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Newton's laws of motion

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Newton's First and Second laws, in Latin, from the original 1687 edition of the Principia

Mathematica.

Classical mechanics

Newton's Second Law

History of ...

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Newtonian mechanics

Lagrangian mechanics

Hamiltonian mechanics

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Newton's laws of motion are threephysical laws that form the basis forclassical

mechanics. They are:

1. In the absence offorce, abody either is at rest or moves in a straight line with

constant speed.

2. A body experiencing a force F experiences anaccelerationa related to F by F =

ma, where m is the mass of the body. Alternatively, force is equal to the timederivative ofmomentum.

3. Whenever a first body exerts a force F on a second body, the second body exerts a

force F on the first body. F and F are equal in magnitude and opposite indirection.[note 1]

These laws describe the relationship between the forces acting on a body and themotion of

that body. They were first compiled by Sir Isaac Newton in his workPhilosophi

Naturalis Principia Mathematica, first published on July 5, 1687.[1] Newton used them toexplain and investigate the motion of many physical objects and systems. [2] For example, in

the third volume of the text, Newton showed that these laws of motion, combined with his

law of universal gravitation, explained Kepler's laws of planetary motion.

First lawThere exists a set ofinertial reference frames relative to which all particles with no

net forceacting on them will move without change in theirvelocity. This law is

often simplified as "A body persists in a state of rest or of uniform motion unlessacted upon by an external force." Newton's first law is often referred to as the law

of inertia.

Second lawObserved from an inertial reference frame, the net force on a particle is equal to the

time rate of change of its linear momentum: F = d(mv)/dt. Since by definition the

mass of a particle is constant, this law is often stated as, "Force equals mass times

acceleration (F = ma): the net force on an object is equal to the mass of the objectmultiplied by its acceleration."

Third law

Whenever a particleA exerts a force on another particleB,B simultaneously exertsa force onA with the same magnitude in the opposite direction. The strong form of

the law further postulates that these two forces act along the same line. This law is
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often simplified into the sentence, "To every action there is an equal and opposite

reaction."

In the given interpretation mass, acceleration, momentum, and (most importantly) force areassumed to be externally defined quantities. This is the most common, but not the only

interpretation: one can consider the laws to be a definition of these quantities.

Some authors interpret the first law as defining what an inertial reference frame is; from

this point of view, the second law only holds when the observation is made from an inertialreference frame, and therefore the first law cannot be proved as a special case of the

second. Other authors do treat the first law as a corollary of the second.[3]The explicit

concept of an inertial frame of reference was not developed until long after Newton's death.

At speeds approaching the speed of lightthe effects ofspecial relativity must be taken intoaccount.[note 2]

Newton's first law

Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in

directum, nisi quatenus a viribus impressis cogitur statum illum mutare.Every bodypersists in its state of being at rest or of moving uniformly straight forward, except

insofar as it is compelled to change its state by force impressed.[4]

Newton's first law is also called the law ofinertia. It states that if the vector sum of all

forces (that is, the net force) acting on an object is zero, then the acceleration of the object

is zero and its velocity is constant. Consequently:

An object that is not moving will not move until a force acts upon it. An object that is moving will not change its velocity until a net force acts upon it.

The first point needs no comment, but the second seems to violate everyday experience.

For example, a hockey puck sliding along ice does not move forever; rather, it slows and

eventually comes to a stop. According to Newton's first law, the puck comes to a stopbecause of a net external force applied in the direction opposite to its motion. This net

external force is due to a frictional force between the puck and the ice, as well as a

frictional force between the puck and the air. If the ice were frictionless and the puck weretraveling in a vacuum, the net external force on the puck would be zero and it would travel

with constant velocity so long as its path were unobstructed.

Implicit in the discussion of Newton's first law is the concept of an inertial reference frame,

which for the purposes of Newtonian mechanics is defined to be areference frame in whichNewton's first law holds true.

