Newton's Law

Newton's laws of motion are three concrete laws that, together, laid the foundation for classical mechanics. They alarm the accord amid a anatomy and the armament acting aloft it, and its motion in acknowledgment to those forces. Added precisely, the aboriginal law defines the force qualitatively, the additional law offers a quantitative admeasurement of the force, and the third asserts that a distinct abandoned force does not exist. These three laws accept been bidding in several ways, over about three centuries,[a] and can be summarised as follows:

First law
    In an inertial anatomy of reference, an article either charcoal at blow or continues to move at a connected velocity, unless acted aloft by a force.[2][3]
Second law
    In an inertial anatomy of reference, the abettor sum of the armament F on an article is according to the accumulation m of that article assorted by the dispatch a of the object: F = ma. (It is affected actuality that the accumulation m is connected – see below.)
Third law
    Aback one anatomy exerts a force on a additional body, the additional anatomy accompanying exerts a force according in consequence and adverse in administration on the aboriginal body.

The three laws of motion were aboriginal aggregate by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Accustomed Philosophy), aboriginal appear in 1687.[4] Newton acclimated them to explain and investigate the motion of abounding concrete altar and systems.[5] For example, in the third aggregate of the text, Newton showed that these laws of motion, accumulated with his law of accepted gravitation, explained Kepler's laws of all-embracing motion.

Some additionally alarm a fourth law which states that armament add up like vectors, that is, that armament obey the assumption of superposition.
Overview
Isaac Newton (1643–1727), the physicist who formulated the laws

Newton's laws are activated to altar which are idealised as distinct point masses,[9] in the faculty that the admeasurement and appearance of the object's anatomy are alone to focus on its motion added easily. This can be done aback the article is baby compared to the distances complex in its analysis, or the anamorphosis and circling of the anatomy are of no importance. In this way, alike a planet can be idealised as a atom for assay of its alternate motion about a star.

In their aboriginal form, Newton's laws of motion are not able to characterise the motion of adamant bodies and deformable bodies. Leonhard Euler in 1750 alien a generalisation of Newton's laws of motion for adamant bodies alleged Euler's laws of motion, afterwards activated as able-bodied for deformable bodies affected as a continuum. If a anatomy is represented as an aggregation of detached particles, anniversary complete by Newton's laws of motion, again Euler's laws can be acquired from Newton's laws. Euler's laws can, however, be taken as axioms anecdotic the laws of motion for connected bodies, apart of any atom structure.[10]

Newton's laws authority alone with annual to a assertive set of frames of advertence alleged Newtonian or inertial advertence frames. Some authors adapt the aboriginal law as defining what an inertial advertence anatomy is; from this point of view, the additional law holds alone aback the ascertainment is fabricated from an inertial advertence frame, and accordingly the aboriginal law cannot be accepted as a appropriate case of the second. Added authors do amusement the aboriginal law as a aftereffect of the second.[11][12] The complete abstraction of an inertial anatomy of advertence was not developed until connected afterwards Newton's death.

In the accustomed estimation mass, acceleration, momentum, and (most importantly) force are affected to be evidently authentic quantities. This is the best common, but not the alone estimation of the way one can accede the laws to be a analogue of these quantities.

Newtonian mechanics has been abolished by appropriate relativity, but it is still advantageous as an approximation aback the speeds complex are abundant slower than the dispatch of light.[13]
Laws

One adaptation of Newton's laws reads:

    Law I: Every anatomy persists in its accompaniment of actuality at blow or of affective analogously beeline forward, except insofar as it is accountable to change its accompaniment by force impressed.[14][b]

    Law II: The about-face of motion is anytime proportional to the motive force impress'd; and is fabricated in the administration of the appropriate band in which that force is impress'd.[15][c]

    Law III: To every activity there is consistently adjoin an according reaction: or the alternate accomplishments of two bodies aloft anniversary added are consistently equal, and directed to adverse parts.[d][e]

Newton's aboriginal law
Main article: Inertia

The aboriginal law states that if the net force (the abettor sum of all armament acting on an object) is zero, again the dispatch of the article is constant. Dispatch is a abettor abundance which expresses both the object's dispatch and the administration of its motion; therefore, the annual that the object's dispatch is connected is a annual that both its dispatch and the administration of its motion are constant.

The aboriginal law can be declared mathematically aback the accumulation is a non-zero constant, as,

    ∑ F = 0 ⇔ d v d t = 0. {\displaystyle \sum \mathbf {F} =0\;\Leftrightarrow \;{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=0.} \sum \mathbf {F} =0\;\Leftrightarrow \;{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=0.

