This article is about the scientific concept. For the main liturgical service in some Christian churches, seeMass (liturgy). For other uses, seeMass (disambiguation).
Mass is anintrinsic property of abody. It was traditionally believed to be related to thequantity ofmatter in a body, until the discovery of theatom andparticle physics. It was found that different atoms and differentelementary particles, theoretically with the same amount of matter, have nonetheless different masses. Mass in modern physics has multipledefinitions which are conceptually distinct, but physically equivalent. Mass can be experimentally defined as ameasure of the body'sinertia, meaning the resistance toacceleration (change ofvelocity) when anet force is applied.[1] The object's mass also determines thestrength of itsgravitational attraction to other bodies.
TheSI base unit of mass is thekilogram (kg). Inphysics, mass isnot the same asweight, even though mass is often determined by measuring the object's weight using aspring scale, rather thanbalance 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.
There are several distinct phenomena that can be used to measure mass. Although some theorists have speculated that some of these phenomena could be independent of each other,[3] current experiments have found no difference in results regardless of how it is measured:
Inertial mass measures an object's resistance to being accelerated by a force (represented by the relationshipF =ma).
Active gravitational mass determines the strength of the gravitational field generated by an object.
Passive gravitational mass measures the gravitational force exerted on an object in a known gravitational field.
The mass of an object determines its acceleration in the presence of an applied force. The inertia and the inertial mass describe this property of physical bodies at the qualitative and quantitative level respectively. According toNewton's second law of motion, if a body of fixed massm is subjected to a single forceF, its accelerationa is given byF/m. A body's mass also determines the degree to which it generates and is affected by agravitational field. If a first body of massmA is placed at a distancer (center of mass to center of mass) from a second body of massmB, each body is subject to an attractive forceFg =GmAmB/r2, whereG =6.67×10−11 N⋅kg−2⋅m2 is the "universalgravitational constant". This is sometimes referred to as gravitational mass.[note 1] Repeated experiments since the 17th century have demonstrated that inertial and gravitational mass are identical; since 1915, this observation has been incorporateda priori in theequivalence principle ofgeneral relativity.
TheInternational System of Units (SI) unit of mass is thekilogram (kg). The kilogram is 1000 grams (g), and was first defined in 1795 as the mass of one cubic decimetre of water at themelting point of ice. However, because precise measurement of a cubic decimetre of water at the specified temperature and pressure was difficult, in 1889 the kilogram was redefined as the mass of a metal object, and thus became independent of the metre and the properties of water, this being a copper prototype of thegrave in 1793, the platinumKilogramme des Archives in 1799, and the platinum–iridiumInternational Prototype of the Kilogram (IPK) in 1889.
Thegrain was the earliestunit of mass and is the smallest unit in theapothecary,avoirdupois, Tower, andtroy systems. The early unit was a grain of wheat or barleycorn used to weigh the precious metals silver and gold. Larger units preserved in stone standards were developed that were used as both units of mass and of monetary currency. Thepound was derived from themina (unit) used by ancient civilizations. A smaller unit was theshekel, and a larger unit was thetalent. The magnitude of these units varied from place to place. The Babylonians and Sumerians had a system in which there were 60 shekels in a mina and 60 minas in a talent. The Roman talent consisted of 100 libra (pound) which were smaller in magnitude than the mina. The troy pound (~373.2 g) used in England and the United States for monetary purposes, like the Roman pound, was divided into 12 ounces, but the Roman uncia (ounce) was smaller. The carat is a unit for measuring gemstones that had its origin in the carob seed, which later was standardized at 1/144 ounce and then 0.2 gram.
Goods of commerce were originally traded by number or volume. When weighing of goods began, units of mass based on a volume of grain or water were developed. The diverse magnitudes of units having the same name, which still appear today in our dry and liquid measures, could have arisen from the various commodities traded. The larger avoirdupois pound for goods of commerce might have been based on volume of water which has a higherbulk density than grain.
