Editing Newton's laws of motion (section)

===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.