There is a class of frames of reference (called inertial frames) relative to which the motion

of a particle not subject to forces is a straight line.[5]

Newton placed the law of inertia first to establish frames of reference for which the otherlaws are applicable.[5][6] To understand why the laws are restricted to inertial frames,

consider a ball at rest inside an airplane on a runway. From the perspective of an observer

within the airplane (that is, from the airplane's frame of reference) the ball will appear tomove backward as the plane accelerates forward. This motion appears to contradict

Newton's second law (F = ma), since, from the point of view of the passengers, there

appears to be no force acting on the ball that would cause it to move. However, Newton's

first law does not apply: the stationary ball does not remain stationary in the absence ofexternal force. Thus the reference frame of the airplane is not inertial, and Newton's second

law does not hold in the form F = ma.[note 3]

History of the first law

Newton's first law is a restatement of what Galileo had already described and Newton gave

credit to Galileo. It differs from Aristotle's view that all objects have a natural place in theuniverse.Aristotle believed that heavy objects like rocks wanted to be at rest on the Earth
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and that light objects like smoke wanted to be at rest in the sky and the stars wanted to

remain in the heavens. However, a key difference between Galileo's idea and Aristotle's is

that Galileo realized that force acting on a body determines acceleration, not velocity. Thisinsight leads to Newton's First Lawno force means no acceleration, and hence the body

will maintain its velocity.

The law of inertia apparently occurred to several different natural philosophers and

scientists independently. The inertia of motion was described in the 3rd century BC by theChinese philosopherMo Tzu, and in the 11th century by the Muslim scientistsAlhazen[7]

and Avicenna.[8] The 17th century philosopherRen Descartesalso formulated the law,

although he did not perform any experiments to confirm it.

The first law was understood philosophically well before Newton's publication of the law.[note 4]

Newton's second law

Newton's second law states that the force applied to a body produces a proportional

acceleration; the relationship between the two is

where F is the force applied, m is the mass of the body, and a is the body's acceleration. If

the body is subject to multiple forces at the same time, then the acceleration is proportional

to the vector sum(that is, the net force):

The second law can also be shown to relate the net force and themomentump of the body:

Therefore, Newton's second law also states that the net force is equal to the time derivativeof the body's momentum:

Consistent with the first law, the time derivative of the momentum is non-zero when the

momentum changes direction, even if there is no change in its magnitude (see time

derivative). The relationship also implies the conservation of momentum: when the net

force on the body is zero, the momentum of the body is constant.

Both statements of the second law are valid only for constant-mass systems,[9][10][11] since

any mass that is gained or lost by the system will cause a change in momentum that is not

the result of an external force. A different equation is necessary forvariable-mass systems.

Newton's second law requires modification if the effects ofspecial relativity are to be taken

into account, since it is no longer true that momentum is the product of inertial mass andvelocity.

Impulse

An impulseI occurs when a force F acts over an interval of time t, and it is given by[12][13]

Since force is the time derivative of momentum, it follows that
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This relation between impulse and momentum is closer to Newton's wording of the second

law.[14]

Impulse is a concept frequently used in the analysis of collisions and impacts.[15]

Variable-mass systems

Variable-mass systems, like a rocket burning fuel and ejecting spent gases, are not closed

and cannot be directly treated by making mass a function of time in the second law. [10] The

reasoning, given inAn Introduction to Mechanics by Kleppner and Kolenkow and othermodern texts, is that Newton's second law applies fundamentally to particles.[11] In classical

mechanics, particles by definition have constant mass. In case of a well-defined system of

particles, Newton's law can be extended by summing over all the particles in the system:

where Fnet is the total external force on the system,Mis the total mass of the system, and

acm is the acceleration of the center of mass of the system.

Variable-mass systems like a rocket or a leaking bucket cannot usually be treated as asystem of particles, and thus Newton's second law cannot be applied directly. Instead, the

general equation of motion for a body whose mass m varies with time by either ejecting or

accreting mass is obtained by rearranging the second law and adding a term to account forthe momentum carried by mass entering or leaving the system:[9]

where u is the relative velocity of the escaping or incoming mass with respect to the center

of mass of the body. Under some conventions, the quantity u dm/dton the left-hand side isdefined as a force (the force exerted on the body by the changing mass, such as rocket

exhaust) and is included in the quantity F. Then, by substituting the definition of

acceleration, the equation becomes

History of the second law

Newton's Latin wording for the second law is:

Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum

lineam rectam qua vis illa imprimitur.

This was translated quite closely in Motte's 1729 translation as:

LAW II: The alteration of motion is ever proportional to the motive force impress'd; and ismade in the direction of the right line in which that force is impress'd.

