Editing Mass (section)

== Newtonian mass ==
{| class="wikitable floatright" 
|-
! colspan=2| Earth's Moon !! rowspan=2| Mass of Earth
|-
! [[Semi-major axis]] !! [[Sidereal orbital period]]
|-
|0.002 569 [[Astronomical unit|AU]]||0.074 802 [[sidereal year]]
|rowspan=3|<math qid=Q11376>1.2\pi^2\cdot10^{-5}\frac{\text{AU}^3}{\text{y}^2}=3.986\cdot10^{14}\frac{\text{m}^3}{\text{s}^2}</math>
|-
! Earth's gravity !! Earth's radius
|-
|9.806 65&nbsp;m/s<sup>2</sup>||6 375&nbsp;km
|}

[[File:GodfreyKneller-IsaacNewton-1689.jpg|Isaac Newton, 1689|upright|thumb]]

[[Robert Hooke]] had published his concept of gravitational forces in 1674, stating that all [[celestial bodies]] have an attraction or gravitating power towards their own centers, and also attract all the other celestial bodies that are within the sphere of their activity. He further stated that gravitational attraction increases by how much nearer the body wrought upon is to its own center.<ref>{{cite book |last=Hooke |first=R. |date=1674 |title=An attempt to prove the motion of the earth from observations |url=https://books.google.com/books?id=JgtPAAAAcAAJ&pg=PA1 |publisher=[[Royal Society]]}}</ref> In correspondence with [[Isaac Newton]] from 1679 and 1680, Hooke conjectured that gravitational forces might decrease according to the double of the distance between the two bodies.<ref>{{cite book |editor-last=Turnbull |editor-first=H.W. |date=1960 |title=Correspondence of Isaac Newton, Volume 2 (1676–1687) |page=297 |publisher=Cambridge University Press}}</ref> 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.<ref>{{cite book |title=Principia |url=https://massless.info/images/Isaac_Newton_Principia_English.pdf |pages=16}}</ref> After being encouraged by Halley, Newton decided to develop his ideas about gravity and publish all of his findings.  In November 1684, Isaac Newton sent a document to Edmund Halley, now lost but presumed to have been titled ''[[De motu corporum in gyrum]]'' (Latin for "On the motion of bodies in an orbit").<ref>
{{cite book |editor-last=Whiteside |editor-first=D.T. |date=2008 |title=The Mathematical Papers of Isaac Newton, Volume VI (1684–1691) |url=https://books.google.com/books?id=lIZ0v23iqRgC|publisher=Cambridge University Press |isbn=978-0-521-04585-8}}</ref> 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]]'' (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.<ref name="principia">{{cite book |author1=Sir Isaac Newton |author2=N.W. Chittenden |title=Newton's Principia: The mathematical principles of natural philosophy |url=https://archive.org/details/newtonsprincipi00chitgoog |page=[https://archive.org/details/newtonsprincipi00chitgoog/page/n43 31]|year=1848 |publisher=D. Adee|isbn=9780520009295 }}</ref>{{rp|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:
: <math>\mathbf{g}=-\mu\frac{\hat{\mathbf{R}}}{|\mathbf{R}|^2}</math>
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.<ref>{{cite web |last=Cuk |first=M. |date=January 2003 |title=Curious About Astronomy: How do you measure a planet's mass? |url=http://curious.astro.cornell.edu/question.php?number=452 |work=Ask an Astronomer |access-date=2011-03-12 |url-status=dead |archive-url=https://web.archive.org/web/20030320113723/http://curious.astro.cornell.edu/question.php?number=452 |archive-date=20 March 2003}}</ref>

=== Newton's cannonball ===
[[File:Newton Cannon.svg|thumb|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,&nbsp;B). At [[orbital speed|intermediate speeds]], it will revolve around Earth along an elliptical orbit (C,&nbsp;D). Beyond the [[escape velocity]], it will leave the Earth without returning (E).]]
{{main|Newton's cannonball}}

