The laws in Newton's law of motion are independent of each other, and the internal logic conforms to self consistent consistency.Its scope of application isclassical mechanicsScope, applicable conditions areparticle、Inertial reference frameas well asmacroscopic、low speedSports problems.Newton's law of motion explainsNewtonian mechanicsThis paper expounds the basic laws of motion in classical mechanics, and is widely used in various fields.[3-4]
Under the action of, the rate of change of its momentum with time is proportional to the external force on the particle, and the direction of the external force is the same;The formula is:
。
According to the definition of momentum,
。
If the mass of the particle does not change with time (i.e
）, the acceleration of the particle motion is proportional to the magnitude of the external force acting on the particle, and the direction of the acceleration is the same as the direction of the external force;The formula is:
The acting force and reaction force between the two interacting particles are always equal in magnitude and opposite in direction, acting on the same straight line;
The formula is:
(where
Represents the force exerted on particle 1 by particle 2,
Represents the reaction force of particle 2 on particle 1).[1][3][8]
（Isaac NewtonOn《Mathematical Principles of Natural Philosophy》The original statement of the law is almost identical with the above statement;The book also gives six inferences based on the above laws.[9]The component form of Newton's law of motion and its application inNatural coordinate systemSee the sub entries for the forms in other coordinate systems.[10]The form of this law may also be changed when it is quoted in various versions of textbooks.[3][11]）
Law characteristics
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Laws in Newton's law of motionmutually independent：
Newton's first law of motion prepared the concept for the subsequent laws and qualitatively clarifiedpowerandmotionRelationship.In particular, the first law states that "objects are not subject toexternal forceThe sum of the external force vector of the object in the second law is zero（Resultant forceThis movement state is different, and the first law cannot be regarded as the second law
The motion of can only be thoroughly explained as inertial motion by the first law itself（staticorUniform linear motion）。The first law is a completely independent basic law. No other law can solve the problem with it. The second and third laws cannot replace the first law at all.[12-13]
Newton's second law of motion introducedInertial mass, which comprehensively and completely depicts the generation of objects due to forceacceleration, and acceleration vsexternal forceandqualityThe quantitative relationship between the second law and the first law constitutes the profound connotation and fundamental reason for the independence of the second law from the third law.[12][14-15]
Newton's third law of motion cannot be deduced from the second law, nor can the second law replace the third law, nor can the first law replace the third law;The third law is also a law proposed by Newton on the basis of Galileo's previous point of view.The correctness of the third law depends on a lot of practice.The third law is actually expressed by forceLaw of conservation of momentumThe law of conservation of momentum is one of the few basic physical laws that are universally held in nature,Momentum conservationIn any physical field (taking into accountelectromagnetic fieldOfmomentumThe momentum of moving electrons and electromagnetic fields is also conserved).[12][16]
The Internal Logic of Newton's Law of MotionConsistent with self consistencyThat is, the three laws are consistent with each other to form an organic whole:
Newton's law of motion presents a progressive relationship on the research object.The first and second laws only study a single object (it can be only one object or one object can be isolated from many objects as the research object) to solve the motion problem after it is not subject to force or many forces;The third law extends the research object, at least the research is the interaction between two objects, which restricts or affects the movement of the research object or other objects.Only the organic combination of the first, second and third laws can solve all the complexitydynamicsThe problem is solved from the dynamics of particlesParticle system、rigid body、fluid、Vibration、waveWaitingDynamicsQuestion.[12][17-19]
Newton's laws of motion are only determined in the first lawInertial reference frameEstablishment.NewtonianAbsolute space-time viewAlthough it is difficult to have a logical cycle (or logical identity) in the inertial system of, it is essential for the theoretical system in the language expression of dynamic forces.All dynamic problems can be solved once the inertial frame is determined.Since no science can achieve absolute truth, mechanics is also a science with a high degree of approximation. Absolute inertial frames do not exist, but approximate inertial frames always exist.Newton's law of motion is only applicable in the inertial system, which shows the consistency of the three laws.[12][19-20]
The first law introduces the concept and explanation of forceinertiaAttributes, which qualitatively reveal the relationship between force and motion, lay the foundation for the second law and prepare necessary concepts;The third law further gives the nature of the force and reveals the mutual restriction mechanism of the motion of objects.The combination of the three laws comprehensively solves the motion problem of any object under the action of complex external forces.Newton's law of motion is an organic whole, a complete theoretical system coming down in one continuous line, and a basic axiom of mechanics. It is deduced from themMomentum theorem、Law of conservation of momentum、Kinetic energy theorem、Law of conservation of mechanical energy、Theorem of moment of momentum、Conservation law of angular momentumAnd further confirmed the compatibility and consistency of the kinetic axiomatic system.[12][17-20]
Deductive Verification
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Main theoretical derivation or experimental verification methods of Newton's law of motion
Methodology Overview
Illustration
Newton's first law of motion
Galileo's ideal inclined plane experiment:
The Circular Demonstration of the Logical Identity of Newton's First Law of Motion[21], the law can be theoretically deduced through ideal experiments[22]。
In reality, when the ball rolls down the slope, its speed increases, and when it rolls up, its speed decreases.It can be seen from this that when the ball rolls along the horizontal plane, the speed should not change.