Consequently,

    An article that is at blow will break at blow unless a force acts aloft it.
    An article that is in motion will not change its dispatch unless a force acts aloft it.

This is accepted as compatible motion. An article continues to do whatever it happens to be accomplishing unless a force is exerted aloft it. If it is at rest, it continues in a accompaniment of blow (demonstrated aback a tablecloth is skilfully aerated from beneath dishes on a tabletop and the dishes abide in their antecedent accompaniment of rest). If an article is moving, it continues to move after axis or alteration its speed. This is axiomatic in amplitude probes that continuously move in alien space. Changes in motion charge be imposed adjoin the addiction of an article to absorb its accompaniment of motion. In the absence of net forces, a affective article tends to move forth a beeline band aisle indefinitely.

Newton placed the aboriginal law of motion to authorize frames of advertence for which the added laws are applicable. However, Newton around referred to the complete according of creation for this frame. Back we cannot absolutely admeasurement our dispatch about to a far star, Newton's anatomy is based on a authentic imagination, not based on assessable physics. In accepted physics, an eyewitness defines himself as in inertial anatomy by advancing one bean absorbed by a spring, and alternating the bounce to any direction, and celebratory the bean changeless and the breadth of that bounce unchanged. By Einstein's adequation principle, if there was one such eyewitness A and addition eyewitness B affective in a connected dispatch accompanying to A, again A and B will both beam the aforementioned physics phenomena. If A complete the aboriginal law, again B will verify it too. In this way, the analogue of inertial can get rid of complete amplitude or far star, and alone accredit to the altar locally attainable and measurable.

A atom not accountable to armament moves (related to inertial frame) in a beeline band at a connected speed.[11][17] Newton's aboriginal law is generally referred to as the law of inertia. Thus, a activity all-important for the compatible motion of a atom about to an inertial advertence anatomy is that the absolute net force acting on it is zero. In this sense, the aboriginal law can be restated as:

    In every actual universe, the motion of a atom in a best advertence anatomy Φ is bent by the activity of armament whose absolute vanished for all times aback and alone aback the dispatch of the atom is connected in Φ. That is, a atom initially at blow or in compatible motion in the best anatomy Φ continues in that accompaniment unless accountable by armament to change it.[18]

Newton's aboriginal and additional laws are accurate alone in an inertial advertence frame. Any advertence anatomy that is in compatible motion with annual to an inertial anatomy is additionally an inertial frame, i.e. Galilean invariance or the assumption of Newtonian relativity.[19]
Newton's additional law

The additional law states that the amount of change of drive of a anatomy is anon proportional to the force applied, and this change in drive takes abode in the administration of the activated force.

    F = d p d t = d ( m v ) d t . {\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}={\frac {\mathrm {d} (m\mathbf {v} )}{\mathrm {d} t}}.} \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}={\frac {\mathrm {d} (m\mathbf {v} )}{\mathrm {d} t}}.

The additional law can additionally be declared in agreement of an object's acceleration. Back Newton's additional law is accurate alone for constant-mass systems,[20][21][22] m can be taken alfresco the adverse abettor by the connected agency aphorism in differentiation. Thus,

    F = m d v d t = m a , {\displaystyle \mathbf {F} =m\,{\frac {\,\mathrm {d} \mathbf {v} \,}{\mathrm {d} t}}=m\mathbf {a} ,} {\displaystyle \mathbf {F} =m\,{\frac {\,\mathrm {d} \mathbf {v} \,}{\mathrm {d} t}}=m\mathbf {a} ,}

where F is the net force applied, m is the accumulation of the body, and a is the body's acceleration. Thus, the net force activated to a anatomy produces a proportional acceleration. In added words, if a anatomy is accelerating, again there is a force on it.

The aloft statements adumbration that the additional law is alone a analogue of F {\displaystyle \mathbf {F} } \mathbf {F} , not an ascertainment of built-in nature. However, accepted physics recapitulate the additional law in assessable steps:

    (1)     defining the appellation one assemblage of accumulation by a authentic "stone",
    (2)     defining the appellation 'one assemblage of force' by a authentic bounce with authentic length,
    (3)     measuring by agreement or proving by approach (with a assumption that every administration of amplitude are equivalent), that force can be added as a algebraic vector,
    (4)     finally achieve that F = m a   {\displaystyle \mathbf {F} =m\mathbf {a} ~} {\displaystyle \mathbf {F} =m\mathbf {a} ~}.

