Editing Newton's laws of motion (section)

== Laws ==

===First law{{anchor|Newton's_first_law}}===
[[File:Skylab and Earth Limb - GPN-2000-001055.jpg|alt=see caption|thumb|Artificial satellites move along curved [[orbit]]s, rather than in straight lines, because of the Earth's [[gravity]].]]
Translated from Latin, Newton's first law reads,
:''Every object perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.''{{refn|group=note|Per Cohen and Whitman.<ref name="Cohen&Whitman" /> For other phrasings, see Eddington<ref>{{Cite book |title=The Nature of the Physical World |first=Arthur |last=Eddington |pages=123–125 |date=1929 |publisher=Macmillan |location=New York |author-link=Arthur Eddington}}</ref> and Frautschi et al.<ref name=":0">{{Cite book|last1=Frautschi|first1=Steven C.|title=The Mechanical Universe: Mechanics and Heat|title-link=The Mechanical Universe|last2=Olenick|first2=Richard P.|last3=Apostol|first3=Tom M.|last4=Goodstein|first4=David L.|date=2007|publisher=Cambridge University Press|isbn=978-0-521-71590-4|edition=Advanced|location=Cambridge [Cambridgeshire]|oclc=227002144|author-link=Steven Frautschi|author-link3=Tom M. Apostol|author-link4=David L. Goodstein}}</ref>{{Rp|page=114}} Andrew Motte's 1729 translation rendered Newton's "nisi quatenus" as ''unless'' instead of ''except insofar,'' which Hoek argues was erroneous.<ref>{{Cite journal |journal=Philosophy of Science |date=2023 |title=Forced Changes Only: A New Take on Inertia |arxiv=2112.02339 |doi=10.1017/psa.2021.38 |pages=60–73 |volume=90 |issue=1 |first=D. |last=Hoek}}</ref><ref>{{Cite journal |journal=Scientific American |date=5 September 2023 |title=Mistranslation of Newton's First Law Discovered after Nearly Nearly 300 Years |pages= |volume= |issue= |first=Stephanie |last=Pappas |url=https://www.scientificamerican.com/article/mistranslation-of-newtons-first-law-discovered-after-nearly-300-years1/}}</ref>}}
Newton's first law expresses the principle of [[inertia]]: the natural behavior of a body is to move in a straight line at constant speed. A body's motion preserves the status quo, but external forces can perturb this.

The modern understanding of Newton's first law is that no [[inertial observer]] is privileged over any other. The concept of an inertial observer makes quantitative the everyday idea of feeling no effects of motion. For example, a person standing on the ground watching a train go past is an inertial observer. If the observer on the ground sees the train moving smoothly in a straight line at a constant speed, then a passenger sitting on the train will also be an inertial observer: the train passenger ''feels'' no motion. The principle expressed by Newton's first law is that there is no way to say which inertial observer is "really" moving and which is "really" standing still. One observer's state of rest is another observer's state of uniform motion in a straight line, and no experiment can deem either point of view to be correct or incorrect. There is no absolute standard of rest.<ref>{{cite book|last=Resnick |first=Robert |author-link=Robert Resnick |title=Introduction to Special Relativity |publisher=Wiley |year=1968 |pages=8–16 |oclc=1120819093}}</ref><ref name=":0"/>{{rp|62–63}}<ref name=":2" />{{rp|7–9}} Newton himself believed that [[absolute space and time]] existed, but that the only measures of space or time accessible to experiment are relative.<ref>{{Cite journal|last=Brading|first=Katherine|author-link=Katherine Brading|date=August 2019|title=A note on rods and clocks in Newton's Principia|url=https://linkinghub.elsevier.com/retrieve/pii/S135521981730120X|journal=[[Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics]]|language=en|volume=67|pages=160–166|doi=10.1016/j.shpsb.2017.07.004|bibcode=2019SHPMP..67..160B |s2cid=125131430 }}</ref>