But in fact, due to the existence ofFrictional resistance, the ball speed will become slower and slower until it finally stops, and the smoother the surface, the farther the ball will roll.It can be seen from this that if there is no frictional resistance, the ball will roll forever.
If the ball rolls down from a static state along a smooth slope, the ball will roll up another slope to the original height and then roll down;After reducing the slope angle, the ball still reaches the same height on the other slope but rolls farther.It can be seen from this that when the inclined plane is laid flat, the ball will roll down forever.
That is, force is not the reason for maintaining the motion (speed) of the object.Once an object has a certain speed and is not subject to external force, it will keep the speed constant and move in a straight line.[23-24]
Galileo's ideal inclined plane experiment[24]
Newton's second law of motion
Verify with the dot timer:
When studying the relationship between the acceleration of the system and the mass and tension of the systemticker-tape timer Fix it on one end of the board, tie the weight and trolley on both ends of the thin line, the thin line crosses the pulley, and the weight of the weight is used as the pull force, so that the trolley dragging the paper tape moves on a flat plane, then the trolley and the weight on it, and the hook tied on the other end of the line form a motion system.
The tensile force and system quality must be noted on the paper tape for each test.
In order to offset friction, two methods are usually adopted as shown in the right figure: inclined sliding method and horizontal pulling method.[25]
Inclined sliding method and horizontal pulling method[25]
Verify on the air cushion guide rail that:
takeAir trackAfter leveling (due to certain bending of the guide rail and resistance between the slider and the guide rail, leveling is generally measured by the equal speed of the slider when it passes through two photoelectric gates in the experiment), measureViscous dampingConstant b.
In order to correct the velocity loss caused by the existence of viscous friction resistance, it is necessary to determine the viscous damping constant.
In order to eliminate viscous damping, the following two methods are usually adopted: inclined guide rail method (as shown in the right figure) and vibration method.[25]
Inclined guideway method[25]
Verify with nonlinear regression method:
Even if Newton's second law of motion is verified on the air cushion guide rail, there will beair resistanceAs the main influencing factor, it affects the experimental measurement accuracy.This requires an attempt to reduce the impact to a negligible level through correction.But often usedOne variable linear regression method, is not enough to explain the quality of the whole regression equation;The binary linear regression method also has some problems.[26]
useNonlinear regressionTo verify the law, first linearize the dynamic model of particle motion, obtain the linear estimate of model parameters, and take it asnon-linear model The nonlinear regression analysis was carried out on the dynamic model with the initial value of.[27]
The nonlinear regression method has verified the correctness of the law, improved the traditional experimental method of verifying the law, and has certain application and promotion value.[27]
Nonlinear fitting diagram of Newton's second law of motion[27]
In addition, the verification of Newton's second law of motion is also based onLabVIEWTeaching platform of[28], based on wireless module andVisual BasicDesign of simulation demonstration experiment based on matlab[29], based onPhotoelectric sensorExperimental device of[30]Etc.
Use twoforce sensor And keep twosensorOn the same plane, let the force measuring hooks of the two sensors hook or touch each other.Through the data acquisition software, two force time plots are obtained, as shown in Figure 1 and Figure 2 on the right;At the same time, get theForceandReaction forceReal time data over time.
Through observation, we can see the corresponding relationship between the acting force and reaction force and time: at any time, the magnitude of these two forces is basically the same.This means that the two forces are equal in magnitude.