According to modern ideas of how Newton was using his terminology,[note 5] this is

understood, in modern terms, as an equivalent of:

The change of momentum of a body is proportional to the impulse impressed

on the body, and happens along the straight line on which that impulse is

impressed.

Motte's 1729 translation of Newton's Latin continued with Newton's commentary on the

second law of motion, reading:

If a force generates a motion, a double force will generate double the motion, a triple forcetriple the motion, whether that force be impressed altogether and at once, or gradually and
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successively. And this motion (being always directed the same way with the generating

force), if the body moved before, is added to or subtracted from the former motion,

according as they directly conspire with or are directly contrary to each other; or obliquelyjoined, when they are oblique, so as to produce a new motion compounded from the

determination of both.

The sense or senses in which Newton used his terminology, and how he understood the

second law and intended it to be understood, have been extensively discussed by historiansof science, along with the relations between Newton's formulation and modern

formulations.[note 6]

Newton's third law: law of reciprocal actions

Newton's third law. The skaters' forces on each other are equal in magnitude, but act in

opposite directions.Lex III: Actioni contrariam semper et qualem esse reactionem: sive corporum

duorum actiones in se mutuo semper esse quales et in partes contrarias dirigi.

''To every action there is always an equal and opposite reaction: or the forces of

two bodies on each other are always equal and are directed in opposite

directions''.

A more direct translation than the one just given above is:

LAW III: 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. Whatever draws or presses another is as much drawn or pressed by that other. If you press

a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied

to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for thedistended rope, by the same endeavour to relax or unbend itself, will draw the horse as

much towards the stone, as it does the stone towards the horse, and will obstruct the

progress of the one as much as it advances that of the other. If a body impinges upon

another, and by its force changes the motion of the other, that body also (because of theequality of the mutual pressure) will undergo an equal change, in its own motion, toward

the contrary part. The changes made by these actions are equal, not in the velocities but in

the motions of the bodies; that is to say, if the bodies are not hindered by any otherimpediments. For, as the motions are equally changed, the changes of the velocities made

toward contrary parts are reciprocally proportional to the bodies. This law takes place also

in attractions, as will be proved in the next scholium.[note 7]

In the above, as usual, motion is Newton's name for momentum, hence his carefuldistinction between motion and velocity.

The Third Law means that all forces are interactions, and thus that there is no such thing as

a unidirectional force. If bodyA exerts a force on bodyB, simultaneously, bodyB exerts a

force of the same magnitude bodyA, both forces acting along the same line. As shown inthe diagram opposite, the skaters' forces on each other are equal in magnitude, but act in

opposite directions. Although the forces are equal, the accelerations are not: the less

massive skater will have a greater acceleration due to Newton's second law. It is important

to note that the action and reaction act on different objects and do not cancel each other out.The two forces in Newton's third law are of the same type (e.g., if the road exerts a forward

frictional force on an accelerating car's tires, then it is also a frictional force that Newton'sthird law predicts for the tires pushing backward on the road).

Newton used the third law to derive the law ofconservation of momentum;[16]however

from a deeper perspective, conservation of momentum is the more fundamental idea

(derived viaNoether's theorem from Galilean invariance), and holds in cases whereNewton's third law appears to fail, for instance when force fields as well as particles carry

momentum, and inquantum mechanics.
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Importance and range of validity

Newton's laws were verified by experiment and observation for over 200 years, and they

are excellent approximations at the scales and speeds of everyday life. Newton's laws ofmotion, together with his law ofuniversal gravitation and the mathematical techniques of

calculus, provided for the first time a unified quantitative explanation for a wide range ofphysical phenomena.

These three laws hold to a good approximation for macroscopic objects under everydayconditions. However, Newton's laws (combined with Universal Gravitation andClassical

Electrodynamics) are inappropriate for use in certain circumstances, most notably at very

small scales, very high speeds (in special relativity, theLorentz factormust be included inthe expression for momentum along withrest mass and velocity) or very strong

gravitational fields. Therefore, the laws cannot be used to explain phenomena such as

conduction of electricity in a semiconductor, optical properties of substances, errors in non-

relativistically corrected GPS systems andsuperconductivity. Explanation of these

phenomena requires more sophisticated physical theory, including General RelativityandRelativistic Quantum Mechanics.

