Force Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.Anti-spam check. Do not fill this in! == Newtonian mechanics == {{main|Newton's laws of motion}} Sir Isaac Newton described the motion of all objects using the concepts of [[inertia]] and force. In 1687, Newton published his magnum opus, ''[[Philosophiæ Naturalis Principia Mathematica]]''.<ref name=uniphysics_ch2/><ref name="Principia">{{Cite book |last=Newton |first=Isaac |title=The Principia Mathematical Principles of Natural Philosophy |publisher=University of California Press |year=1999 |isbn=978-0-520-08817-7 |location=Berkeley |author-link=Isaac Newton}} This is a recent translation into English by [[I. Bernard Cohen]] and Anne Whitman, with help from Julia Budenz.</ref> In this work Newton set out three laws of motion that have dominated the way forces are described in physics to this day.<ref name="Principia"/> The precise ways in which Newton's laws are expressed have evolved in step with new mathematical approaches.<ref>{{cite book |last=Howland |first=R. A. |title=Intermediate dynamics a linear algebraic approach |date=2006 |publisher=Springer |location=New York |isbn=978-0387280592 |pages=255–256 |edition=Online-Ausg.}}</ref> === First law === {{main|Newton's first law}} Newton's first law of motion states that the natural behavior of an object at rest is to continue being at rest, and the natural behavior of an object moving at constant speed in a straight line is to continue moving at that constant speed along that straight line.<ref name="Principia"/> The latter follows from the former because of the [[principle of relativity|principle that the laws of physics are the same]] for all [[inertial frame of reference|inertial observers]], i.e., all observers who do not feel themselves to be in motion. An observer moving in tandem with an object will see it as being at rest. So, its natural behavior will be to remain at rest with respect to that observer, which means that an observer who sees it moving at constant speed in a straight line will see it continuing to do so.<ref name="mermin2005">{{cite book|first=N. David |last=Mermin |author-link=N. David Mermin |title=It's About Time: Understanding Einstein's Relativity |publisher=Princeton University Press |year=2005 |isbn=978-0-691-21877-9}}</ref>{{rp|1–7}} <!-- This location for image in the text seems to line up with the next heading on desktop --> [[File:GodfreyKneller-IsaacNewton-1689.jpg|upright|thumb|[[Sir Isaac Newton]] in 1689. His ''Principia'' presented his three laws of motion in geometrical language, whereas modern physics uses [[differential calculus]] and [[Vector (mathematics and physics)|vector]]s.]] === Second law === {{main|Newton's second law}} According to the first law, motion at constant speed in a straight line does not need a cause. It is ''change'' in motion that requires a cause, and Newton's second law gives the quantitative relationship between force and change of motion. [[Newton's second law]] states that the net force acting upon an object is equal to the [[time derivative|rate]] at which its [[momentum]] changes with [[time]]. If the mass of the object is constant, this law implies that the [[acceleration]] of an object is directly [[Proportionality (mathematics)|proportional]] to the net force acting on the object, is in the direction of the net force, and is inversely proportional to the [[mass]] of the object.<ref name="openstax-university-physics" />{{rp|pages=204–207}} A modern statement of Newton's second law is a vector equation: <math display="block" qid=Q104212301>\vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t},</math> where <math> \vec{p}</math> is the momentum of the system, and <math> \vec{F}</math> is the net ([[Vector (geometric)#Addition and subtraction|vector sum]]) force.