This experimental scheme is not only applicable to quantitativeInteraction forceIt is also suitable for quantifying the interaction force on the vertical plane or the same plane at any angle with the vertical direction. As long as it is on the same plane with two forces, it can accurately simulate the force and reaction force, which reflects that the force and reaction force between two objects are always equal and act on the same line,More intuitively and effectively highlight the universality of Newton's third law of motion.[31]
Take a thin glass tube about 15cm long with openings at both ends. The diameter of the tube is about 3mm (it can let matches in and out).
Put two matches into the tube, make the match heads contact each other in the middle of the tube, and then put them flat.Use an alcohol lamp to aim at the match head for heating, and the glass tube will be heated soon.When the match head reaches the ignition point, it burns rapidly, and the gas compresses each other. Two match sticks fly out from the openings of the two pipes at the same time, and it is observed that the combustion continues.
The flying distance from two match heads is approximately equal, which indicates that the force between objects is mutual.This directly verified Newton's third law of motion.[32]
Observation legal experiment[32]
(There are many kinds of verification experiments for Newton's law of motion. This section only selects several important or typical experiments as examples. With the use of modern experimental facilities, the original experimental methods will be improved or supplemented.[33-35]）
Applicable conditions
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Newton's law of motion is based onNewtonian mechanicsBasic assumptions of: ①spaceIt is absolute and can be regarded as an abstract space in mathematics, independent of the filling materials in the space; ②timeIt is continuous, uniform and endless; ③Time and space are independent; ④Time has nothing to do with movement state; ⑤The mass of an object has nothing to do with its state of motion.General relativityThere is a breakthrough in the first assumption,Special relativityBreaking through the third, fourth and fifth assumptions.[6][36]Therefore:
Newton's law of motion only applies toparticleThe object referred to in Newton's law of motion is a particle.yesParticle system, the second law of Newton's law of motion is generally usedIsolation method, or Newton's second law of motion for the particle system.[3][14][37]For force nonConstant forceSituations such as time, velocity, or position dependent forces, applyintegralNewton's law of motion can also be used.[38]
Newton's law of motion only applies toInertial reference frame。Stationary or moving in a uniform straight line with respect to an isolated particleReference systemIs the inertial reference system.[4][39]In the non inertial reference system, Newton's law of motion is not applicable, because objects not subject to external forces may also have acceleration in this reference system, which is contrary to Newton's first law of motion;Newton's law of motion is applicable only in inertial reference systems.But through the introduction of inertial force, the second law of Newton's law of motion can be expressed inNon inertial frameWhere applicable, that is, the mechanical equation is used
Solve mechanical problems, where
Is the resultant force on the object measured in the inertial system,
Is the inertial force measured in a non inertial system（
Is the acceleration of non inertial system).[14][40-42]
Newton's law of motion only appliesmacroscopicproblem。When the motion linearity of the object under investigation can beDe Broglie waveCompared with quasi time, it can be seen from the uncertainty relationship of particle motion that the momentum and position of the object can no longer be accurately known at the same time, so Newton's dynamic equation can not be solved due to the lack of accurate initial conditions, that is, the classical description method has failed or needs to be modified due to the uncertainty relationship of particle motion.For an action close to or less thanPlanck constantH=6.6 × 10 ⁻ ³ ≮ J · s microscopic particles (or adimensionNear or less thanatomObjects with linearity α=10 ⁻8315m[10]）, must usequantum mechanics。[39]For quantum mechanicsHilbert spaceThe concept of state vector in replaces the concepts of position and momentum (or speed) to describe the state of an object (i.ewave function）, usingSchrodinger equationInstead of Newton's dynamic equation (i.e. containingforce fieldSpecific form of Newton's second law of motion).useState vectorreplacepositionandmomentumThe reason is that the accurate information of position and momentum cannot be known at the same time due to the uncertainty principle[105], but the probability distribution of position and momentum can be known;The limitation of the uncertainty principle to the measurement accuracy lies in a certain relationship between the probability distribution of the two.[14][43-44]