===Second law{{anchor|Newton's_second_law}}===
:''The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed.''<ref name=":0" />{{Rp|page=114}}
By "motion", Newton meant the quantity now called [[momentum]], which depends upon the amount of matter contained in a body, the speed at which that body is moving, and the direction in which it is moving.<ref>{{Cite book |last=Feather |first=Norman |title=An Introduction to the Physics of Mass, Length, and Time |publisher=University Press |year=1959 |location=United Kingdom |pages=126–128}}</ref> In modern notation, the momentum of a body is the product of its mass and its velocity:
<math display="block">\mathbf{p} = m\mathbf{v} \, ,</math>
where all three quantities can change over time.
In common cases the mass <math>m</math> does not change with time and the derivative acts only upon the velocity. Then force equals the product of the mass and the time derivative of the velocity, which is the acceleration:<ref>{{cite book|last1=Resnick |first1=Robert |last2=Halliday |first2=David |year=1966 |title=Physics |chapter=Section 5-4: Mass; Newton's Second Law |publisher=John Wiley & Sons |lccn=66-11527}}</ref>
<math display="block">\mathbf{F} = m \frac{d\mathbf{v}}{dt} = m\mathbf{a} \, .</math>
As the acceleration is the second derivative of position with respect to time, this can also be written
<math display="block">\mathbf{F} = m\frac{d^2\mathbf{s}}{dt^2} .</math>

Newton's second law, in modern form, states that the time derivative of the momentum is the force:<ref name="Kleppner"/>{{rp|loc=4.1}}
<math display="block">\mathbf{F} = \frac{d\mathbf{p}}{dt} \, .</math>
When applied to [[variable mass system|systems of variable mass]], the equation above is only valid only for a fixed set of particles. Applying the derivative as in 
<math display="block">\mathbf{F} = m \frac{\mathrm{d} \mathbf{v}} {\mathrm{d}t} + \mathbf{v}\frac{\mathrm{d} m} {\mathrm{d}t} \ \ \mathrm{(incorrect)}</math>
can lead to incorrect results.<ref name=Plastino-1992>{{Cite journal |last1=Plastino |first1=A.R. |last2=Muzzio |first2=J.C. |last3=Etkina |year=1992 |title=On the use and abuse of Newton's second law for variable mass problems |journal=[[Celestial Mechanics and Dynamical Astronomy]] |language=en |volume=53 |issue=3 |pages=227–232 |doi=10.1007/BF00052611 |bibcode=1992CeMDA..53..227P |issn=0923-2958 }}</ref> For example, the momentum of a water jet system must include the momentum of the ejected water:<ref>{{cite book|last1=Arnold |first1=Sommerfeld |year=1952 |title=Mechanics|publisher=Academic Press |isbn=978-0-12-654668-2}}</ref>
<math display="block">\mathbf{F}_{\mathrm{ext}}  = {\mathrm{d} \mathbf{p} \over \mathrm{d}t} - \mathbf{v}_{\mathrm{eject}} \frac{\mathrm{d} m}{\mathrm{d}t}.</math>

[[File:Free body1.3.svg|right|thumb|A [[free body diagram]] for a block on an inclined plane, illustrating the [[normal force]] perpendicular to the plane (''N''), the downward force of gravity (''mg''), and a force ''f'' along the direction of the plane that could be applied, for example, by friction or a string]] The forces acting on a body [[Euclidean vector#Addition and subtraction|add as vectors]], and so the total force on a body depends upon both the magnitudes and the directions of the individual forces.<ref name="Kleppner"/>{{rp|58}} When the net force on a body is equal to zero, then by Newton's second law, the body does not accelerate, and it is said to be in [[mechanical equilibrium]]. A state of mechanical equilibrium is ''stable'' if, when the position of the body is changed slightly, the body remains near that equilibrium. Otherwise, the equilibrium is ''unstable.''<ref name=":0" />{{rp|121}}<ref name="Kleppner"/>{{rp|174}}

A common visual representation of forces acting in concert is the [[free body diagram]], which schematically portrays a body of interest and the forces applied to it by outside influences.<ref>{{Cite journal |last1=Rosengrant |first1=David |author2-link=Alan Van Heuvelen |last2=Van Heuvelen |first2=Alan |last3=Etkina |first3=Eugenia|author3-link=Eugenia Etkina |date=2009-06-01 |title=Do students use and understand free-body diagrams? |journal=[[Physical Review Special Topics - Physics Education Research]] |language=en |volume=5 |issue=1 |pages=010108 |doi=10.1103/PhysRevSTPER.5.010108 |bibcode=2009PRPER...5a0108R |issn=1554-9178|doi-access=free }}</ref> For example, a free body diagram of a block sitting upon an [[inclined plane]] can illustrate the combination of gravitational force, [[Normal force|"normal" force]], friction, and string tension.{{refn|group=note|One textbook observes that a block sliding down an inclined plane is what "some cynics view as the dullest problem in all of physics".<ref name="Kleppner"/>{{rp|70}} Another quips, "Nobody will ever know how many minds, eager to learn the secrets of the universe, found themselves studying inclined planes and pulleys instead, and decided to switch to some more interesting profession."<ref name=":0"/>{{rp|173}}}}