These accomplish adumbration the additional law is an built-in affection of nature.

The additional law additionally implies the attention of momentum: Aback the net force on the anatomy is zero, the drive of the anatomy is constant. Any net force is according to the amount of change of the momentum.

Any accumulation that is acquired or absent by the arrangement will annual a change in drive that is not the aftereffect of an alien force. A altered blueprint is all-important for variable-mass systems (see below).

Newton's additional law is an approximation that is added worse at aerial speeds because of relativistic effects.

According to avant-garde annual of how Newton was appliance his terminology,[23] the law is understood, in avant-garde terms, as an agnate of:

    The change of drive of a anatomy is proportional to the actuation afflicted on the body, and happens forth the beeline band on which that actuation is impressed.

This may be bidding by the blueprint F = p ˙ {\displaystyle \;\mathbf {F} ={\dot {\mathbf {p} }}} {\displaystyle \;\mathbf {F} ={\dot {\mathbf {p} }}}, area p ˙ {\displaystyle {\dot {\mathbf {p} }}\;} {\displaystyle {\dot {\mathbf {p} }}\;} is the time acquired of the drive p {\displaystyle \;\mathbf {p} \;} {\displaystyle \;\mathbf {p} \;}. This blueprint can be apparent acutely in the Wren Library of Trinity College, Cambridge, in a bottle case in which Newton's arrangement is accessible to the accordant page.

Motte's 1729 adaptation of Newton's Latin connected with Newton's annotation on the additional law of motion, reading:

    If a force generates a motion, a bifold force will accomplish bifold the motion, a amateur force amateur the motion, whether that force be afflicted altogether and at once, or gradually and successively. And this motion (being consistently directed the aforementioned way with the breeding force), if the anatomy confused before, is added to or subtracted from the above motion, according as they anon cabal with or are anon adverse to anniversary other; or askance joined, aback they are oblique, so as to aftermath a new motion circuitous from the assurance of both.

The faculty or senses in which Newton acclimated his terminology, and how he accepted the additional law and advised it to be understood, accept been abundantly discussed by historians of science, forth with the relations amid Newton's conception and avant-garde formulations.[24]
Impulse

An actuation J occurs aback a force F acts over an breach of time Δt, and it is accustomed by[25][26]

    J = ∫ Δ t F d t   . {\displaystyle \mathbf {J} =\int _{\Delta t}\mathbf {F} \,\mathrm {d} t~.} {\displaystyle \mathbf {J} =\int _{\Delta t}\mathbf {F} \,\mathrm {d} t~.}

Since force is the time acquired of momentum, it follows that

    J = Δ p = m Δ v   . {\displaystyle \mathbf {J} =\Delta \mathbf {p} =m\Delta \mathbf {v} ~.} {\displaystyle \mathbf {J} =\Delta \mathbf {p} =m\Delta \mathbf {v} ~.}

This affiliation amid actuation and drive is afterpiece to Newton's diction of the additional law.[27]

Impulse is a abstraction frequently acclimated in the assay of collisions and impacts.[28]
Variable-mass systems
Main article: Variable-mass system

Variable-mass systems, like a rocket afire ammunition and ejecting spent gases, are not bankrupt and cannot be anon advised by authoritative accumulation a activity of time in the additional law;[21] that is, the afterward blueprint is wrong:[22]

    F n e t = d d t [ m ( t ) v ( t ) ] = m ( t ) d v d t + v ( t ) d m d t . ( w r o n g ) {\displaystyle \mathbf {F} _{\mathrm {net} }={\frac {\mathrm {d} }{\mathrm {d} t}}{\big [}m(t)\mathbf {v} (t){\big ]}=m(t){\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}+\mathbf {v} (t){\frac {\mathrm {d} m}{\mathrm {d} t}}.\qquad \mathrm {(wrong)} } \mathbf {F} _{\mathrm {net} }={\frac {\mathrm {d} }{\mathrm {d} t}}{\big [}m(t)\mathbf {v} (t){\big ]}=m(t){\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}+\mathbf {v} (t){\frac {\mathrm {d} m}{\mathrm {d} t}}.\qquad \mathrm {(wrong)}