Newton's law of motion only appliesLow speed problem。If the velocity v of the object is equal tolight speedC=3.0 × 10 ⁸ m/s must be used when approachingSpecial relativity。[39]Newton's law of motion forGalileo transformationyescovariantOf, but forLorentz transformationIt is not covariant, so it cannot be compatible with special relativity.When an object moves at a high speed, the definitions of mechanical variables such as force and speed need to be modified todynamicsThe equation can meet the requirements of Lorentz covariant, and it will also be linked withclassical mechanicsThere are differences.[14][45-49]
Newton's law of motion hasIntrinsic randomness。It contains far more "uncertain behaviors" than "certain behaviors" given by it, especially inConservative systemandDissipative systemMedium.[50-51]
(The three laws in Newton's law of motion are independent and have their own scope of application. Minor changes in the different expressions of each law will also change their own scope of application. For specific expressions and the scope of application of the corresponding extended form, please refer to each independent term.[12][52-58]）
Development History
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In the 5th century BC, ancient Greek philosophersDemocritus(Leucippus, 500 BC-440 BC)Epicurus(Epicurus, 341 BC-270 BC) believed that: "WhenatomWhen they are carried forward in the void and nothing collides with them, they must move at the same speed. "This is just a result of conjecture or speculation.[22]
In the 4th century BC, ancient Greek philosophersAristotle（Aristotle，384 BC-322 BC) pointed out that static is the natural state of an object. If there is no force, there will be no movement（Force is the reason for maintaining the movement of objects）。This view is wrong because it misses the key point that "force can make an object stop moving and can also make an object start moving".[59]But for the first time, he proposed the relationship between force and motion, and made some contributions to the development of mechanics.[7]
In the 6th century, Greek scholar Philopenos（J.Philoponus）He was critical of Aristotle's theory of movement.He believed that the projectile itself had some power to push the object forward until it was exhausted. This view later developed into the "impact theory" of the 14th century.[22]
French philosopher in the 14th centuryBridan（Jean Buridan，1295-1358？）、Albert, Nicole Orisim（Nicole Oresme，1320？-1382) and others put forward the "Impulse Theory". They believed that: "When the driver pushes an object to move, he will apply a certain impulse or power to it. The higher the speed is, the greater the impulse is. When the impulse is exhausted, the object will stop." This theory is an Italian physicistGalileo Galilei （Galileo Galilei，1564-1642) and British physicistsIsaac Newton（Isaac Newton，1643-1727).[22]
In the 17th century, Galileo repeatedly proposed in his works similar toPrinciple of inertiaThe statement of.In 1632 and 1638, respectively《Dialogue Concerning the Two Chief World Systems 》And《A Dialogue on Two New Sciences of Mechanics and Position Motion》The ideal slope experiment is recorded in（A small ball rolls to the horizontal along the inclined platform, the smoother the surface, the farther the ball rolls[60]）And reasoned that "if there is a sufficiently long and absolutely smooth surface, there will be nothing（friction）It can block the movement of the ball, so the ball continues to move or until there is something（external force）It can be concluded that "objects in their natural state will maintain their original motion rather than tend to stop".This conclusion breaks the old idea that "force is the reason for maintaining the motion of objects" for about 1300 years since Aristotle, but it still has its influence.This conclusion is very close to the law of inertia（Newton's first law of motion, also known as the law of inertia, was first discovered by Galileo[3]）。[7][60-61]
In 1644, French physicist Rene Descartes（Rene Descartes，1596-1650) made up for Galileo's shortcomings in Philosophical Principles.[62]He clearly pointed out that unless an object is affected by external factors, it will always remain stationary or in motion, and specifically stated that an inertia moving object will never make itself move towards a curve, but only keep moving on a straight line.He expressed this basic principle as two laws: ① Each individual particle of matter will continue to remain the same state until it collides with other particles and is forced to change this state; ②All the motions themselves follow a straight line.However, Descartes did not establish the system that he tried to establish to deduce various natural phenomena, but his thoughts had a certain impact on Newton's summary of such laws.Descartes' greatest contribution lies in his first recognition that force is the reason for changing the state of motion of objects.[7][63]
In 1662, Galileo pointed out that "the speed of an object moving at any speed can be kept unchanged as long as the external causes of acceleration or deceleration are removed." Descartes also believed that "when there is no external action, particles either move at a uniform speed or remain stationary." Newton took this assumption asNewton's first law of motionHe further extended Galileo's thought to powerful occasions and proposedNewton's second law of motion。[7][64]