Newton's second law is sometimes presented as a ''definition'' of force, i.e., a force is that which exists when an inertial observer sees a body accelerating. This is sometimes regarded as a potential [[Tautology (logic)|tautology]] — acceleration implies force, force implies acceleration. However, Newton's second law not only merely defines the force by the acceleration: forces exist as separate from the acceleration produced by the force in a particular system. The same force that is identified as producing acceleration to an object can then be applied to any other object, and the resulting accelerations (coming from that same force) will always be inversely proportional to the mass of the object.  What Newton's Second Law states is that all the effect of a force onto a system can be reduced to two pieces of information: the magnitude of the force, and it's direction, and then goes on to specify what the effect is.

Beyond that, an equation detailing the force might also be specified, like [[Newton's law of universal gravitation]]. By inserting such an expression for <math>\mathbf{F}</math> into Newton's second law, an equation with predictive power can be written.{{refn|group=note|For example, José and Saletan (following [[Ernst Mach|Mach]] and [[Leonard Eisenbud|Eisenbud]]<ref name="Eisenbud">{{cite journal|first=Leonard |last=Eisenbud |author-link=Leonard Eisenbud |year=1958 |title=On the Classical Laws of Motion |journal=[[American Journal of Physics]] |volume=26 |issue=3 |pages=144–159 |doi=10.1119/1.1934608|bibcode=1958AmJPh..26..144E }}</ref>) take the conservation of momentum as a fundamental physical principle and treat <math>\mathbf{F} = m\mathbf{a}</math> as a definition of "force".<ref name=":2" />{{Rp|page=9}} See also Frautschi et al.,<ref name=":0" />{{Rp|page=134}} as well as Feynman, Leighton and Sands,<ref name="FLS">{{Cite book |last1=Feynman |first1=Richard P. |title=The Feynman Lectures on Physics, Volume 1 |title-link=The Feynman Lectures on Physics |last2=Leighton |first2=Robert B. |last3=Sands |first3=Matthew L. |date=1989 |publisher=Addison-Wesley Pub. Co |isbn=0-201-02010-6 |location=Reading, Mass. |oclc=531535 |author-link=Richard Feynman |author-link2=Robert B. Leighton |author-link3=Matthew Sands |orig-date=1965}}</ref>{{Rp|location=12-1}} who argue that the second law is incomplete without a specification of a force by another law, like the law of gravity. Kleppner and Kolenkow argue that the second law is incomplete without the third law: an observer who sees one body accelerate without a matching acceleration of some other body to compensate would conclude, not that a force is acting, but that they are not an inertial observer.<ref name="Kleppner"/>{{rp|60}} Landau and Lifshitz bypass the question by starting with the Lagrangian formalism rather than the Newtonian.<ref name="Landau"/>}} Newton's second law has also been regarded as setting out a research program for physics, establishing that important goals of the subject are to identify the forces present in nature and to catalogue the constituents of matter.<ref name=":0" />{{Rp|page=134}}<ref name="FLS" />{{Rp|location=12-2}}

However, forces can often be measured directly with no acceleration being involved, such as through [[weighing scale]]s. By postulating a physical object that can be directly measured independently from acceleration, Newton made a objective physical statement with the second law alone, the predictions of which can be verified even if no force law is given.