<ref name="openstax-university-physics" />{{rp|page=399}} If a body is in equilibrium, there is zero ''net'' force by definition (balanced forces may be present nevertheless). In contrast, the second law states that if there is an ''unbalanced'' force acting on an object it will result in the object's momentum changing over time.<ref name="Principia"/> In common engineering applications the mass in a system remains constant allowing as simple algebraic form for the second law. By the definition of momentum, <math display="block">\vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t} = \frac{\mathrm{d}\left(m\vec{v}\right)}{\mathrm{d}t},</math> where ''m'' is the [[mass]] and <math> \vec{v}</math> is the [[velocity]].<ref name=FeynmanVol1/>{{rp|((9-1,9-2))}} If Newton's second law is applied to a system of [[Newton's Laws of Motion#Open systems|constant mass]], ''m'' may be moved outside the derivative operator. The equation then becomes <math display="block">\vec{F} = m\frac{\mathrm{d}\vec{v}}{\mathrm{d}t}.</math> By substituting the definition of [[acceleration]], the algebraic version of [[Newton's second law]] is derived: <math display="block" qid=Q2397319>\vec{F} =m\vec{a}.</math> === Third law === {{main|Newton's third law}} Whenever one body exerts a force on another, the latter simultaneously exerts an equal and opposite force on the first. In vector form, if <math>\vec{F}_{1,2}</math> is the force of body 1 on body 2 and <math>\vec{F}_{2,1}</math> that of body 2 on body 1, then <math display="block">\vec{F}_{1,2}=-\vec{F}_{2,1}.</math> This law is sometimes referred to as the ''action-reaction law'', with <math> \vec{F}_{1,2}</math> called the ''action'' and <math> -\vec{F}_{2,1}</math> the ''[[Reaction (physics)|reaction]]''. Newton's Third Law is a result of applying [[symmetry]] to situations where forces can be attributed to the presence of different objects. The third law means that all forces are ''interactions'' between different bodies.<ref>{{cite journal |title=Newton's third law revisited |first=C. |last=Hellingman |journal=Phys. Educ. |volume=27 |year=1992 |issue=2 |pages=112–115 |quote=Quoting Newton in the ''Principia'': It is not one action by which the Sun attracts Jupiter, and another by which Jupiter attracts the Sun; but it is one action by which the Sun and Jupiter mutually endeavour to come nearer together. |doi=10.1088/0031-9120/27/2/011 |bibcode=1992PhyEd..27..112H |s2cid=250891975 }}</ref><ref>{{Cite book |last1=Resnick |first1=Robert |title=Physics. 1 |last2=Halliday |first2=David |last3=Krane |first3=Kenneth S. |date=2002 |isbn=978-0-471-32057-9 |edition=5|quote="Any single force is only one aspect of a mutual interaction between ''two'' bodies."}}</ref> and thus that there is no such thing as a unidirectional force or a force that acts on only one body. In a system composed of object 1 and object 2, the net force on the system due to their mutual interactions is zero: <math display="block">\vec{F}_{1,2}+\vec{F}_{2,1}=0.</math> More generally, in a [[closed system]] of particles, all internal forces are balanced. The particles may accelerate with respect to each other but the [[center of mass]] of the system will not accelerate. If an external force acts on the system, it will make the center of mass accelerate in proportion to the magnitude of the external force divided by the mass of the system.<ref name=FeynmanVol1 />{{rp|((19-1))}}<ref name=Kleppner /> Combining Newton's Second and Third Laws, it is possible to show that the [[Conservation of momentum|linear momentum of a system is conserved]] in any [[closed system]]. In a system of two particles, if <math> \vec{p}_1</math> is the momentum of object 1 and <math> \vec{p}_{2}</math> the momentum of object 2, then <math display="block">\frac{\mathrm{d}\vec{p}_1}{\mathrm{d}t} + \frac{\mathrm{d}\vec{p}_2}{\mathrm{d}t}= \vec{F}_{1,2} + \vec{F}_{2,1} = 0.</math> Using similar arguments, this can be generalized to a system with an arbitrary number of particles. In general, as long as all forces are due to the interaction of objects with mass, it is possible to define a system such that net momentum is never lost nor gained.