In 1664, Newton receivedcollisionThe influence of Descartes, who studied the problem earlier, also began to study the inelasticity of two spheresrigid bodyThe collision problem of.one thousand six hundred and sixty-five-In 1666, Newton studied the collision between two spherical rigid bodies.He didn't focus onmomentumandMomentum conservationThe aspect focuses on the interaction between objects.For the collision of two rigid bodies, he proposed: "In the time when they move towards each other (that is, the moment when they collide)Their pressure is at the maximum value,... their entire movement is blocked by the pressure between them at this moment,... as long as these two objects do not yield to each other, they will hold the same strong pressure between them,... they will move away from each other as much as they did before bouncing back. "[65]
one thousand six hundred and sixty-eight-In 1669, Dutch physicistChristian Huygens（Christiaan Huygens，1629-1695), Wallis（willis）And British physicistsChristopher Ryan（Christopher Wren，1632-1723)collisionA lot of research has been done and some important conclusions have been drawn.[66]Among them, Huygens' work is more outstanding. He proved that the momentum of two hardware bodies in the same direction remains unchanged during collision, corrected the error that Cartesian did not consider the directionality of momentum, and proposed for the first time theMomentum conservation。Newton formally proposedNewton's third law of motionThey affirmed their work and pointed out their limitations.Newton believed that: "The theory of Ryan and Huygens is based on the premise of absolute hard objects, and more positive results can be obtained by using ideal elastic bodies, and the theory of nonIdeal elastomerFor example, the compressed wooden ball, steel ball and glass ball are tested, and the results are consistent after the error is eliminated. "[65]
French physicist in 1673Mariotte（EdmeMarotte，1620-1684) use twoSimple pendulumBy doing collision experiments, the instantaneous velocity before and after the collision was measured ingeniously.Newton also repeated the experiment, and he further discussedair resistanceAnd the improvement methods, and the results were revised.[61]
Since August 1684, physicists in BritainEdmundo Halley（EdmondHalley，1656-1742), Newton began to write《Mathematical Principles of Natural Philosophy》, systematize manuscripts and reconsider some issues.In November 1685, two volumes of monographs were formed.On July 5, 1687, The Principle was published in Latin.Newton's law of motion is put forward in the section of axioms or laws of motion in the introduction of Principles, which breaks away from the shackles of old ideas.In 1713, The Principle was published in the second edition;In 1725, the third edition was published.[7][67-71]
German physicist in the second half of the 19th centuryGustav Robert Kirhoff（Gustav Robert Kirchhoff，1824-1887), Austrian and Czech physicistsErnst Mach（Ernst Mach，1838-1916), American physicist Eisenbold（L. Eisenbud）Auston, American physicist（N. Austern）They all discussed the expression of Newton's law of motion, and put forward their own amendments.Among them, Mach's "Mechanics in Development"[72]In, Newton's law of motion has been comprehensively investigated and analyzed;Eisenbaud's Law of Motion on Experience[73]The Expression of Newtonian Mechanics[74]A similar new expression is also proposed.[75]However, some of these amendments were questioned by Swiss and American physicistsAlbert Einstein（Albert Einstein，1879-1955), etc.[76]
Since 1905, Einstein'srelativityOverthrew most of the scientific system established by Newton.Einstein pointed out that Newton's law of motion exceededclassical mechanicsScope orparticle、Inertial reference frameas well asmacroscopic, low speed movement, etc.This part has gone beyond the discussion scope of the development history of Newton's law of motion. For further development, please refer toSpecial relativity、General relativityEtc.[76-77]
application area
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Newton's law of motionphysicsAnd other disciplinesOn, widely used:
Newton's law of motion can be solveddynamicsQuestion.[14][78]
When the force condition of an object is known to solve the motion state, or when the motion state is known to solve the force condition, the three conditions (conclusions) of the force condition, acceleration, and motion state are transformed in turn. The solution ideas of the two types of problems are the same, which can be basically divided into the following three steps: ① determine the research object, conduct force analysis or motion state change analysis; ②Establish an appropriate coordinate system, list Newton's law of motion equations, and properly supplement other equations; ③Solve the equation and discuss.[79]
In addition to the field of dynamics, Newton's law of motion has applications in other branches of physics.[80-81]
，[82]More intuitive and understandable.[83]For density
Let the volume force density it receives be
, pressure gradient force is
Newton's second law of motion has a special expression in fluid mechanics[84]：
。[85](where
Is gradient operator
。[86]
It may also be recorded as
, at this time
express