===Third law{{anchor|Newton's_third_law}}===
:''To every action, there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.''<ref name=":0" />{{Rp|page=116}}
[[File:Iridium-1 Launch (32312419215).jpg|thumb|upright|[[Rocket]]s work by creating unbalanced high pressure that pushes the rocket upwards while exhaust gas exits through an open nozzle.<ref>{{cite book|last1=Warren|first1=J. W.|title=Understanding force: an account of some aspects of teaching the idea of force in school, college and university courses in engineering, mathematics and science|date=1979|publisher=Murray|location=London|isbn=978-0-7195-3564-2|pages=[https://archive.org/details/understandingfor0000warr/page/28 28–29]|url=https://archive.org/details/understandingfor0000warr/page/28}}</ref>]]
In other words, if one body exerts a force on a second body, the second body is also exerting a force on the first body, of equal magnitude in the opposite direction. Overly brief paraphrases of the third law, like "action equals [[Reaction (physics)|reaction]]" might have caused confusion among generations of students: the "action" and "reaction" apply to different bodies. For example, consider a book at rest on a table. The Earth's gravity pulls down upon the book. The "reaction" to that "action" is ''not'' the support force from the table holding up the book, but the gravitational pull of the book acting on the Earth.{{refn|group=note|See, for instance, Moebs et al.,<ref>{{cite book|first1=William |last1=Moebs |display-authors=etal |title=University Physics, Volume 1 |chapter=5.5 Newton's Third Law |chapter-url=https://openstax.org/books/university-physics-volume-1/pages/5-5-newtons-third-law |publisher=OpenStax |isbn=978-1-947172-20-3 |year=2023 |page=220}}</ref> Gonick and Huffman,<ref>{{cite book|first1=Larry |last1=Gonick |author-link1=Larry Gonick |first2=Art |last2=Huffman |title=The Cartoon Guide to Physics |year=1991 |isbn=0-06-273100-9 |publisher=HarperPerennial |page=50}}</ref> Low and Wilson,<ref>{{Cite journal|last1=Low|first1=David J.|last2=Wilson|first2=Kate F.|date=January 2017|title=The role of competing knowledge structures in undermining learning: Newton's second and third laws|url=http://scitation.aip.org/content/aapt/journal/ajp/85/1/10.1119/1.4972041|journal=[[American Journal of Physics]] |language=en|volume=85|issue=1|pages=54–65|doi=10.1119/1.4972041|bibcode=2017AmJPh..85...54L |issn=0002-9505}}</ref> Stocklmayer et al.,<ref name="Stocklmayer">{{Cite journal|last1=Stocklmayer|first1=Sue|author-link1=Susan Stocklmayer |last2=Rayner|first2=John P.|last3=Gore|first3=Michael M.|date=October 2012|title=Changing the Order of Newton's Laws—Why & How the Third Law Should be First|url=http://aapt.scitation.org/doi/10.1119/1.4752043|journal=[[The Physics Teacher]] |language=en|volume=50|issue=7|pages=406–409|doi=10.1119/1.4752043|bibcode=2012PhTea..50..406S |issn=0031-921X}}</ref> Hellingman,<ref>{{Cite journal|last=Hellingman|first=C.|date=March 1992|title=Newton's third law revisited|url=https://iopscience.iop.org/article/10.1088/0031-9120/27/2/011|journal=[[Physics Education]] |volume=27|issue=2|pages=112–115|doi=10.1088/0031-9120/27/2/011|bibcode=1992PhyEd..27..112H |s2cid=250891975 |issn=0031-9120}}</ref> and Hodanbosi.<ref>{{Cite web|url=https://www.grc.nasa.gov/www/k-12/WindTunnel/Activities/third_law_motion.html#:~:text=DESCRIPTION:+A+set+of+mathematics,with+Newton%27s+Laws+of+Motion.&text=The+book+lying+on+the,the+book+remains+at+rest.|title=Third Law of Motion|website=www.grc.nasa.gov |first=Carol |last=Hodanbosi |editor-first=Jonathan G. |editor-last=Fairman |date=August 1996}}</ref>}}