Newton's cannonball was a [[thought experiment]] used to bridge the gap between Galileo's gravitational acceleration and Kepler's elliptical orbits.  It appeared in Newton's 1728 book ''A Treatise of the System of the World''.  According to Galileo's concept of gravitation, a dropped stone falls with constant acceleration down towards the Earth.  However, Newton explains that when a stone is thrown horizontally (meaning sideways or perpendicular to Earth's gravity) it follows a curved path.  "For a stone projected is by the pressure of its own weight forced out of the rectilinear path, which by the projection alone it should have pursued, and made to describe a curve line in the air; and through that crooked way is at last brought down to the ground.  And the greater the velocity is with which it is projected, the farther it goes before it falls to the Earth."<ref name=principia />{{rp|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."<ref>{{cite book |last1=Newton |first1=Isaac |author-link=Isaac Newton |title=A Treatise of the System of the World |date=1728 |publisher=F. Fayram |location=London |page=6 |url=https://books.google.com/books?id=rEYUAAAAQAAJ&q=ball&pg=PR1 |access-date=4 May 2022}}</ref>

=== Universal gravitational mass ===
[[File:Universal gravitational mass.jpg|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.|left|thumb]]
In contrast to earlier theories (e.g. [[celestial spheres]]) which stated that the heavens were made of entirely different material, Newton's theory of mass was groundbreaking partly because it introduced [[Newton's law of universal gravitation|universal gravitational mass]]: every object has gravitational mass, and therefore, every object generates a gravitational field.  Newton further assumed that the strength of each object's gravitational field would decrease according to the square of the distance to that object. If a large collection of small objects were formed into a giant spherical body such as the Earth or Sun, Newton calculated the collection would create a gravitational field proportional to the total mass of the body,<ref name="principia"/>{{rp|397}} and inversely proportional to the square of the distance to the body's center.<ref name="principia"/>{{rp|221}}<ref group="note">These two properties are very useful, as they allow spherical collections of objects to be treated exactly like large individual objects.</ref>

For example, according to Newton's theory of universal gravitation, each carob seed produces a gravitational field.  Therefore, if one were to gather an immense number of carob seeds and form them into an enormous sphere, then the gravitational field of the sphere would be proportional to the number of carob seeds in the sphere.  Hence, it should be theoretically possible to determine the exact number of carob seeds that would be required to produce a gravitational field similar to that of the Earth or Sun.  In fact, by [[unit conversion]] it is a simple matter of abstraction to realize that any traditional mass unit can theoretically be used to measure gravitational mass.

[[File:Cavendish Experiment.png|thumb|right|Vertical section drawing of Cavendish's torsion balance instrument including the building in which it was housed. The large balls were hung from a frame so they could be rotated into position next to the small balls by a pulley from outside. Figure 1 of Cavendish's paper.]]

Measuring gravitational mass in terms of traditional mass units is simple in principle, but extremely difficult in practice.  According to Newton's theory, all objects produce gravitational fields and it is theoretically possible to collect an immense number of small objects and form them into an enormous gravitating sphere.  However, from a practical standpoint, the gravitational fields of small objects are extremely weak and difficult to measure. Newton's books on universal gravitation were published in the 1680s, but the first successful measurement of the Earth's mass in terms of traditional mass units, the [[Cavendish experiment]], did not occur until 1797, over a hundred years later. [[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.{{clarify|reason="mass in kilograms" does not specify an experimental method. What's the distinction between the methods?|date=January 2014}}