Inquantum mechanicsconcepts such as force, momentum, and position are defined bylinearoperatorsthat operate on thequantum state; at speeds that are much lower than the

speed of light, Newton's laws are just as exact for these operators as they are for classical

objects. At speeds comparable to the speed of light, the second law holds in the original

form F = dp/dt, which says that the force is the derivative of the momentum of the objectwith respect to time, but some of the newer versions of the second law (such as the constant

mass approximation above) do not hold at relativistic velocities.

Relationship to the conservation laws

In modern physics, the laws ofconservationofmomentum,energy, andangular momentumare of more general validity than Newton's laws, since they apply to both light and matter,

and to both classical and non-classical physics.

This can be stated simply, "Momentum, energy and angular momentum cannot be created

or destroyed."

Because force is the time derivative of momentum, the concept of force is redundant and

subordinate to the conservation of momentum, and is not used in fundamental theories (e.g.

quantum mechanics,quantum electrodynamics,general relativity, etc.). The standardmodelexplains in detail how the three fundamental forces known asgauge forcesoriginateout of exchange byvirtual particles. Other forces such asgravity and fermionic degeneracy

pressure also arise from the momentum conservation. Indeed, the conservation of4-

momentum in inertial motion via curved space-time results in what we call gravitationalforce in general relativity theory. Application of space derivative (which is a momentum

operatorin quantum mechanics) to overlapping wave functions of pair offermions

(particles with semi-integerspin) results in shifts of maxima of compound wavefunctionaway from each other, which is observable as "repulsion" of fermions.

Newton stated the third law within a world-view that assumed instantaneous action at a

distance between material particles. However, he was prepared for philosophical criticismof thisaction at a distance, and it was in this context that he stated the famous phrase "Ifeign no hypotheses". In modern physics, action at a distance has been completely

eliminated, except for subtle effects involving quantum entanglement. However in modern

engineering in all practical applications involving the motion of vehicles and satellites, theconcept of action at a distance is used extensively.

Conservation of energywas discovered nearly two centuries after Newton's lifetime, the

long delay occurring because of the difficulty in understanding the role of microscopic and

invisible forms of energy such as heat and infra-red light.
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.org/wiki/Spin_(physics)http://en.wikipedia.org/wiki/Action_at_a_distance_(physics)http://en.wikipedia.org/wiki/Hypotheses_non_fingohttp://en.wikipedia.org/wiki/Hypotheses_non_fingohttp://en.wikipedia.org/wiki/Quantum_entanglementhttp://en.wikipedia.org/wiki/Conservation_of_energy


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Kinetic theory (or the kinetic orkinetic-molecular theory of gases) is the theory thatgases are made up of a large number of small particles (atoms ormolecules), all of which

are in constant, randommotion. The rapidly moving particles constantly collide with each

other and with the walls of the container. Kinetic theory explains macroscopic properties ofgases, such as pressure, temperature, or volume, by considering their molecular

composition and motion. Essentially, the theory posits that pressure is due not to static

repulsion between molecules, as was Isaac Newton's conjecture, but due to collisions

between molecules moving at different velocities.

While the particles making up a gas are too small to be visible, the jittering motion of

pollen grains or dust particles which can be seen under a microscope, known asBrownian

motion, results directly from collisions between the particle and air molecules. Thisexperimental evidence for kinetic theory, pointed out byAlbert Einstein in 1905, is

generally seen as having confirmed the existence of atoms and molecules.

Postulates

The theory for ideal gases makes the following assumptions:

The gas consists of very small particles, all with non-zero mass.

The number of molecules is large such that statistical treatment can be applied.
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These molecules are in constant, random motion. The rapidly moving particles

constantly collide with the walls of the container.

The collisions of gas particles with the walls of the container holding them areperfectly elastic.

The interactions among molecules are negligible. They exert no forces on one

another except during collisions. The total volume of the individual gas molecules added up is negligible compared

to the volume of the container. This is equivalent to stating that the average distance

separating the gas particles is large compared to theirsize.

The molecules are perfectly spherical in shape, and elastic in nature.

The averagekinetic energy of the gas particles depends only on the temperature of

the system.

Relativistic effects are negligible. Quantum-mechanical effects are negligible. This means that the inter-particle

distance is much larger than the thermal de Broglie wavelength and the molecules

are treated as classicalobjects.

The time during collision of molecule with the container's wall is negligible ascomparable to the time between successive collisions.