The stone, quarter, hundredweight, and ton were larger units of mass used in Britain. Today only the stone continues in customary use for measuring personal body weight. The present stone is 14 pounds (~6.35 kg), but an earlier unit appears to have been 16 pounds (~7.25 kg). The other units were multiples of 2, 8, and 160 times the stone, or 28, 112, and 2240 pounds (~12.7 kg, 50.8 kg, 1016 kg), respectively. The hundredweight was approximately equal to two talents. The "long ton" is equal to 2240 pounds (1016.047 kg), the "short ton" is equal to 2000 pounds (907.18474 kg), and the tonne (or metric ton) (t) is equal to 1000 kg (or 1 megagram).
Definitions
Inphysical science, one may distinguish conceptually between at least seven different aspects ofmass, or seven physical notions that involve the concept ofmass.[6] Every experiment to date has shown these seven values to beproportional, 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 oroperationally defined:
Inertial mass is a measure of an object's resistance to acceleration when aforce 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 greaterinertia.
Active gravitational mass[note 4] is a measure of the strength of an object'sgravitational flux (gravitational flux is equal to thesurface integral of gravitational field over an enclosing surface). Gravitational field can be measured by allowing a small "test object" to fall freely and measuring itsfree-fall acceleration. For example, an object in free-fall near theMoon is subject to a smaller gravitational field, and hence accelerates more slowly, than the same object would if it were in free-fall near the Earth. The gravitational field near the Moon is weaker because the Moon has less active gravitational mass.
Passive gravitational mass is a measure of the strength of an object's interaction with agravitational 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.
According torelativity, mass is nothing else than therest energy of a system of particles, meaning the energy of that system in areference frame where it has zeromomentum. Mass can be converted into other forms of energy according to the principle ofmass–energy equivalence. This equivalence is exemplified in a large number of physical processes includingpair production,beta decay andnuclear fusion. Pair production and nuclear fusion are processes in which measurable amounts of mass are converted tokinetic energy or vice versa.
Curvature ofspacetime is a relativistic manifestation of the existence of mass. Suchcurvature 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 preciseatomic 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 calledgravitational time dilation. Other forms of curvature have been measured using theGravity Probe B satellite.
Quantum mass manifests itself as a difference between an object's quantumfrequency and itswave number. The quantum mass of a particle is proportional to the inverseCompton wavelength and can be determined through various forms ofspectroscopy. In relativistic quantum mechanics, mass is one of the irreducible representation labels of the Poincaré group.
Mass and weight of a given object onEarth andMars. Weight varies due to different amount ofgravitational acceleration whereas mass stays the same.
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 theEarth's gravitational field at different places, thedistinction 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 inkilograms) refers to an intrinsic property of an object, whereas "weight" (measured innewtons) measures an object's resistance to deviating from its current course offree fall, which can be influenced by the nearby gravitational field. No matter how strong the gravitational field, objects in free fall areweightless, though they still have mass.[7]
The force known as "weight" is proportional to mass andacceleration 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 theEarth or theMoon. 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 weightW of an object is related to its massm byW =mg, whereg =9.80665 m/s2 is the acceleration due toEarth'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 theproper 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 weightW of an object is related to its massm by the equationW = –ma, wherea 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).
Although inertial mass, passive gravitational mass and active gravitational mass are conceptually distinct, no experiment has ever unambiguously demonstrated any difference between them. Inclassical 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.
The particular equivalence often referred to as the "Galilean equivalence principle" or the "weak equivalence principle" has the most important consequence for freely falling objects. Suppose an object has inertial and gravitational massesm andM, respectively. If the only force acting on the object comes from a gravitational fieldg, the force on the object is:
Given this force, the acceleration of the object can be determined by Newton's second law:
Putting these together, the gravitational acceleration is given by:
This says that the ratio of gravitational to inertial mass of any object is equal to some constantKif 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 constantK can be taken as 1 by defining our units appropriately.