Given two objects A and B, of masses ''M''<sub>A</sub> and ''M''<sub>B</sub>, separated by a [[Displacement (vector)|displacement]] '''R'''<sub>AB</sub>, Newton's law of gravitation states that each object exerts a gravitational force on the other, of magnitude
: <math>\mathbf{F}_{\text{AB}}=-GM_{\text{A}}M_{\text{B}}\frac{\hat{\mathbf{R}}_{\text{AB}}}{|\mathbf{R}_{\text{AB}}|^2}\ </math>,
where ''G'' is the universal [[gravitational constant]]. The above statement may be reformulated in the following way: if ''g'' is the magnitude at a given location in a gravitational field, then the gravitational force on an object with gravitational mass ''M'' is
: <math>F=Mg</math>.
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 (device)|spring]] beneath the weighing pan, as per [[Hooke's law]], and the scales are [[calibration|calibrated]] to take ''g'' into account, allowing the mass ''M'' to be read off. Assuming the gravitational field is equivalent on both sides of the balance, a [[Beam balance|balance]] 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, [[Adhémar Jean Claude Barré de Saint-Venant|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.<ref>{{cite journal |last1=Coelho |first1=Ricardo Lopes |title=On the Concept of Force: How Understanding its History can Improve Physics Teaching |journal=Science & Education |date=January 2010 |volume=19 |issue=1 |pages=91–113 |doi=10.1007/s11191-008-9183-1|bibcode=2010Sc&Ed..19...91C |s2cid=195229870 }}</ref> (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.)<ref>{{cite web |last1=Gibbs |first1=Yvonne |title=Teachers Learn the Difference Between Mass and Weight Even in Space |url=https://www.nasa.gov/centers/armstrong/features/teachers-learn-the-difference-between-mass-and-weight-even-in-space |website=NASA |access-date=20 March 2023 |date=31 March 2017 |archive-date=20 March 2023 |archive-url=https://web.archive.org/web/20230320070434/https://www.nasa.gov/centers/armstrong/features/teachers-learn-the-difference-between-mass-and-weight-even-in-space/ |url-status=dead }}</ref> 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".<ref>{{cite journal |last1=Hecht |first1=Eugene |title=There Is No Really Good Definition of Mass |journal=The Physics Teacher |date=January 2006 |volume=44 |issue=1 |pages=40–45 |doi=10.1119/1.2150758|bibcode=2006PhTea..44...40H }}</ref>

[[File:Massmeter.jpg|thumb|Massmeter, a device for measuring the inertial mass of an astronaut in weightlessness. The mass is calculated via the oscillation period for a spring with the astronaut attached ([[Tsiolkovsky State Museum of the History of Cosmonautics]]).]]
''Inertial mass'' is the mass of an object measured by its resistance to acceleration. This definition has been championed by [[Ernst Mach]]<ref>Ernst Mach, "Science of Mechanics" (1919)</ref><ref name=belkind>Ori Belkind, "Physical Systems: Conceptual Pathways between Flat Space-time and Matter" (2012) Springer (''Chapter 5.3'')</ref> and has since been developed into the notion of [[Operationalization|operationalism]] by [[Percy W. Bridgman]].<ref>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.</ref><ref>{{cite journal | last1 = Gillies | first1 = D.A. | year = 1972 | title = PDF | url = https://www.ucl.ac.uk/sts/staff/gillies/documents/1972a_Operationalism.pdf | journal = Synthese | volume = 25 | pages = 1–24 | doi = 10.1007/BF00484997 | s2cid = 239369276 | access-date = 10 April 2016 | archive-date = 26 April 2016 | archive-url = https://web.archive.org/web/20160426201827/https://www.ucl.ac.uk/sts/staff/gillies/documents/1972a_Operationalism.pdf | url-status = dead }}</ref> The simple [[classical mechanics]] definition of mass differs slightly from the definition in the theory of [[special relativity]], but the essential meaning is the same.

In classical mechanics, according to [[Newton's second law]], we say that a body has a mass ''m'' if, at any instant of time, it obeys the equation of motion
: <math>\mathbf{F}=m \mathbf{a},</math>
where '''F''' is the resultant [[force]] acting on the body and '''a''' is the [[acceleration]] of the body's centre of mass.<ref group="note">In its original form, Newton's second law is valid only for bodies of constant mass.</ref> For the moment, we will put aside the question of what "force acting on the body" actually means.