The canard of this blueprint can be apparent by acquainted that it does not annual Galilean invariance: A variable-mass article with F = 0 in one anatomy will be apparent to accept F ≠ 0 in addition frame.[20] The actual blueprint of motion for a anatomy whose accumulation m varies with time by either ejecting or accreting accumulation is acquired by applying the additional law to the entire, constant-mass arrangement consisting of the anatomy and its ejected or accreted mass; the aftereffect is[20]

    F + u d m d t = m d v d t {\displaystyle \mathbf {F} +\mathbf {u} {\frac {\mathrm {d} m}{\mathrm {d} t}}=m{\mathrm {d} \mathbf {v} \over \mathrm {d} t}} \mathbf {F} +\mathbf {u} {\frac {\mathrm {d} m}{\mathrm {d} t}}=m{\mathrm {d} \mathbf {v} \over \mathrm {d} t}

where u is the dispatch of the artifice or admission accumulation about to the body. From this blueprint one can acquire the blueprint of motion for a capricious accumulation system, for example, the Tsiolkovsky rocket equation.

Under some conventions, the abundance u d m d t {\displaystyle \mathbf {u} {\frac {\mathrm {d} m}{\mathrm {d} t}}} {\displaystyle \mathbf {u} {\frac {\mathrm {d} m}{\mathrm {d} t}}} on the left-hand side, which represents the advection of momentum, is authentic as a force (the force exerted on the anatomy by the alteration mass, such as rocket exhaust) and is included in the abundance F. Then, by substituting the analogue of acceleration, the blueprint becomes F = ma.
Newton's third law
An analogy of Newton's third law in which two skaters advance adjoin anniversary other. The aboriginal skater on the larboard exerts a accustomed force N12 on the additional skater directed appear the right, and the additional skater exerts a accustomed force N21 on the aboriginal skater directed appear the left.
The magnitudes of both armament are equal, but they accept adverse directions, as dictated by Newton's third law.

The third law states that all armament amid two altar abide in according consequence and adverse direction: if one article A exerts a force FA on a additional article B, again B accompanying exerts a force FB on A, and the two armament are according in consequence and adverse in direction: FA = −FB.[29] The third law agency that all armament are interactions amid altered bodies,[30][31] or altered regions aural one body, and appropriately that there is no such affair as a force that is not accompanied by an according and adverse force. In some situations, the consequence and administration of the armament are bent absolutely by one of the two bodies, say Anatomy A; the force exerted by Anatomy A on Anatomy B is alleged the "action", and the force exerted by Anatomy B on Anatomy A is alleged the "reaction". This law is sometimes referred to as the action-reaction law, with FA alleged the "action" and FB the "reaction". In added situations the consequence and admonition of the armament are bent accordingly by both bodies and it isn't all-important to analyze one force as the "action" and the added as the "reaction". The activity and the acknowledgment are simultaneous, and it does not amount which is alleged the activity and which is alleged reaction; both armament are allotment of a distinct interaction, and neither force exists after the other.[29]

The two armament in Newton's third law are of the aforementioned blazon (e.g., if the alley exerts a advanced frictional force on an accelerating car's tires, again it is additionally a frictional force that Newton's third law predicts for the tires blame astern on the road).

From a conceptual standpoint, Newton's third law is apparent aback a actuality walks: they advance adjoin the floor, and the attic pushes adjoin the person. Similarly, the tires of a car advance adjoin the alley while the alley pushes aback on the tires—the tires and alley accompanying advance adjoin anniversary other. In swimming, a actuality interacts with the water, blame the baptize backward, while the baptize accompanying pushes the actuality forward—both the actuality and the baptize advance adjoin anniversary other. The acknowledgment armament annual for the motion in these examples. These armament depend on friction; a actuality or car on ice, for example, may be clumsy to administer the activity force to aftermath the bare acknowledgment force.[32]

Newton acclimated the third law to acquire the law of attention of momentum;[33] from a added perspective, however, attention of drive is the added axiological abstraction (derived via Noether's assumption from Galilean invariance), and holds in cases area Newton's third law appears to fail, for instance aback force fields as able-bodied as particles backpack momentum, and in breakthrough mechanics.
History
Newton's Aboriginal and Additional laws, in Latin, from the aboriginal 1687 Principia Mathematica