The first experiments demonstrating the universality of free-fall were—according to scientific 'folklore'—conducted byGalileo obtained by dropping objects from theLeaning Tower of Pisa. This is most likely apocryphal: he is more likely to have performed his experiments with balls rolling down nearly frictionlessinclined planes to slow the motion and increase the timing accuracy. Increasingly precise experiments have been performed, such as those performed byLoránd Eötvös,[8] using thetorsion balance pendulum, in 1889. As of 2008[update], no deviation from universality, and thus from Galilean equivalence, has ever been found, at least to the precision 10−6. More precise experimental efforts are still being carried out.[9]
Astronaut David Scott performs the feather and hammer drop experiment on the Moon.
The universality of free-fall only applies to systems in which gravity is the only acting force. All other forces, especiallyfriction andair 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 infree-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 avacuum, 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, asDavid Scott did on the surface of theMoon duringApollo 15.
A stronger version of the equivalence principle, known as theEinstein equivalence principle or thestrong equivalence principle, lies at the heart of thegeneral theory of relativity. Einstein's equivalence principle states that within sufficiently small regions of spacetime, it is impossible to distinguish between a uniform acceleration and a uniform gravitational field. Thus, the theory postulates that the force acting on a massive object caused by a gravitational field is a result of the object's tendency to move in a straight line (in other words its inertia) and should therefore be a function of its inertial mass and the strength of the gravitational field.
Intheoretical physics, amass generation mechanism is a theory which attempts to explain the origin of mass from the most fundamental laws ofphysics. 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 thegravitational interaction but a theory of the latter has not been yet reconciled with the currently popular model ofparticle physics, known as theStandard Model.
The concept ofamount is very old andpredates recorded history. The concept of "weight" would incorporate "amount" and acquire a double meaning that was not clearly recognized as such.[10]
What we now know as mass was until the time of Newton called “weight.” ... A goldsmith believed that an ounce of gold was a quantity of gold. ... But the ancients believed that a beam balance also measured “heaviness” which they recognized through their muscular senses. ... Mass and its associated downward force were believed to be the same thing.
— K. M. Browne, The pre-Newtonian meaning of the word “weight”
Humans, at some early era, realized that the weight of a collection of similar objects wasdirectly proportional to the number of objects in the collection:
whereW is the weight of the collection of similar objects andn is the number of objects in the collection. Proportionality, by definition, implies that two values have a constantratio:
, or equivalently
An early use of this relationship is abalance scale, which balances the force of one object's weight against the force of another object's weight. The two sides of a balance scale are close enough that the objects experience similar gravitational fields. Hence, if they have similar masses then their weights will also be similar. This allows the scale, by comparing weights, to also compare masses.
Consequently, historical weight standards were often defined in terms of amounts. The Romans, for example, used thecarob seed (carat orsiliqua) as a measurement standard. If an object's weight was equivalent to1728 carob seeds, then the object was said to weigh one Roman pound. If, on the other hand, the object's weight was equivalent to144 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:
In 1600 AD,Johannes Kepler sought employment withTycho Brahe, who had some of the most precise astronomical data available. Using Brahe's precise observations of the planet Mars, Kepler spent the next five years developing his own method for characterizing planetary motion. In 1609, Johannes Kepler published his three laws of planetary motion, explaining how the planets orbit the Sun. In Kepler's final planetary model, he described planetary orbits as following elliptical paths with the Sun at a focal point of theellipse. Kepler discovered that thesquare of theorbital period of each planet is directlyproportional to thecube of thesemi-major axis of its orbit, or equivalently, that theratio of these two values is constant for all planets in theSolar System.[note 5]
On 25 August 1609,Galileo Galilei demonstrated his first telescope to a group of Venetian merchants, and in early January 1610, Galileo observed four dim objects near Jupiter, which he mistook for stars. However, after a few days of observation, Galileo realized that these "stars" were in fact orbiting Jupiter. These four objects (later named theGalilean 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.
Galilean free fall
Galileo Galilei (1636)Distance traveled by a freely falling ball is proportional to the square of the elapsed time.