This equation illustrates how mass relates to the [[inertia]] of a body. Consider two objects with different masses. If we apply an identical force to each, the object with a bigger mass will experience a smaller acceleration, and the object with a smaller mass will experience a bigger acceleration. We might say that the larger mass exerts a greater "resistance" to changing its state of motion in response to the force.

However, this notion of applying "identical" forces to different objects brings us back to the fact that we have not really defined what a force is. We can sidestep this difficulty with the help of [[Newton's third law]], which states that if one object exerts a force on a second object, it will experience an equal and opposite force. To be precise, suppose we have two objects of constant inertial masses ''m''<sub>1</sub> and ''m''<sub>2</sub>. We isolate the two objects from all other physical influences, so that the only forces present are the force exerted on ''m''<sub>1</sub> by ''m''<sub>2</sub>, which we denote '''F'''<sub>12</sub>, and the force exerted on ''m''<sub>2</sub> by ''m''<sub>1</sub>, which we denote '''F'''<sub>21</sub>. Newton's second law states that
: <math>
\begin{align}
\mathbf{F_{12}} & =m_1\mathbf{a}_1,\\
\mathbf{F_{21}} & =m_2\mathbf{a}_2,
\end{align}</math>
where '''a'''<sub>1</sub> and '''a'''<sub>2</sub> are the accelerations of ''m''<sub>1</sub> and ''m''<sub>2</sub>, 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
: <math qid=Q3235565>\mathbf{F}_{12}=-\mathbf{F}_{21};</math>
and thus
: <math>m_1=m_2\frac{|\mathbf{a}_2|}{|\mathbf{a}_1|}\!.</math>

If {{abs|'''a'''<sub>1</sub>}} is non-zero, the fraction is well-defined, which allows us to measure the inertial mass of ''m''<sub>1</sub>. In this case, ''m''<sub>2</sub> is our "reference" object, and we can define its mass ''m'' as (say) 1&nbsp;kilogram. Then we can measure the mass of any other object in the universe by colliding it with the reference object and measuring the accelerations.

Additionally, mass relates a body's [[momentum]] '''p''' to its linear [[velocity]] '''v''':
: <math qid=Q41273>\mathbf{p}=m\mathbf{v}</math>,
and the body's [[kinetic energy]] ''K'' to its velocity:
: <math qid=Q46276>K=\dfrac{1}{2}m|\mathbf{v}|^2</math>.

The primary difficulty with Mach's definition of mass is that it fails to take into account the [[potential energy]] (or [[binding energy]]) needed to bring two masses sufficiently close to one another to perform the measurement of mass.<ref name=belkind/> This is most vividly demonstrated by comparing the mass of the [[proton]] in the nucleus of [[deuterium]], to the mass of the proton in free space (which is greater by about 0.239%—this is due to the binding energy of deuterium). Thus, for example, if the reference weight ''m''<sub>2</sub> is taken to be the mass of the neutron in free space, and the relative accelerations for the proton and neutron in deuterium are computed, then the above formula over-estimates the mass ''m''<sub>1</sub> (by 0.239%) for the proton in deuterium.  At best, Mach's formula can only be used to obtain ratios of masses, that is, as ''m''<sub>1</sub>&nbsp;/&nbsp;''m''<sub>2</sub> = {{abs|'''a'''<sub>2</sub>}}&nbsp;/&nbsp;{{abs|'''a'''<sub>1</sub>}}.  An additional difficulty was pointed out by [[Henri Poincaré]], which is that the measurement of instantaneous acceleration is impossible: unlike the measurement of time or distance, there is no way to measure acceleration with a single measurement; one must make multiple measurements (of position, time, etc.) and perform a computation to obtain the acceleration.  Poincaré termed this to be an "insurmountable flaw" in the Mach definition of mass.<ref>Henri Poincaré. "[https://brocku.ca/MeadProject/Poincare/Poincare_1905_07.html Classical Mechanics]". Chapter 6 in Science and Hypothesis. London: Walter Scott Publishing (1905): 89-110.</ref>