Newton's third law relates to a more fundamental principle, the [[conservation of momentum]]. The latter remains true even in cases where Newton's statement does not, for instance when [[Force field (physics)|force fields]] as well as material bodies carry momentum, and when momentum is defined properly, in [[quantum mechanics]] as well.{{refn|group=note|See, for example, Frautschi et al.<ref name=":0" />{{rp|356}}}} In Newtonian mechanics, if two bodies have momenta <math>\mathbf{p}_1</math> and <math>\mathbf{p}_2</math> respectively, then the total momentum of the pair is <math>\mathbf{p} = \mathbf{p}_1 + \mathbf{p}_2</math>, and the rate of change of <math>\mathbf{p}</math> is <math display="block">\frac{d\mathbf{p}}{dt} = \frac{d\mathbf{p}_1}{dt} + \frac{d\mathbf{p}_2}{dt}.</math> By Newton's second law, the first term is the total force upon the first body, and the second term is the total force upon the second body. If the two bodies are isolated from outside influences, the only force upon the first body can be that from the second, and vice versa. By Newton's third law, these forces have equal magnitude but opposite direction, so they cancel when added, and <math>\mathbf{p}</math> is constant. Alternatively, if <math>\mathbf{p}</math> is known to be constant, it follows that the forces have equal magnitude and opposite direction.

===Candidates for additional laws===
Various sources have proposed elevating other ideas used in classical mechanics to the status of Newton's laws. For example, in Newtonian mechanics, the total mass of a body made by bringing together two smaller bodies is the sum of their individual masses. [[Frank Wilczek]] has suggested calling attention to this assumption by designating it "Newton's Zeroth Law".<ref>{{cite web|first=Frank |last=Wilczek |author-link=Frank Wilczek |title=The Origin of Mass |website=MIT Physics Annual 2003 |url=https://physics.mit.edu/wp-content/uploads/2021/01/physicsatmit_03_wilczek_originofmass.pdf |year=2003 |access-date=2022-01-13}}</ref> Another candidate for a "zeroth law" is the fact that at any instant, a body reacts to the forces applied to it at that instant.<ref>{{Cite journal |last1=Scherr |first1=Rachel E.|author1-link=Rachel Scherr |last2=Redish |first2=Edward F. |date=2005-01-01 |title=Newton's Zeroth Law: Learning from Listening to Our Students |url=https://aapt.scitation.org/doi/10.1119/1.1845990 |journal=[[The Physics Teacher]] |volume=43 |issue=1 |pages=41–45 |doi=10.1119/1.1845990 |bibcode=2005PhTea..43...41S |issn=0031-921X}}</ref> Likewise, the idea that forces add like vectors (or in other words obey the [[superposition principle]]), and the idea that forces change the energy of a body, have both been described as a "fourth law".{{refn|group=note|For the former, see Greiner,<ref>{{cite book|last1=Greiner|first1=Walter|title=Classical Mechanics: Point Particles and Relativity|url=https://archive.org/details/springer_10.1007-b97649|date=2003|page=135|publisher=Springer|location=New York|isbn=978-0-387-21851-9}}</ref> or Wachter and Hoeber.<ref>{{cite book|last1=Wachter|first1=Armin|last2=Hoeber|first2=Henning|title=Compendium of theoretical physics|date=2006|page=6|publisher=Springer|location=New York|isbn=978-0-387-25799-0}}</ref> For the latter, see Tait<ref>{{Cite book|last=Tait|first=Peter Guthrie|author-link=Peter Guthrie Tait|date=1889|chapter=Mechanics |title=Encyclopædia Britannica |title-link=Encyclopædia Britannica |edition=9th |pages=715–716 |volume=15 |chapter-url=https://en.wikisource.org/wiki/Page:Encyclop%C3%A6dia_Britannica,_Ninth_Edition,_v._15.djvu/747}}</ref> and Heaviside.<ref>{{Cite journal|last=Heaviside|first=Oliver|author-link=Oliver Heaviside|date=August 1905|title=The Transverse Momentum of an Electron|journal=[[Nature (journal)|Nature]]|language=en|volume=72|issue=1870|pages=429|doi=10.1038/072429a0|bibcode=1905Natur..72Q.429H |s2cid=4016382 |issn=0028-0836|doi-access=free}}</ref> }}

Moreover, some texts organize the basic ideas of Newtonian mechanics into different postulates, other than the three laws as commonly phrased, with the goal of being more clear about what is empirically observed and what is true by definition.<ref name=":2" />{{Rp|page=9}}<ref name="Eisenbud"/>

===Examples===
The study of the behavior of massive bodies using Newton's laws is known as Newtonian mechanics. Some example problems in Newtonian mechanics are particularly noteworthy for conceptual or historical reasons.