Sometime prior to 1638, Galileo turned his attention to the phenomenon of objects in free fall, attempting to characterize these motions. Galileo was not the first to investigate Earth's gravitational field, nor was he the first to accurately describe its fundamental characteristics. However, Galileo's reliance on scientific experimentation to establish physical principles would have a profound effect on future generations of scientists. It is unclear if these were just hypothetical experiments used to illustrate a concept, or if they were real experiments performed by Galileo,[11] but the results obtained from these experiments were both realistic and compelling. A biography by Galileo's pupilVincenzo Viviani stated that Galileo had droppedballs of the same material, but different masses, from theLeaning Tower of Pisa to demonstrate that their time of descent was independent of their mass.[note 6] 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.[12]
A later experiment was described in Galileo'sTwo 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, polishedgroove. 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 variousangles to slow the acceleration enough so that the elapsed time could be measured. The ball was allowed to roll a known distance down the ramp, and the time taken for the ball to move the known distance was measured. The time was measured using a water clock described as follows:
a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.[13]
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 had shown that objects in free fall under the influence of the Earth's gravitational field have a constant acceleration, and Galileo's contemporary, Johannes Kepler, had shown that the planets follow elliptical paths under the influence of the Sun's gravitational mass. However, Galileo's free fall motions and Kepler's planetary motions remained distinct during Galileo's lifetime.
Mass as distinct from weight
According to K. M. Browne: "Kepler formed a [distinct] concept of mass ('amount of matter' (copia materiae)), but called it 'weight' as did everyone at that time."[10] Finally, in 1686, Newton gave this distinct concept its own name. In the first paragraph ofPrincipia, Newton defined quantity of matter as “density and bulk conjunctly”, and mass as quantity of matter.[14]
The quantity of matter is the measure of the same, arising from its density and bulk conjunctly. ... It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight.
— Isaac Newton, Mathematical principles of natural philosophy, Definition I.
Robert Hooke had published his concept of gravitational forces in 1674, stating that allcelestial 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 its own center.[15] In correspondence withIsaac Newton from 1679 and 1680, Hooke conjectured that gravitational forces might decrease according to the double of the distance between the two bodies.[16] Hooke urged Newton, who was a pioneer in the development ofcalculus, 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.[17] 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 titledDe motu corporum in gyrum (Latin for "On the motion of bodies in an orbit").[18] Halley presented Newton's findings to theRoyal Society of London, with a promise that a fuller presentation would follow. Newton later recorded his ideas in a three-book set, entitledPhilosophiæ Naturalis Principia Mathematica (English: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.[19]: 31
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:
whereg 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, andR 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.[20]
Newton's cannonball
A cannon on top of a very high mountain shoots a cannonball horizontally. If the speed is low, the cannonball quickly falls back to Earth (A, B). Atintermediate speeds, it will revolve around Earth along an elliptical orbit (C, D). Beyond theescape velocity, it will leave the Earth without returning (E).
Newton's cannonball was athought experiment used to bridge the gap between Galileo's gravitational acceleration and Kepler's elliptical orbits. It appeared in Newton's 1728 bookA 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."[19]: 513 Newton further reasons that if an object were "projected in an horizontal direction from the top of a high mountain" with sufficient velocity, "it would reach at last quite beyond the circumference of the Earth, and return to the mountain from which it was projected."[21]
Universal gravitational mass
An apple experiences gravitational fields directed towards every part of the Earth; however, the sum total of these many fields produces a single gravitational field directed towards the Earth's center.
In contrast to earlier theories (e.g.celestial spheres) which stated that the heavens were made of entirely different material, Newton's theory of mass was groundbreaking partly because it introduceduniversal 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,[19]: 397 and inversely proportional to the square of the distance to the body's center.[19]: 221 [note 7]
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, byunit conversion it is a simple matter of abstraction to realize that any traditional mass unit can theoretically be used to measure gravitational mass.
Vertical section drawing of Cavendish's torsion balance instrument including the building in which it was housed. The large balls were hung from a frame so they could be rotated into position next to the small balls by a pulley from outside. Figure 1 of Cavendish's paper.
