empirical fact.
Albert Einstein developed his general theory of relativity starting from the assumption that this correspondence between inertial and (passive) gravitational mass is not accidental: that no experiment will ever detect a difference between them (the weak version of the equivalence principle). However, in the resulting theory, gravitation is not a force and thus not subject to Newton's third law, so "the equality of inertial and active gravitational mass [...] remains as puzzling as ever".
Inertial mass is the mass of an object measured by its resistance to acceleration.
To understand what the inertial mass of a body is, one begins with classical mechanics and Newton's Laws of Motion. Later on, we will see how our classical definition of mass must be altered if we take into consideration the theory of special relativity, which is more accurate than classical mechanics. However, the implications of special relativity will not change the meaning of "mass" in any essential way.
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
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 A and B, with constant inertial masses mA and mB. We isolate the two objects from all other physical influences, so that the only forces present are the force exerted on A by B, which we denote FAB, and the force exerted on B by A, which we denote FBA. Newton's second law states that
where aA and aB are the accelerations of A and B, 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
Note that our requirement that aA be non-zero ensures that the fraction is well-defined.
This is, in principle, how we would measure the inertial mass of an object. We choose a "reference" object and define its mass mB 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.
The Newtonian concept of gravitational mass rests on Newton's law of gravitation. Let us suppose we have two objects A and B, separated by a distance rAB. The law of gravitation states that if A and B have gravitational masses MA and MB respectively, then each object exerts a gravitational force on the other, of magnitude
where G is the universal gravitational constant. The above statement may be reformulated in the following way: if g is the acceleration of a reference mass 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. A balance measures gravitational mass; only the spring scale measures weight.
The equivalence of inertial and gravitational masses is sometimes referred to as the Galilean equivalence principle or weak equivalence principle. The most important consequence of this equivalence principle applies to freely falling objects. Suppose we have 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 and the gravitational law yields the acceleration
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 to be 1 by defining our units appropriately.)
The first experiments demonstrating the universality of free-fall were conducted by Galileo. It is commonly stated that Galileo obtained his results by dropping objects from the Leaning Tower of Pisa, but this is most likely apocryphal; actually, he performed his experiments with balls rolling down inclined planes. Increasingly precise experiments have been performed, such as those performed by Loránd Eötvös, using the torsion balance pendulum, in 1889. As of 2008, no deviation from universality, and thus from Galilean equivalence, has ever been found, at least to the accuracy 10−12. More precise experimental efforts are still being carried out.
The universality of free-fall only applies to systems in which gravity is the only acting force. All other forces, especially friction 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 vacuum, as David Scott did on the surface of the Moon during Apollo 15.
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 straght line (in other words its inertia) and should therefore be a function of its inertial mass and the strength of the gravitational field.
The term mass in special relativity usually refers to the rest mass of the object, which is the Newtonian mass as measured by an observer moving along with the object. The invariant mass is another name for the rest mass of single particles. However, the more general invariant mass (calculated with a more complicated formula) may also be applied to systems of particles in relative motion, and because of this, is usually reserved for systems which consist of widely separated high-energy particles. The invariant mass of systems is the same for all observers and inertial frames, and cannot be destroyed, and is thus conserved, so long as the system is closed. In this case, "closure" implies that an idealized boundary is drawn around the system, and no mass/energy is allowed across it.
In as much as energy is conserved in closed systems in relativity, the mass of a system is also a quantity which is conserved: this means it does not change over time, even as some types of particles are converted to others. For any given observer, the mass of any system is separately conserved and cannot change over time, just as energy is separately conserved and cannot change over time. The incorrect popular idea that mass may be converted to (massless) energy in relativity is due to the fact that some matter particles may in some cases be converted to types of energy which are not matter (such as light, kinetic energy, and the potential energy in magnetic, electric, and other fields). However, this confuses "matter" (a non-conserved and ill-defined thing) with mass (which is well-defined and is conserved). Even if not considered "matter," all types of energy still continue to exhibit mass in relativity. 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. "Matter" particles may not be conserved in reactions in relativity, but closed-system mass always is.
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 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.
In bound systems, the binding energy must (often) be subtracted from the mass of the unbound system, simply because this energy has mass, and this mass is subtracted from the system when it is given off, at the time it is bound. Mass is not conserved in this process because the system is not closed during the binding process. A familiar example is the binding energy of atomic nuclei, which appears as other types of energy (such as gamma rays) when the nuclei are formed, and (after being given off) results in nuclides which have less mass than the free particles (nucleons) of which they are composed.