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner /> === Defining "force" === Some textbooks use Newton's second law as a ''definition'' of force.<ref>{{Cite book |last1=Landau |first1=L. D. |author-link=Lev Landau |last2=Akhiezer |author2-link=Aleksander Akhiezer |first2=A. I. |last3=Lifshitz |first3=A. M. |author3-link=Evgeny Lifshitz |title=General Physics; mechanics and molecular physics |publisher=Pergamon Press |year=1967 |location=Oxford |edition= |isbn=978-0-08-003304-4 |url-access=registration |url=https://archive.org/details/generalphysicsme0000land_d9j0 }} Translated by: J. B. Sykes, A. D. Petford, and C. L. Petford. {{LCCN|67-30260}}. In section 7, pp. 12–14, this book defines force as ''dp/dt''.</ref><ref>{{Cite book |last1=Kibble |first1=Tom W. B. |last2=Berkshire |first2=Frank H. |title=Classical Mechanics |publisher=Imperial College Press |year=2004 |location=London |edition=5th |isbn=1860944248}} According to page 12, "[Force] can of course be introduced, by defining it through Newton's second law".</ref><ref>{{Cite book |last1=de Lange |first1=O. L. |last2=Pierrus |first2=J. |title=Solved Problems in Classical Mechanics |publisher=Oxford University Press |year=2010 |location=Oxford |edition= |isbn=978-0-19-958252-5}} According to page 3, "[Newton's second law of motion] can be regarded as defining force".</ref><ref>{{Cite book|last1=José|first1=Jorge V.|url=https://www.worldcat.org/oclc/857769535|title=Classical dynamics: A Contemporary Approach|last2=Saletan|first2=Eugene J.|date=1998|publisher=Cambridge University Press|isbn=978-1-139-64890-5|location=Cambridge [England]|oclc=857769535|author-link=Jorge V. José |page=9}}</ref> However, for the equation <math>\vec{F} = m\vec{a}</math> for a constant mass <math>m</math> to then have any predictive content, it must be combined with further information.<ref>{{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|page=134}}</ref><ref name=FeynmanVol1 />{{rp|((12-1))}} Moreover, inferring that a force is present because a body is accelerating is only valid in an inertial frame of reference.<ref name=Kleppner />{{rp|59}} The question of which aspects of Newton's laws to take as definitions and which to regard as holding physical content has been answered in various ways,<ref name="thornton-marion">{{cite book|first1=Stephen T. |last1=Thornton |first2=Jerry B. |last2=Marion |title=Classical Dynamics of Particles and Systems |edition=5th |publisher=Thomson Brooks/Cole |isbn=0-534-40896-6 |year=2004 |pages=49–50}}</ref><ref name=":0">{{cite book |author-last1=Landau |author-first1=Lev D. |title=Mechanics |author-last2=Lifshitz |author-first2=Evgeny M. |date=1969 |publisher=[[Pergamon Press]] |isbn=978-0-080-06466-6 |edition=2nd |series=[[Course of Theoretical Physics]] |volume=1 |translator-last1=Sykes |translator-first1=J. B. |author-link1=Lev Landau |author-link2=Evgeny Lifshitz |translator-last2=Bell |translator-first2=J. S. |translator-link2=John Stewart Bell}}</ref>{{Rp|pages=vii}} which ultimately do not affect how the theory is used in practice.<ref name="thornton-marion"/> Notable physicists, philosophers and mathematicians who have sought a more explicit definition of the concept of force include [[Ernst Mach]] and [[Walter Noll]].<ref>{{cite book |last=Jammer |first=Max |author-link=Max Jammer |title=Concepts of Force: A study in the foundations of dynamics |year=1999 |publisher=Dover Publications |location=Mineola, NY |isbn=978-0486406893 |pages=220–222 |edition=Facsim.}}</ref><ref>{{cite web |first=Walter |last=Noll |title=On the Concept of Force |url=http://www.math.cmu.edu/~wn0g/Force.pdf |publisher=Carnegie Mellon University |date=April 2007 |access-date=28 October 2013}}</ref> Summary: Please note that all contributions to Christianpedia may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here. 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