Measuring gravitational mass in terms of traditional mass units is simple in principle, but extremely difficult in practice. According to Newton's theory, all objects produce gravitational fields and it is theoretically possible to collect an immense number of small objects and form them into an enormous gravitating sphere. However, from a practical standpoint, the gravitational fields of small objects are extremely weak and difficult to measure. Newton's books on universal gravitation were published in the 1680s, but the first successful measurement of the Earth's mass in terms of traditional mass units, theCavendish experiment, did not occur until 1797, over a hundred years later.Henry Cavendish found that the Earth's density was 5.448 ± 0.033 times that of water. As of 2009, the Earth's mass in kilograms is only known to around five digits of accuracy, whereas its gravitational mass is known to over nine significant figures.[clarification needed]
Given two objects A and B, of massesMA andMB, separated by adisplacementRAB, Newton's law of gravitation states that each object exerts a gravitational force on the other, of magnitude
,
whereG is the universalgravitational constant. The above statement may be reformulated in the following way: ifg is the magnitude at a given location in a gravitational field, then the gravitational force on an object with gravitational massM is
.
This is the basis by which masses are determined byweighing. In simplespring scales, for example, the forceF is proportional to the displacement of thespring beneath the weighing pan, as perHooke's law, and the scales arecalibrated to takeg into account, allowing the massM to be read off. Assuming the gravitational field is equivalent on both sides of the balance, abalance measures relative weight, giving the relative gravitation mass of each object.
Inertial mass
Mass was traditionally believed to be a measure of the quantity of matter in a physical body, equal to the "amount of matter" in an object. For example,Barre´ de Saint-Venant argued in 1851 that every object contains a number of "points" (basically, interchangeable elementary particles), and that mass is proportional to the number of points the object contains.[22] (In practice, this "amount of matter" definition is adequate for most of classical mechanics, and sometimes remains in use in basic education, if the priority is to teach the difference between mass from weight.)[23] This traditional "amount of matter" belief was contradicted by the fact that different atoms (and, later, different elementary particles) can have different masses, and was further contradicted by Einstein's theory of relativity (1905), which showed that the measurable mass of an object increases when energy is added to it (for example, by increasing its temperature or forcing it near an object that electrically repels it.) This motivates a search for a different definition of mass that is more accurate than the traditional definition of "the amount of matter in an object".[24]
Massmeter, a device for measuring the inertial mass of an astronaut in weightlessness. The mass is calculated via the oscillation period for a spring with the astronaut attached (Tsiolkovsky State Museum of the History of Cosmonautics).
Inertial mass is the mass of an object measured by its resistance to acceleration. This definition has been championed byErnst Mach[25][26] and has since been developed into the notion ofoperationalism byPercy W. Bridgman.[27][28] The simpleclassical mechanics definition of mass differs slightly from the definition in the theory ofspecial relativity, but the essential meaning is the same.