The term relativistic mass is also used, and this is the total quantity of energy in a body or system (divided by c2). The relativistic mass (of a body or system of bodies) includes a contribution from the kinetic energy of the body, and is larger the faster the body moves, so unlike the invariant mass, the relativistic mass depends on the observer's frame of reference. However, for given single frames of reference and for closed systems, the relativistic mass is also a conserved quantity.
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 pedagogically useful.
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Mass is a measure of the amount of matter in an object.
MASS (0.E. maesse; Fr. messe; Ger. Messe; Ital. messa; from eccl. Lat. missa), a name for the Christian eucharistic service, practically confined since the Reformation to that of the Roman Catholic Church. The various orders for the celebration of Mass are dealt with under Liturgy; a detailed account of the Roman order is given under Missal; and the general development of the eucharistic service, including the Mass, is described in the article Eucharist. The present article is confined (I) to the consideration of certain special meanings which have become attached to the word Mass and are the subject of somewhat acute controversy, (2) to the Mass in music.
The origin of the word missa, as applied to the Eucharist, is obscure. The first to discuss the matter is Isidore of Seville (Etym. vi. 19), who mentions an "evening office" (oi%icium vespertinum), a "morning office" (officium matutinum), and an office called missa. Of the latter he says: "Missa tempore sacrificii est, quando catechumeni foras mittuntur, clamante levita ` si quis catechumenus remansit, exeat foras.' Et inde missa,' quia sacramentis altaris interesse non possunt, qui nondum regenerati sunt" ("The missa is at the time of the sacrifice, when the catechumens are sent out, the deacon crying, ` If any catechumen remain, let him go forth.'" Hence missa, because those who are as yet unregenerate - i.e. unbaptized - may not be present at the sacraments of the altar). This derivation of the word Mass, which would connect it with the special formula of dismissal still preserved in the Roman liturgy - Ite, missa est- once generally accepted, is now disputed. It is pointed out that the word missa long continued to be applied to any church service, and more particularly to the lections (see Du Cange for numerous examples), and it is held that such services received their name of missal from the solemn form of dismissal with which it was customary to conclude them; thus, in the 4th century Pilgrimage of Etheria (Silvia) the word missa is used indiscriminately of the Eucharist, other services, and the ceremony of dismissal. F. Kattenbusch (Herzog-Hauck, Realencyklop. s. " Messe") ingeniously, but with little evidence, suggests that the word may have had a double origin and meaning: (I) in the sense of dimissio, " dismissal"; (2) in that of commissio, " commission," "official duty," i.e. the exact Latin equivalent of the Greek Xarovp-yia (see Liturgy), and hence the conflicting use of the term. It is, however, far more probable that it was a general term that gradually became crystallized as applying to that service in which the dismissal represented a more solemn function. In the narrower sense of "Mass" it is first found in St Ambrose 20, 4, ed. Ballerini): "Missam facere coepi. Dum offero..." which evidently identifies the missa with the sacrifice. It continued, however, to be used loosely, though its tendency to become proper only to the principal Christian service is clear from a passage in the 12th homily of Caesarius, bishop of Arles (d. 542): "If you will diligently attend, you will recognize that missae are not celebrated when the divine readings are recited in the church, but when gifts are offered and the Body and Blood of the Lord are consecrated." The complete service (missa ad integrum), the bishop goes on to say, cannot be had at home by reading and prayer, but only in the house of God, where, besides the Eucharist, "the divine word is preached and the blessing is given to the people." Whatever its origin, the word Mass had by the time of the Reformation been long applied only to the Eucharist; and, though in itself a perfectly colourless term, and used as such during the earlier stages of the 16th century controversies concerning the Eucharist, it soon became identified with that sacrificial aspect of the sacrament of the altar which it was the chief object of the Reformers to overthrow. In England, so late as the first Prayer-book of Edward VI., it remained one of the official designations of the Eucharist, which is there described as "The Supper of the Lorde and holy Communion, commonly called the Masse." This, however, like the service itself, represented a compromise which the more extreme