The equations of motion of the molecules are time-reversible.

More modern developments relax these assumptions and are based on the Boltzmannequation. These can accurately describe the properties of dense gases, because they include

the volume of the molecules. The necessary assumptions are the absence of quantum

effects, molecular chaos and small gradients in bulk properties. Expansions to higher ordersin the density are known as virial expansions. The definitive work is the book by Chapman

and Enskog but there have been many modern developments and there is an alternative

approach developed by Grad based on moment expansions.[citation needed] In the other limit, for

extremely rarefied gases, the gradients in bulk properties are not small compared to themean free paths. This is known as the Knudsen regime and expansions can be performed in

the Knudsen number.

The kinetic theory has also been extended to include inelastic collisions in granular matterby Jenkins and others.[citation needed]

[edit

] Factors

[edit] Pressure

Pressure is explained by kinetic theory as arising from the force exerted by gas moleculesimpacting on the walls of the container. Consider a gas ofNmolecules, each of mass m,

enclosed in a cuboidal container of volume V=L3. When a gas molecule collides with thewall of the container perpendicular to thex coordinate axis and bounces off in the opposite

direction with the same speed (anelastic collision), then the momentum lost by the particle

and gained by the wall is:

where vx is thex-component of the initial velocity of the particle.

The particle impacts one specific side wall once every

(whereL is the distance between opposite walls).

The forcedue to this particle is:
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The total force on the wall is

where the bar denotes an average over theNparticles. Because of the Pythagorean

, we can rewrite the force as

.

This force is exerted on an areaL2. Therefore the pressure of the gas is

where V=L3 is the volume of the box. The fraction n=N/Vis the number density of the gas

(the mass density=nm is less convenient for theoretical derivations on atomic level).Using n, we can rewrite the pressure as

This is a first non-trivial result of the kinetic theory because it relates pressure, a

macroscopicproperty, to the average (translational) kinetic energy per moleculewhich is amicroscopic property.

[edit] Temperature and kinetic energy

From theideal gas law

(1)

where is theBoltzmann constant, and

the absolutetemperature,

and from the above result

we have

then the temperature

takes the form

(2)

which leads to the expression of the kinetic energy of a molecule

The kinetic energy of the system is N time that of a molecule

The temperature becomes
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(3)

Eq.(3)1 is one important result of the kinetic theory: The average molecular kinetic energyis proportional to the absolute temperature.

From Eq.(1) and Eq.(3)1, we have

(4)

Thus, the product of pressure and volume permole is proportional to the average

(translational) molecular kinetic energy.

Eq.(1) and Eq.(4) are called the "classical results", which could also be derived fromstatistical mechanics; for more details, see[1].

Since there are degrees of freedom (dofs) in a monoatomic-gas system with

particles, the kinetic energy per dof is

(5)

In the kinetic energy per dof, the constant of proportionality of temperature is 1/2 times

Boltzmann constant. This result is related to theequipartition theorem.

As noted in the article onheat capacity, diatomic gases should have 7 degrees of freedom,but the lighter gases act as if they have only 5.

Thus the kinetic energy per kelvin (monatomicideal gas) is:

per mole: 12.47 J

per molecule: 20.7 yJ = 129 eV

Atstandard temperature (273.15 K), we get:

per mole: 3406 J

per molecule: 5.65 zJ = 35.2 meV

[edit] Number of collisions with wall

One can calculate the number of atomic or molecular collisions with a wall of a container

per unit area per unit time.

Assuming an ideal gas, a derivation[2] results in an equation for total number of collisions

per unit time per area:

[edit] RMS speeds of molecules

From the kinetic energy formula it can be shown that
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with v in m/s, Tin kelvins, andR is thegas constant. The molar mass is given as kg/mol.

The most probable speed is 81.6% of the rms speed, and the mean speeds 92.1%

(distribution of speeds).

[edit] History

In 1738 Daniel Bernoulli publishedHydrodynamica, which laid the basis for the kinetic

theory of gases. In this work, Bernoulli positioned the argument, still used to this day, that

gases consist of great numbers of molecules moving in all directions, that their impact on asurface causes the gas pressure that we feel, and that what we experience as heatis simply

the kinetic energy of their motion. The theory was not immediately accepted, in part

because conservation of energy had not yet been established, and it was not obvious tophysicists how the collisions between molecules could be perfectly elastic.