In classical mechanics, according toNewton's second law, we say that a body has a massm if, at any instant of time, it obeys the equation of motion
whereF is the resultantforce acting on the body anda is theacceleration of the body's centre of mass.[note 8] 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 theinertia 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 ofNewton'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 massesm1 andm2. We isolate the two objects from all other physical influences, so that the only forces present are the force exerted onm1 bym2, which we denoteF12, and the force exerted onm2 bym1, which we denoteF21. Newton's second law states that
wherea1 anda2 are the accelerations ofm1 andm2, 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
and thus
If |a1| is non-zero, the fraction is well-defined, which allows us to measure the inertial mass ofm1. In this case,m2 is our "reference" object, and we can define its massm 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'smomentump to its linearvelocityv:
The primary difficulty with Mach's definition of mass is that it fails to take into account thepotential energy (orbinding energy) needed to bring two masses sufficiently close to one another to perform the measurement of mass.[26] This is most vividly demonstrated by comparing the mass of theproton in the nucleus ofdeuterium, to the mass of the proton in free space (which is greater by about 0.239%—this is due to the binding energy of deuterium). Thus, for example, if the reference weightm2 is taken to be the mass of the neutron in free space, and the relative accelerations for the proton and neutron in deuterium are computed, then the above formula over-estimates the massm1 (by 0.239%) for the proton in deuterium. At best, Mach's formula can only be used to obtain ratios of masses, that is, asm1 / m2 = |a2| / |a1|. An additional difficulty was pointed out byHenri Poincaré, which is that the measurement of instantaneous acceleration is impossible: unlike the measurement of time or distance, there is no way to measure acceleration with a single measurement; one must make multiple measurements (of position, time, etc.) and perform a computation to obtain the acceleration. Poincaré termed this to be an "insurmountable flaw" in the Mach definition of mass.[29]
Typically, the mass of objects is measured in terms of the kilogram, which since 2019 is defined in terms of fundamental constants of nature. The mass of an atom or other particle can be compared more precisely and more conveniently to that of another atom, and thus scientists developed thedalton (also known as the unified atomic mass unit). By definition, 1 Da (onedalton) is exactly one-twelfth of the mass of acarbon-12 atom, and thus, a carbon-12 atom has a mass of exactly 12 Da.
In some frameworks ofspecial relativity, physicists have used different definitions of the term. In these frameworks, two kinds of mass are defined:rest mass (invariant mass),[note 9] andrelativistic mass (which increases with velocity). Rest mass is the Newtonian mass as measured by an observer moving along with the object.Relativistic mass is the total quantity of energy in a body or system divided byc2. The two are related by the following equation:
The invariant mass of systems is the same for observers in all inertial frames, while the relativistic mass depends on the observer'sframe of reference. In order to formulate the equations of physics such that mass values do not change between observers, it is convenient to use rest mass. The rest mass of a body is also related to its energyE and the magnitude of its momentump by therelativistic energy-momentum equation:
So long as the system isclosed with respect to mass and energy, both kinds of mass are conserved in any given frame of reference. The conservation of mass holds even as some types of particles are converted to others. Matter particles (such as atoms) may be converted to non-matter particles (such as photons of light), but this does not affect the total amount of mass or energy. Although things like heat may not be matter, all types of energy still continue to exhibit mass.[note 10][30] Thus, mass and energy do not change into one another in relativity; rather, both are names for the same thing, and neither mass nor energyappear without the other.
Both rest and relativistic mass can be expressed as an energy by applying the well-known relationshipE =mc2, yieldingrest energy and "relativistic energy" (total system energy) respectively:
The "relativistic" mass and energy concepts are related to their "rest" counterparts, but they do not have the same value as their rest counterparts in systems where there is a net momentum. Because the relativistic mass isproportional to the energy, it has gradually fallen into disuse among physicists.[31] There is disagreement over whether the concept remains usefulpedagogically.[32][33][34]
In bound systems, thebinding energy must often be subtracted from the mass of the unbound system, because binding energy commonly leaves the system at the time it is bound. The mass of the system changes in this process merely because the system was not closed during the binding process, so the energy escaped. For example, the binding energy ofatomic nuclei is often lost in the form of gamma rays when the nuclei are formed, leavingnuclides which have less mass than the free particles (nucleons) of which they are composed.
Mass–energy equivalence also holds in macroscopic systems.[35] For example, if one takes exactly one kilogram of ice, and applies heat, the mass of the resulting melt-water will be more than a kilogram: it will include the mass from thethermal energy (latent heat) used to melt the ice; this follows from theconservation of energy.[36] This number is small but not negligible: about 3.7 nanograms. It is given by thelatent heat of melting ice (334 kJ/kg) divided by the speed of light squared (c2 ≈9×1016 m2/s2).
However, it turns out that it is impossible to find an objective general definition for the concept ofinvariant mass in general relativity. At the core of the problem is thenon-linearity of theEinstein field equations, making it impossible to write the gravitational field energy as part of thestress–energy tensor in a way that is invariant for all observers. For a given observer, this can be achieved by thestress–energy–momentum pseudotensor.[37]
where the "mass" parameterm is now simply a constant associated with thequantum described by the wave function ψ.