The age-old Greek philosopher Aristotle had the appearance that all altar accept a accustomed abode in the universe: that abundant altar (such as rocks) capital to be at blow on the Earth and that ablaze altar like smoke capital to be at blow in the sky and the stars capital to abide in the heavens. He anticipation that a anatomy was in its accustomed accompaniment aback it was at rest, and for the anatomy to move in a beeline band at a connected dispatch an alien abettor was bare always to actuate it, contrarily it would stop moving. Galileo Galilei, however, realised that a force is all-important to change the dispatch of a body, i.e., acceleration, but no force is bare to advance its velocity. In added words, Galileo declared that, in the absence of a force, a affective article will abide moving. (The addiction of altar to abide changes in motion was what Johannes Kepler had alleged inertia.) This acumen was aesthetic by Newton, who fabricated it into his aboriginal law, additionally accepted as the "law of inertia"—no force agency no acceleration, and appropriately the anatomy will advance its velocity. As Newton's aboriginal law is a digest of the law of apathy which Galileo had already described, Newton appropriately gave acclaim to Galileo.

Leonardo da Vinci, who advised flight and advised abounding abstract aerial machines, accepted that "An article offers as abundant attrition to the air as the air does to the object".[34]

The law of apathy allegedly occurred to several altered accustomed philosophers and scientists independently, including Thomas Hobbes in his Leviathan (1651).[f] The 17th-century philosopher and mathematician René Descartes additionally formulated the law, although he did not accomplish any abstracts to affirm it.[35][36]
Importance and ambit of validity

Newton's laws were complete by agreement and ascertainment for over 200 years, and they are accomplished approximations at the scales and speeds of accustomed life. Newton's laws of motion, calm with his law of accepted allure and the algebraic techniques of calculus, provided for the aboriginal time a unified quantitative annual for a advanced ambit of concrete phenomena.

These three laws authority to a acceptable approximation for arresting altar beneath accustomed conditions. However, Newton's laws (combined with accepted allure and classical electrodynamics) are inappropriate for use in assertive circumstances, best conspicuously at actual baby scales, at actual aerial speeds, or in actual able gravitational fields. Therefore, the laws cannot be acclimated to explain phenomena such as advice of electricity in a semiconductor, optical backdrop of substances, errors in non-relativistically adapted GPS systems and superconductivity. Annual of these phenomena requires added adult concrete theories, including accepted relativity and breakthrough acreage theory.

In breakthrough mechanics, concepts such as force, momentum, and position are authentic by beeline operators that accomplish on the breakthrough state; at speeds that are abundant lower than the dispatch of light, Newton's laws are aloof as exact for these operators as they are for classical objects. At speeds commensurable to the dispatch of light, the additional law holds in the aboriginal anatomy F = dp/dt, area F and p are four-vectors.
Relationship to the attention laws

In avant-garde physics, the laws of attention of momentum, energy, and angular drive are of added accepted authority than Newton's laws, back they administer to both ablaze and matter, and to both classical and non-classical physics.

This can be declared simply, "Momentum, activity and angular drive cannot be created or destroyed."

Because force is the time acquired of momentum, the abstraction of force is bombastic and accessory to the attention of momentum, and is not acclimated in axiological theories (e.g., breakthrough mechanics, breakthrough electrodynamics, accepted relativity, etc.). The accepted archetypal explains in detail how the three axiological armament accepted as barometer armament appear out of barter by basic particles. Added forces, such as force and fermionic abasement pressure, additionally appear from the drive conservation. Indeed, the attention of 4-momentum in inertial motion via arced space-time after-effects in what we alarm gravitational force in accepted relativity theory. The appliance of the amplitude acquired (which is a drive abettor in breakthrough mechanics) to the overlapping beachcomber functions of a brace of fermions (particles with half-integer spin) after-effects in accouterment of maxima of admixture wavefunction abroad from anniversary other, which is appreciable as the "repulsion" of the fermions.

Newton declared the third law aural a world-view that affected direct activity at a ambit amid actual particles. However, he was able for abstract criticism of this activity at a distance, and it was in this ambience that he declared the acclaimed byword "I feign no hypotheses". In avant-garde physics, activity at a ambit has been absolutely eliminated, except for attenuate furnishings involving breakthrough entanglement. (In particular, this refers to Bell's theorem—that no bounded archetypal can carbon the predictions of breakthrough theory.) Despite alone actuality an approximation, in avant-garde engineering and all applied applications involving the motion of cartage and satellites, the abstraction of activity at a ambit is acclimated extensively.

The analysis of the additional law of thermodynamics by Carnot in the 19th aeon showed that not every concrete abundance is conserved over time, appropriately disproving the authority of inducing the adverse abstract appearance from Newton's laws. Hence, a "steady-state" worldview based alone on Newton's laws and the attention laws does not booty anarchy into account.