Other pioneers of the kinetic theory (which were neglected by their contemporaries) were

Mikhail Lomonosov(1747),[3]Georges-Louis Le Sage (ca. 1780, published 1818),[4]John

Herapath (1816)[5]

and John James Waterston(1843),[6]

which connected their research withthe development ofmechanical explanations of gravitation. In 1856 August Krnig

(probably after reading a paper of Waterston) created a simple gas-kinetic model, which

only considered the translational motion of the particles. [7]

In 1857 Rudolf Clausius, according to his own words independently of Krnig, developed asimilar, but much more sophisticated version of the theory which included translational and

contrary to Krnig also rotational and vibrational molecular motions. In this same work he

introduced the concept ofmean free path of a particle.[8] In 1859, after reading a paper byClausius, James Clerk Maxwell formulated theMaxwell distribution of molecular

velocities, which gave the proportion of molecules having a certain velocity in a specific

range. This was the first-ever statistical law in physics.[9]

In his 1873 thirteen page article'Molecules', Maxwell states: we are told that an 'atom' is a material point, invested andsurrounded by 'potential forces' and that when 'flying molecules' strike against a solid body

in constant succession it causes what is calledpressureof air and other gases.[10] In 1871,

Ludwig Boltzmann generalized Maxwell's achievement and formulated theMaxwellBoltzmann distribution. Also the logarithmicconnection between entropy andprobability

was first stated by him.

In the beginning of twentieth century, however, atoms were considered by many physicists

to be purely hypothetical constructs, rather than real objects. An important turning pointwas Albert Einstein's (1905)[11] and Marian Smoluchowski's (1906)[12] papers on Brownian

motion, which succeeded in making certain accurate quantitative predictions based on thekinetic theory.

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Teori kinetik (atau kinetik-molekuler atau teori kinetik gas) adalah teori yang menyatakan

bahwa gas terdiri dari sejumlah besar partikel kecil (atom atau molekul), semua yang terus-

menerus, gerakan acak. Partikel bergerak dengan cepat terus-menerus berbenturan dengansatu sama lain dan dengan dinding wadah. Menjelaskan teori kinetik gas sifat-sifat

makroskopik, seperti tekanan, temperatur, atau volume, dengan mempertimbangkan

komposisi molekul mereka dan gerak. Pada dasarnya, teori ini berpendapat bahwa tekananitu disebabkan bukan statis tolakan antara molekul, seperti Isaac Newton's dugaan, namun

karena tumbukan antara molekul yang bergerak pada kecepatan yang berbeda.
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Sementara partikel yang membentuk gas terlalu kecil untuk dapat dilihat, maka gerakan

jittering serbuk sari atau partikel debu yang dapat dilihat di bawah mikroskop, yang dikenal

sebagai gerakan Brown, hasil langsung dari tumbukan antara partikel dan molekul udara.Ini bukti eksperimental untuk teori kinetik, ditunjukkan oleh Albert Einstein pada tahun

1905, umumnya dilihat sebagai memiliki mengukuhkan keberadaan atom dan molekul.

Postulat

Teori gas ideal membuat asumsi sebagai berikut:

gas terdiri dari partikel-partikel yang sangat kecil, semua dengan nol non-massa. Jumlah molekul besar sehingga perlakuan statistik dapat diterapkan.

Molekul-molekul ini terus-menerus, gerakan acak. Partikel bergerak dengan cepat terus-

menerus berbenturan dengan dinding wadah.

Para tabrakan partikel gas dengan dinding wadah memegang mereka adalah elastissempurna.

Interaksi antara molekul dapat diabaikan. Mereka tidak mengerahkan pasukan satu sama

lain kecuali selama tumbukan.

Total volume molekul gas individu dijumlahkan diabaikan dibandingkan dengan volumewadah. Ini sama dengan menyatakan bahwa jarak rata-rata yang memisahkan partikel gas

besar dibandingkan dengan ukuran mereka. Molekul-molekul berbentuk bulat sempurna, dan elastis di alam.

rata-rata energi kinetik partikel gas hanya bergantung pada suhu sistem.

efek relativistik diabaikan. Quantum-efek mekanis dapat diabaikan. Ini berarti bahwa jarak antar partikel jauh lebih

besar daripada panjang gelombang de Broglie termal dan molekul diperlakukan sebagai

benda klasik.