In theStandard Model ofparticle physics as developed in the 1960s, this term arises from the coupling of the field ψ to an additional field Φ, theHiggs field. In the case of fermions, theHiggs mechanism results in the replacement of the termmψ in the Lagrangian with. This shifts theexplanandum of the value for the mass of each elementary particle to the value of the unknowncoupling constantGψ.
Atachyonic field, or simplytachyon, is aquantum field with animaginary mass.[38] Althoughtachyons (particles that movefaster than light) are a purely hypothetical concept not generally believed to exist,[38][39]fields with imaginary mass have come to play an importantrole in modern physics[40][41][42] and are discussed in popular books on physics.[38][43] Under no circumstances do any excitations ever propagate faster than light in such theories—the presence or absence of a tachyonic mass has no effect whatsoever on the maximum velocity of signals (there is no violation ofcausality).[44] While thefield may have imaginary mass, any physical particles do not; the "imaginary mass" shows that the system becomes unstable, and sheds the instability by undergoing a type ofphase transition calledtachyon condensation (closely related to second order phase transitions) that results insymmetry breaking incurrent models ofparticle physics.
The term "tachyon" was coined byGerald Feinberg in a 1967 paper,[45] but it was soon realized that Feinberg's model in fact did not allow forsuperluminal speeds.[44] Instead, the imaginary mass creates an instability in the configuration:- any configuration in which one or more field excitations are tachyonic will spontaneously decay, and the resulting configuration contains no physical tachyons. This process is known as tachyon condensation. Well known examples include thecondensation of theHiggs boson inparticle physics, andferromagnetism incondensed matter physics.
Although the notion of a tachyonicimaginary mass might seem troubling because there is no classical interpretation of an imaginary mass, the mass is not quantized. Rather, thescalar field is; even for tachyonicquantum fields, thefield operators atspacelike separated points stillcommute (or anticommute), thus preserving causality. Therefore, information still does not propagate faster than light,[45] and solutions grow exponentially, but not superluminally (there is no violation ofcausality).Tachyon condensation drives a physical system that has reached a local limit and might naively be expected to produce physical tachyons, to an alternate stable state where no physical tachyons exist. Once the tachyonic field reaches the minimum of the potential, its quanta are not tachyons any more but rather are ordinary particles with a positive mass-squared.[46]
This is a special case of the general rule, where unstable massive particles are formally described as having acomplex mass, with the real part being their mass in the usual sense, and the imaginary part being thedecay rate innatural units.[46] However, inquantum field theory, a particle (a "one-particle state") is roughly defined as a state which is constant over time; i.e., aneigenvalue of theHamiltonian. Anunstable particle is a state which is only approximately constant over time; If it exists long enough to be measured, it can be formally described as having a complex mass, with the real part of the mass greater than its imaginary part. If both parts are of the same magnitude, this is interpreted as aresonance appearing in a scattering process rather than a particle, as it is considered not to exist long enough to be measured independently of the scattering process. In the case of a tachyon, the real part of the mass is zero, and hence no concept of a particle can be attributed to it.
In aLorentz invariant theory, the same formulas that apply to ordinary slower-than-light particles (sometimes called "bradyons" in discussions of tachyons) must also apply to tachyons. In particular theenergy–momentum relation:
(wherep is the relativisticmomentum of the bradyon andm is itsrest mass) should still apply, along with the formula for the total energy of a particle:
This equation shows that the total energy of a particle (bradyon or tachyon) contains a contribution from its rest mass (the "rest mass–energy") and a contribution from its motion, the kinetic energy.Whenv is larger thanc, the denominator in the equation for the energy is"imaginary", as the value under theradical is negative. Because the totalenergy must bereal, the numerator mustalso be imaginary: i.e. therest massm must be imaginary, as a pure imaginary number divided by another pure imaginary number is a real number.