reformers would not tolerate, and in the second Prayer-book, together with such language in the canon as might imply the doctrine of transubstantiation and of the sacrifice, the word Mass also disappears. That this abolition of the word Mass, as implying the offering of Christ's Body and Blood by the priest for the living and the dead was deliberate is clear from the language of those who were chiefly responsible for the change. Bishops Ridley and Latimer, the two most conspicuous champions of "the new religion," denounced "the Mass" with unmeasured violence; Latimer said of "Mistress Missa" that "the devil hath brought her in again"; Ridley said: "I do not take the Mass as it is at this day for the communion of the Church, but for a popish device," &c. (Works, ed. Parker Soc., pp. 121, 120), and again: "In the stead of the Lord's holy table they give the people, with much solemn disguising, a thing which they call their mass; but in deed and in truth it is a very masking and mockery of the true Supper of the Lord, or rather I may call it a crafty juggling, whereby these false thieves and jugglers have bewitched the minds of the simple people ... unto pernicious idolatory" (ib. p. 409). This language is reflected in the 31st of the Articles of Religion of the Church of England: "Wherefore the sacrifices of Masses, in which it was commonly said that the Priest did offer Christ for the quick and the dead, to have remission of pain and guilt, were blasphemous fables and dangerous deceits." Clearly the word Mass had ceased to be a colourless term generally applicable to the eucharistic service; it was, in fact, not only proscribed officially, but in the common language of English people it passed entirely out of use except in the sense in which it is defined in Johnson's Dictionary, i.e. that of the "Service of the Romish Church at the celebration of the Eucharist." In connexion with the Catholic reaction in the Church of England, which had its origin in the "Oxford Movement" of the 19th century, efforts have been made by some of the clergy to reintroduce the term "Mass" for the Holy Communion in the English Church.
See Du Cange, Glossarium, s.v. " Missa"; F. Kattenbusch in Herzog-Hauck, Realencyklopadie (ed. 1903), s.v. " Messe, dogmengeschichtlich"; for the facts as to the use of the word "Mass" at the time of the Reformation see the article by J. H. Round in the Nineteenth Century for May 1897. (W. A. P.) Mass, In Music: I. Polyphonic Masses. - The composition of musical settings of the Mass plays a part in the history of music which is of special importance up to and including the 1 6th century. As an art-form the musical Mass is governed to a peculiar degree by the structure of its text. It so happens that the supremely important parts of the Mass are those which have the smallest number of words, namely the Kyrie, important as being the opening prayer; the Sanctus and Benedictus, embodying the central acts and ideas of the service; and the Agnus Dei, the prayer with which it concludes. The 16th-century methods were specially fitted for highly developed music when words were few and embodied ideas of such important emotional significance or finality that they could be constantly repeated without losing force. Now the texts of the Gloria and Credo were more voluminous than any others which 16th-century composers attempted to handle in a continuous scheme. The practical limits of the church service made it impossible to break them up by setting each clause to a separate movement, a method by which 16th-century music composers contrived to set psalms and other long texts to compositions lasting an hour or longer. Accordingly, Palestrina and his great contemporaries and predecessors treated the Gloria and Credo in a style midway in polyphonic organization and rhythmic breadth between that of the elaborate motet (adopted in the Sanctus) and the homophonic reciting style of the Litany. The various ways in which this special style could be modified by the scale of the work, and contrasted with the broader and more elaborate parts, gave the Mass (even in its merely technical aspects) a range which made it to the 16th-century composer what the symphony is to the great instrumental classics. Moreover, as being inseparably associated with the highest act of worship, it inspired composers in direct proportion to their piety and depth of mind. Of course there were many false methods of attacking the art-problem, and many other relationships, true and false, between the complexity of the settings of the various parts of the Mass and of motets. The story of the action of the council of Trent on the subject of corruption of church music is told elsewhere (see Music and Palestrina); and it has been recently paralleled by a decree of Pope Pius X., which has restored the 16th-century polyphonic Mass to a permanent place in the Roman Catholic Church music.