====Uniformly accelerated motion====
{{Main|Free fall|Projectile motion}}
[[Image:Bouncing ball strobe edit.jpg|thumb|upright=1.3|A [[bouncing ball]] photographed at 25 frames per second using a [[stroboscope|stroboscopic flash]]. In between bounces, the ball's height as a function of time is close to being a [[parabola]], deviating from a parabolic arc because of air resistance, spin, and deformation into a non-spherical shape upon impact.]]
If a body falls from rest near the surface of the Earth, then in the absence of air resistance, it will accelerate at a constant rate. This is known as [[free fall]]. The speed attained during free fall is proportional to the elapsed time, and the distance traveled is proportional to the square of the elapsed time.<ref>{{Cite journal |last=Nicodemi |first=Olympia |author-link=Olympia Nicodemi |date=2010-02-01 |title=Galileo and Oresme: Who Is Modern? Who Is Medieval? |url=https://doi.org/10.4169/002557010X479965 |journal=[[Mathematics Magazine]] |volume=83 |issue=1 |pages=24–32 |doi=10.4169/002557010X479965 |s2cid=122113958 |issn=0025-570X}}</ref> Importantly, the acceleration is the same for all bodies, independently of their mass. This follows from combining Newton's second law of motion with his [[Newton's law of universal gravitation|law of universal gravitation]]. The latter states that the magnitude of the gravitational force from the Earth upon the body is
<math display="block">F = \frac{GMm}{r^2} ,</math>
where <math>m</math> is the mass of the falling body, <math>M</math> is the mass of the Earth, <math>G</math> is Newton's constant, and <math>r</math> is the distance from the center of the Earth to the body's location, which is very nearly the radius of the Earth. Setting this equal to <math>ma</math>, the body's mass <math>m</math> cancels from both sides of the equation, leaving an acceleration that depends upon <math>G</math>, <math>M</math>, and <math>r</math>, and <math>r</math> can be taken to be constant. This particular value of acceleration is typically denoted <math>g</math>:
<math display="block">g = \frac{GM}{r^2} \approx \mathrm{9.8 ~m/s^2}.</math>

If the body is not released from rest but instead launched upwards and/or horizontally with nonzero velocity, then free fall becomes [[projectile motion]].<ref>{{cite web|url=https://webhome.phy.duke.edu/~schol/phy361/faqs/faq3/ |first=Kate |last=Scholberg |author-link=Kate Scholberg |access-date=2022-01-16 |title=Frequently Asked Questions: Projectile Motion |website=Physics 361 |year=2020}}</ref> When air resistance can be neglected, projectiles follow [[parabola]]-shaped trajectories, because gravity affects the body's vertical motion and not its horizontal. At the peak of the projectile's trajectory, its vertical velocity is zero, but its acceleration is <math>g</math> downwards, as it is at all times. Setting the wrong vector equal to zero is a common confusion among physics students.<ref>{{Cite journal |last1=Carli |first1=Marta |last2=Lippiello |first2=Stefania |last3=Pantano |first3=Ornella |last4=Perona |first4=Mario |last5=Tormen |first5=Giuseppe |date=2020-03-19 |title=Testing students ability to use derivatives, integrals, and vectors in a purely mathematical context and in a physical context |journal=[[Physical Review Physics Education Research]] |language=en |volume=16 |issue=1 |pages=010111 |doi=10.1103/PhysRevPhysEducRes.16.010111 |bibcode=2020PRPER..16a0111C |s2cid=215832738 |issn=2469-9896|doi-access=free |hdl=11577/3340932 |hdl-access=free }}</ref>