^When a distinction is necessary, the active and passive gravitational masses may be distinguished.
^The dalton is convenient for expressing the masses of atoms and molecules.
^These are used mainly in the United States except in scientific contexts where SI units are usually used instead.
^The distinction between "active" and "passive" gravitational mass does not exist in the Newtonian view of gravity as found inclassical mechanics, and can safely be ignored for many purposes. In most practical applications, Newtonian gravity is assumed because it is usually sufficiently accurate, and is simpler than General Relativity; for example, NASA uses primarily Newtonian gravity to design space missions, although "accuracies are routinely enhanced by accounting for tiny relativistic effects".www2.jpl.nasa.gov/basics/bsf3-2.php The distinction between "active" and "passive" is very abstract, and applies to post-graduate level applications of General Relativity to certain problems in cosmology, and is otherwise not used. There is, nevertheless, an important conceptual distinction in Newtonian physics between "inertial mass" and "gravitational mass", although these quantities are identical; the conceptual distinction between these two fundamental definitions of mass is maintained for teaching purposes because they involve two distinct methods of measurement. It was long considered anomalous that the two distinct measurements of mass (inertial and gravitational) gave an identical result. The property, observed by Galileo, that objects of different mass fall with the same rate of acceleration (ignoring air resistance), shows that inertial and gravitational mass are the same.
^This constant ratio was later shown to be a direct measure of the Sun's active gravitational mass; it has units of distance cubed per time squared, and is known as thestandard gravitational parameter:
^At the time when Viviani asserts that the experiment took place, Galileo had not yet formulated the final version of his law of free fall. He had, however, formulated an earlier version that predicted that bodiesof the same material falling through the same medium would fall at the same speed. SeeDrake, S. (1978).Galileo at Work. University of Chicago Press. pp. 19–20.ISBN978-0-226-16226-3.
^These two properties are very useful, as they allow spherical collections of objects to be treated exactly like large individual objects.
^In its original form, Newton's second law is valid only for bodies of constant mass.
^It is possible to make a slight distinction between "rest mass" and "invariant mass". For a system of two or more particles, none of the particles are required be at rest with respect to the observer for the system as a whole to be at rest with respect to the observer. To avoid this confusion, some sources will use "rest mass" only for individual particles, and "invariant mass" for systems.
^For example, a nuclear bomb in an idealized super-strong box, sitting on a scale, would in theory show no change in mass when detonated (although the inside of the box would become much hotter). In such a system, the mass of the box would change only if energy were allowed to escape from the box as light or heat. However, in that case, the removed energy would take its associated mass with it. Letting heat or radiation out of such a system is simply a way to remove mass. Thus, mass, like energy, cannot be destroyed, but only moved from one place to another.
References
^Bray, Nancy (28 April 2015)."Science".NASA. Archived fromthe original on 30 May 2023. Retrieved20 March 2023.Mass can be understood as a measurement of inertia, the resistance of an object to be set in motion or stopped from motion.
^abOri Belkind, "Physical Systems: Conceptual Pathways between Flat Space-time and Matter" (2012) Springer (Chapter 5.3)
^P.W. Bridgman,Einstein's Theories and the Operational Point of View, in: P.A. Schilpp, ed.,Albert Einstein: Philosopher-Scientist, Open Court, La Salle, Ill., Cambridge University Press, 1982, Vol. 2, pp. 335–354.
^abcLisa Randall,Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions, p.286: "People initially thought of tachyons as particles travelling faster than the speed of light...But we now know that a tachyon indicates an instability in a theory that contains it. Regrettably for science fiction fans, tachyons are not real physical particles that appear in nature."
^Tipler, Paul A.; Llewellyn, Ralph A. (2008).Modern Physics (5th ed.). New York: W.H. Freeman & Co. p. 54.ISBN978-0-7167-7550-8.... so existence of particles v > c ... Called tachyons ... would present relativity with serious ... problems of infinite creation energies and causality paradoxes.
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