The next definite stage in the musical history of the Mass was attained by the Neapolitan composers who were first to reach musical coherence after the monodic revolution at the beginning of the 17th century. The fruit of their efforts came to maturity in the Masses of Mozart and Haydn. By this time the resources of music were such that the long and varied text of the Gloria and Credo inevitably either overbalanced the scheme or met with an obviously perfunctory treatment. It is almost impossible, without asceticism of a radically inartistic kind, to treat with the resources of instrumental music and free harmony such passages as that from the Crucifixus to the Resurrexit, without an emotional contrast which inevitably throws any natural treatment of the Sanctus into the background, and makes the A gnus Dei an inadequate conclusion to the musical scheme. So unfavourable were the conditions of 18th-century music for the formation of a good ecclesiastical style that only a very small proportion of Mozart's and Haydn's Mass music may be said to represent their ideas of religious music at all. The best features of their Masses are those that combine faithfulness to the Neapolitan forms with a contrapuntal richness such as no Neapo litan composer ever achieved. Thus Mozart's most perfect as well as most ecclesiastical example is his extremely terse Mass in F, written at the age of seventeen, which is scored simply for fourpart chorus and solo voices accompanied by the organ with a largely independent bass and by two violins mostly in independent real parts. This scheme, with the addition of a pair of trumpets and drums and, occasionally, oboes, forms the normal orchestra of 18th-century Masses developed or degenerated from this model. Trombones often played with the three lower voices, a practice of high antiquity surviving from a time when there were soprano trombones or cornetti (Zincken, a sort of treble serpent) to play with the sopranos.
The enormous dramatic development in the symphonic music of Beethoven made the problem of the Mass with orchestral accompaniment almost insoluble. This makes it all the more remarkable that Beethoven's second and only important Mass (in D, Op. 123) is not only the most dramatic ever penned but is, perhaps, the last classical Mass that is thoughtfully based upon the liturgy, and is not a mere musical setting of what happens to be a liturgic text. It was intended for the installation of Beethoven's friend, the archduke Rudolph, as archbishop of Olmiitz; and, though not ready until two years after that occasion, it shows the most careful consideration of the meaning of a church service, no doubt of altogether exceptional length and pomp, but by no means impossible for its unique occasion. Immense as was Beethoven's dramatic force, it was equalled by his power of sublime repose; and he was accordingly able once more to put the supreme moment of the music where the service requires it to be, viz. in the Sanctus and Benedictus. In the Agnus Dei the circumstances of the time gave him something special to say which has never so imperatively demanded utterance since. Europe had been shattered by the Napoleonic wars. Beethoven read the final prayer of the Mass as a "prayer for inward and outward peace," and, giving it that title, organized it on the basis of a contrast between terrible martial sounds and the triumph of peaceful themes, in a scheme none the less spiritual and sublime because those who first heard it had derived their notions of the horror of war from living in Vienna during its bombardment. Critics who have lived in London during the relief of Mafeking have blamed Beethoven for his realism.
Schubert's Masses show rather the influence of Beethoven's not very impressive first Mass, which they easily surpass in interest, though they rather pathetically show an ignorance cf the meaning of the Latin words. The last two Masses are later than Beethoven's Mass in D and contain many remarkable passages. It is evident from them that a dramatic treatment of the Agnus Dei was "in the air"; all the more so, since Schubert does not imitate Beethoven's realism.
Music with Latin words is not excluded from the Lutheran Church, and the Kyrie and Gloria are frequently sung in succession and entitled a Mass. Thus the Four Short Masses of Bach are called short, not because they are on a small scale, which is far from being the case, but because they consist only of the Kyrie and Gloria. Bach's method is to treat each clause of his text as a separate movement, alternating choruses with groups of arias; a method which was independently adopted by Mozart in those larger masses in which he transcends the Neapolitan type, such as the great unfinished Mass in,C minor. This method, in the case of an entire Mass, results in a length far too great for a Roman Catholic service; and Bach's B minor Mass, which is such a setting of the entire test, must be regarded as a kind of oratorio. It thus has obviously nothing to do with the Roman liturgy; but as an independent setting of the text it is one of the most sublime and profoundly religious works in all art; and its singular perfection as a design is nowhere more evident than in its numerous adaptations of earlier works.