====Uniform circular motion====
{{Main|Circular motion}}
[[File:Binary system orbit q=3 e=0.gif|thumb|Two objects in uniform circular motion, orbiting around the [[barycenter]] (center of mass of both objects)]]
When a body is in uniform circular motion, the force on it changes the direction of its motion but not its speed. For a body moving in a circle of radius <math>r</math> at a constant speed <math>v</math>, its acceleration has a magnitude<math display="block">a = \frac{v^2}{r}</math>and is directed toward the center of the circle.{{refn|group=note|Among the many textbook explanations of this are Frautschi et al.<ref name=":0" />{{Rp|page=104}} and Boas.<ref name="Boas">{{Cite book |last=Boas |first=Mary L. |title=Mathematical Methods in the Physical Sciences |title-link=Mathematical Methods in the Physical Sciences |date=2006 |publisher=Wiley |isbn=978-0-471-19826-0 |edition=3rd |location=Hoboken, NJ |oclc=61332593 |author-link=Mary L. Boas}}</ref>{{Rp|page=287}}}} The force required to sustain this acceleration, called the [[centripetal force]], is therefore also directed toward the center of the circle and has magnitude <math>mv^2/r</math>. Many [[orbit]]s, such as that of the Moon around the Earth, can be approximated by uniform circular motion. In such cases, the centripetal force is gravity, and by Newton's law of universal gravitation has magnitude <math>GMm/r^2</math>, where <math>M</math> is the mass of the larger body being orbited. Therefore, the mass of a body can be calculated from observations of another body orbiting around it.<ref>{{Cite book |last=Brown |first=Mike |title-link=How I Killed Pluto and Why It Had It Coming |title=How I Killed Pluto and Why It Had It Coming |date=2010 |publisher=Spiegel & Grau |isbn=978-0-385-53108-5 |edition=1st |location=New York |oclc=495271396 |author-link=Mike Brown (astronomer)}}</ref>{{Rp|page=130}}

[[Newton's cannonball]] is a [[thought experiment]] that interpolates between projectile motion and uniform circular motion. A cannonball that is lobbed weakly off the edge of a tall cliff will hit the ground in the same amount of time as if it were dropped from rest, because the force of gravity only affects the cannonball's momentum in the downward direction, and its effect is not diminished by horizontal movement. If the cannonball is launched with a greater initial horizontal velocity, then it will travel farther before it hits the ground, but it will still hit the ground in the same amount of time. However, if the cannonball is launched with an even larger initial velocity, then the curvature of the Earth becomes significant: the ground itself will curve away from the falling cannonball. A very fast cannonball will fall away from the inertial straight-line trajectory at the same rate that the Earth curves away beneath it; in other words, it will be in orbit (imagining that it is not slowed by air resistance or obstacles).<ref>{{Cite journal |last1=Topper |first1=D. |last2=Vincent |first2=D. E. |date=1999-01-01 |title=An analysis of Newton's projectile diagram |url=https://iopscience.iop.org/article/10.1088/0143-0807/20/1/018 |journal=[[European Journal of Physics]] |volume=20 |issue=1 |pages=59–66 |doi=10.1088/0143-0807/20/1/018 |bibcode=1999EJPh...20...59T |s2cid=250883796 |issn=0143-0807}}</ref>

====Harmonic motion====
{{Main|Harmonic oscillator}}
[[Image:Animated-mass-spring.gif|right|frame|An undamped [[spring–mass system]] undergoes simple harmonic motion.]]
Consider a body of mass <math>m</math> able to move along the <math>x</math> axis, and suppose an equilibrium point exists at the position <math>x = 0</math>. That is, at <math>x = 0</math>, the net force upon the body is the zero vector, and by Newton's second law, the body will not accelerate. If the force upon the body is proportional to the displacement from the equilibrium point, and directed to the equilibrium point, then the body will perform [[simple harmonic motion]]. Writing the force as <math>F = -kx</math>, Newton's second law becomes
<math display="block">m\frac{d^2 x}{dt^2} = -kx \, .</math>
This differential equation has the solution
<math display="block">x(t) = A \cos \omega t + B \sin \omega t \, </math>
where the frequency <math>\omega</math> is equal to <math>\sqrt{k/m}</math>, and the constants <math>A</math> and <math>B</math> can be calculated knowing, for example, the position and velocity the body has at a given time, like <math>t = 0</math>.