Change of directional pressure.[87]）In addition, based on the phenomenological model obtained by modifying the linear law of stress and strainNon Newtonian fluidThe constitutive equation of can be obtained by establishing the momentum balance equation of kinetic energy mass gas diffusion transport based on Newton's law of motion, that is, the universal momentum transport law applicable to non Newtonian flow. This way can also clarify that the essence of some non Newtonian flow phenomena is inertia in the process of energy mass movement.[88]
stayelectromagnetic inductionParallel guide rail model with medium capacitance load[89]The conductor bars connected to different loads will have different motion forms.When capacitive reactance is connected, the capacitor is charged, and the conductor bar is always subject toAmberiIt can flexibly use Newton's law of motion and the same linear vector synthesis for specific physical processes[90]Method to determine the motion state of the rod.[91]
Newton's law of motionDaily life and production practiceThere are also many applications and help:
stayMachinery manufacturingIn the field, Newton's law of motion can help to develop safe and efficient mechanical structures or products.The expression of normal pressure gradient derived from Newton's second law of motion
，[92]Can better explain the wing lift[83]；According to Newton's third law of motion
, which can be designed based on the principle of "air suction (entrainment)"Tailpipe[93]。The above two research results can be widely used to guide the manufacturing and design of aircraft, rockets, vehicles and other moving machinery, and will greatly help to improve their propulsion efficiency.[38][60]
In the field of information sociology, the derivation of relevant concepts of information sociology and the adaptation of theorems can be completed by using the thinking method of Newton's law of motion for reference, and the new laws obtained can guide the current situation and trend of library and information work.[94]
stayMental health educationIn the field, Newton's three laws of motion can respectively correspond to three educational links of "determination", "self-cultivation" and "advocating benevolence".In Newtonian mechanics, the three laws are independent of each other and have consistency, integrity and compatibility within the system;In pedagogy, these three links are complementary and harmonious.This has a positive enlightenment on guiding science and engineering students to reshape, optimize and adjust their psychological quality and state.[95-96]
In the financial field, Newton's law of motion can also be used to explain and predict financial development trends.If insharesIn market investment, there are three laws corresponding to Newton's law of motion one by one: ① except for external reasons, the stock price maintains the original trend; ②The growth rate of share price changes in proportion to the market; ③Every buyer is a seller.The forecast and data[97]Basically accurate.[98]
In the field of animation production, because Newton's law of motion indicates that the role of force is the fundamental cause of all movements, and animation is the film and television art that makes the screen move, Newton's law of motion occupies an important position in the animation art and is an indispensable research object in animation.[99]For example, the weight of objects on the screen depends entirely on the distance between animations when they are forced to move, not on the beauty and fidelity of the animation script itself.This requires reasonable use of Newton's law of motion to enhance the realism of animation.[100]
Law influence
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Newton's law of motion isDynamicsThe important law in is researchclassical mechanicseven to the extent thatphysicsThe basic laws of motion in classical mechanics are expounded.[64]The applicable scope of this law isNewton's first law of motionGivenInertial reference frameAnd make people meaningful to the research of physical problems and the measurement of physical quantities.[23][42]
Newton's law of motion refutes theAristotlePeople's wrong views on the concept of force lay the foundation for establishing a correct concept of force.[101]This law was the first scientificInertial mass、powerThe qualitative definition of several basic concepts in classical mechanics such as Newton's law of motionParticle mechanicsThe system principle laid the conceptual foundation.[1][4][8]
The first law of Newton's law of motion is the premise and foundation of other principles, which lays the conceptual foundation of classical mechanics, thus making it the premise of the first principle in the theoretical system.[101]The second law andMomentum theorem、Functional principleThe relationship between the change of object motion state and the external action is determined.[102]The Third Law andLaw of conservation of momentumAnd so on, relating the motion of related objects;andLaw of universal gravitation, createdCelestial mechanicsIt is the first time for people to have an accurate understanding of the operating laws of the sun, moon and stars;Gives the rightnatural forceThe universal statement of ", which reveals the law of interaction between two objects, provides a theoretical basis for solving mechanical problems and transforming research objects.".[48][103-104]