The most interesting of all these adaptations is the setting of the words: "Et expecto resurrectionem mortuorum et vitam venturi saeculi. - Amen." Obviously the greatest difficulty in any elaborate instrumental setting of the Credo is the inevitable anti-climax after the Resurrexit. Bach contrives to give this anti-climax a definite artistic value; all the more from the fact that his Crucifixus and Resurrexit, and the contrast between them, are among the most sublime and directly impressive things in all music. To the end of his Resurrexit chorus he appends an orchestral ritornello, summing up the material of the chorus in the most formal possible way, and thereby utterly destroying all sense of finality as a member of a large group, while at the same time not in the least impairing the force and contrast of the whole - that contrast having ineffaceably asserted itself at the moment when it occurred. After this the aria "Et in spiritum sanctum," in which the next dogmatic clauses are enshrined like relics in a casket, furnishes a beautiful decorative design on which the listener can repose his mind; and then comes the voluminous ecclesiastical fugue, Confiteor unum baptisma, leading, as through the door and world-wide spaces of the Catholic Church, to that veil which is not all darkness to the eye of faith. At the words "Et expecto resurrectionem mortuorum" the music plunges suddenly into a slow series of some of the most sublime and mysterious modulations ever written, until it breaks out as suddenly into a vivace e allegro of broad but terse design, which comes to its climax very rapidly and ends as abruptly as possible, the last chord being carefully written as a short note without a pause. This gives the utmost possible effect of finality to the whole Credo, and contrasts admirably with the coldly formal instrumental end of the Resurrexit three movements further back. Now, such subtleties seem as if they must be unconscious on the part of the composer; yet here Bach is so far aware of his reasons that his vivace e allegro is an arrangement of the second chorus of a church cantata, Gott man lobet dich in der Stille; and in the cantata the chorus has introductory and final symphonies and a middle section with a da capo! 5. The Requiem. - The Missa pro defunctis or Requiem Mass has a far less definite musical history than the ordinary Mass; and such special musical forms as it has produced have little in common with each other. The text of the Dies Irae so imperatively demands either a very dramatic elaboration or none at all, that even in the 16th century it could not possibly be set to continuous music on the lines of the Gloria and Credo. Fortunately, however, the Gregorian canto fermo associated with it is of exceptional beauty and symmetry; and the great 16th century masters either, like Palestrina, left it to be sung as plain-chant, or obviated all occasion for dramatic expression by setting it in versicles (like their settings of the Magnificat and other canticles) for two groups of voices alternatively, or for the choir in alternation with the plain chant of the priests.
With modern orchestral conditions the text seems positively to demand an unecclesiastical, not to say sensational, style, and probably the only instrumental Requiem Masses which can be said to be great church music are the sublime unfinished work of Mozart (the antecedents of which would be a very interesting subject) and the two beautiful works by Cherubini. These latter, however, tend to be funereal rather than uplifting. The only other artistic solution of the problem is to follow Berlioz, Verdi and Dvorak in the complete renunciation of all ecclesiastical style.
Brahms's Deutsches requiem has nothing to do with the Mass for the dead, being simply a large choral work on a text compiled from the Bible by the composer. (D. F. T.)
From Old English masse, messe, Old English mæsse. Late Latin missa, from Latin mittere, missum, to send, dismiss: compare French messe. In the ancient churches, the public services at which the catechumens were permitted to be present were called missa catechumenorum, ending with the reading of the Gospel. Then they were dismissed with these words: "Ite, missa est", the congregation is dismissed. After that the sacrifice proper began. At its close the same words were said to those who remained. So the word gave the name of Mass to the sacrifice in the Catholic Church. Compare Christmas, Lammas, Mess a dish, Missal
Mass (plural Masses)
Mass is the amount of matter in a body. An object has the same mass where ever it is. You can think of mass as the amount of stuff that is in an object, in a way that does not depend on how much space it takes up.
The SI unit of mass is the kilogram, written as kg. There are also many other units of weight (mass with gravity pulling on it): tonnes (t), pounds (lb.), and ounces (oz.). Other units of mass used in science and engineering include slugs, atomic mass units, Planck masses, Solar masses, and eV/c2. The last unit is based on the electron volt (eV), which is usually used as a unit of energy.
Physicists do not know what gives an object mass. The Standard Model of particle physics thinks that mass is caused by the Higgs field. Physicists are trying to find the Higgs field now using the Large Hadron Collider at CERN.
In physics, Special Relativity shows that the mass of an object becomes bigger when the object moves very fast. As the speed gets close to the speed of light the mass becomes very big. The whole energy (E) of the body is
where m - mass of body, v - speed of body, c - speed of light.
Some things that do not have mass on their own act like they have mass because of their movement. This is true for light - a light photon has no mass, but its energy can act as mass when it hits something. Technically, light does not have mass, but it does have momentum. From its momentum you can figure out what mass would produce the same momentum at the speed of light, but this is an "as if" kind of mass, not real mass.