One reason that the harmonic oscillator is a conceptually important example is that it is good approximation for many systems near a stable mechanical equilibrium.{{refn|group=note|Among the many textbook treatments of this point are Hand and Finch<ref name="hand-finch">{{Cite book|last1=Hand|first1=Louis N.|url=https://www.worldcat.org/oclc/37903527|title=Analytical Mechanics|last2=Finch|first2=Janet D.|date=1998|publisher=Cambridge University Press|isbn=0-521-57327-0|location=Cambridge|oclc=37903527}}</ref>{{Rp|page=81}} and also Kleppner and Kolenkow.<ref name="Kleppner">{{Cite book|last1=Kleppner|first1=Daniel|url=https://books.google.com/books?id=Hmqvhu7s4foC|title=An introduction to mechanics|last2=Kolenkow|first2=Robert J.|date=2014|publisher=Cambridge University Press|isbn=978-0-521-19811-0|edition=2nd|location=Cambridge|oclc=854617117}}</ref>{{Rp|page=103}}}} For example, a [[pendulum]] has a stable equilibrium in the vertical position: if motionless there, it will remain there, and if pushed slightly, it will swing back and forth. Neglecting air resistance and friction in the pivot, the force upon the pendulum is gravity, and Newton's second law becomes <math display="block">\frac{d^2\theta}{dt^2} = -\frac{g}{L} \sin\theta,</math>where <math>L</math> is the length of the pendulum and <math>\theta</math> is its angle from the vertical. When the angle <math>\theta</math> is small, the [[Sine and cosine|sine]] of <math>\theta</math> is nearly equal to <math>\theta</math> (see [[small-angle approximation]]), and so this expression simplifies to the equation for a simple harmonic oscillator with frequency <math>\omega = \sqrt{g/L}</math>.

A harmonic oscillator can be ''damped,'' often by friction or viscous drag, in which case energy bleeds out of the oscillator and the amplitude of the oscillations decreases over time. Also, a harmonic oscillator can be ''driven'' by an applied force, which can lead to the phenomenon of [[resonance]].<ref>{{Cite journal|last1=Billah|first1=K. Yusuf|last2=Scanlan|first2=Robert H.|date=1991-02-01|title=Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks|url=http://www.ketchum.org/billah/Billah-Scanlan.pdf|journal=[[American Journal of Physics]] |volume=59|issue=2|pages=118–124|doi=10.1119/1.16590|issn=0002-9505|bibcode=1991AmJPh..59..118B}}</ref>

====Objects with variable mass====
{{main|Variable-mass system}}
[[File:Space Shuttle Atlantis launches from KSC on STS-132 side view.jpg|thumb|Rockets, like the [[Space Shuttle Atlantis|Space Shuttle ''Atlantis'']], expel mass during operation. This means that the mass being pushed, the rocket and its remaining onboard fuel supply, is constantly changing.]]
Newtonian physics treats matter as being neither created nor destroyed, though it may be rearranged. It can be the case that an object of interest gains or loses mass because matter is added to or removed from it. In such a situation, Newton's laws can be applied to the individual pieces of matter, keeping track of which pieces belong to the object of interest over time. For instance, if a rocket of mass <math>M(t)</math>, moving at velocity <math>\mathbf{v}(t)</math>, ejects matter at a velocity <math>\mathbf{u}</math> relative to the rocket, then<ref name=Plastino-1992/>
<math display="block">\mathbf{F} = M \frac{d\mathbf{v}}{dt} - \mathbf{u} \frac{dM}{dt} \, </math>
where <math>\mathbf{F}</math> is the net external force (e.g., a planet's gravitational pull).<ref name="Kleppner" />{{rp|139}}

====Fan and sail====
[[File:Newtonssailboat.jpg|thumb|A boat equipped with a fan and a sail]]
The fan and sail example is a situation studied in discussions of Newton's third law.<ref name="j940">{{cite journal | last=Wilson | first=Jerry D. | title=LETTERS: Newton's Sailboat | journal=[[The Physics Teacher]] | volume=10 | issue=6 | date=1972-09-01 | issn=0031-921X | doi=10.1119/1.2352231 | pages=300| bibcode=1972PhTea..10..300W }}</ref> In the situation, a [[fan (machine)|fan]] is attached to a cart or a [[sailboat]] and blows on its sail. From the third law, one would reason that the force of the air pushing in one direction would cancel out the force done by the fan on the sail, leaving the entire apparatus stationary. However, because the system is not entirely enclosed, there are conditions in which the vessel will move; for example, if the sail is built in a manner that redirects the majority of the airflow back towards the fan, the net force will result in the vessel moving forward.<ref name="Stocklmayer"/><ref name="l303">{{cite journal | last=Clark | first=Robert Beck | title=The answer is obvious, Isn't it? | journal=The Physics Teacher | volume=24 | issue=1 | date=1986-01-01 | issn=0031-921X | doi=10.1119/1.2341931 | pages=38–39| bibcode=1986PhTea..24...38C }}</ref>