Waktu selama benturan molekul dengan dinding wadah yang diabaikan sebagai sebandingdengan waktu antara tumbukan berturut-turut.

The persamaan gerak dari molekul adalah waktu-reversibel.

Santai perkembangan lebih modern dan asumsi ini didasarkan pada persamaan Boltzmann.Ini dapat secara akurat menggambarkan sifat-sifat padat gas, karena mereka termasuk

volume molekul. Asumsi yang diperlukan adalah tidak adanya efek kuantum, molekul

kekacauan dan gradien kecil dalam massal properti. Perluasan perintah yang lebih tinggidalam kepadatan virial dikenal sebagai ekspansi. Kerja definitif buku oleh Chapman dan

Enskog tetapi ada banyak perkembangan modern dan ada pendekatan alternatif yang

dikembangkan oleh Grad didasarkan pada saat ekspansi. [Rujukan?] Dalam batasan lain,

karena sangat langka gas, maka gradien dalam massal properti tidak kecil jikadibandingkan dengan jalan bebas rata-rata. Hal ini dikenal sebagai rezim dan ekspansi

Knudsen dapat dilakukan di nomor Knudsen.

Teori kinetik juga telah diperluas untuk mencakup tumbukan inelastis dalam masalah rincioleh Jenkins dan lain-lain. [Rujukan?]

[sunting] Faktor-faktor

[sunting] TekananTekanan dijelaskan dengan teori kinetik yang timbul dari gaya yang diberikan oleh molekul

gas yang berdampak pada dinding wadah. Pertimbangkan gas N molekul, masing-masing

bermassa m, tertutup dalam wadah cuboidal volume V = L3. Ketika molekul gas

bertabrakan dengan dinding wadah tegak lurus terhadap sumbu koordinat x dan memantulke arah berlawanan dengan kecepatan yang sama (tumbukan elastik), maka momentum

yang hilang oleh partikel dan dinding yang diperoleh adalah:

mana vx adalah x-komponen kecepatan awal partikel.

Dampak partikel salah satu dinding sisi tertentu sekali setiap

(di mana L adalah jarak antara dinding yang berseberangan).

Gaya karena partikel ini adalah:

Total gaya pada dinding

mana bar menunjukkan rata-rata di atas N partikel. Karena Pythagoras, kita dapat menulis

ulang gaya sebagai.

Gaya ini diberikan pada suatu daerah L2. Oleh karena itu tekanan gas


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di mana V = L3 adalah volume kotak. Fraksi n = N / V adalah jumlah gas kepadatan

(densitas massa = nm kurang nyaman bagi teoretis turunan pada tingkat atom).

Menggunakan n, kita dapat menulis ulang tekanan sebagai

Ini adalah pertama sepele non-akibat dari teori kinetik karena hal ini berkaitan tekanan,

makroskopik properti, dengan rata-rata (translasi) energi kinetik per molekul yangmerupakan properti mikroskopis.

[sunting] Suhu dan energi kinetik

Dari hukum gas ideal

(1)

di mana adalah konstanta Boltzmann, dan suhu absolut,

dan dari hasil di ataskita

maka suhu mengambil bentuk

(2)yang mengarah pada ekspresi dari energi kinetik dari sebuah molekul

Energi kinetik dari sistem adalah N waktu yang dari molekul

Suhu menjadi

(3)

Persamaan. (3) 1 adalah salah satu hasil penting dari teori kinetik: Rata-rata energi kinetik

molekul sebanding dengan suhu absolut.

Dari Persamaan. (1) dan Persamaan. (3) 1, kita telah

(4)

Dengan demikian, hasil dari tekanan dan volume per mol adalah sebanding dengan rata-rata (translasi) energi kinetik molekul.

Persamaan. (1) dan Persamaan. (4) disebut sebagai "hasil klasik", yang juga dapat

diturunkan dari mekanika statistik; untuk rincian lebih lanjut, lihat [1].Karena terdapat derajat kebebasan (dofs) dalam sistem monoatomic gas dengan partikel,

energi kinetik per dof adalah

(5)Dalam energi kinetik per dof, konstanta proporsionalitas suhu adalah 1 / 2 kali Boltzmann

konstan. Hasil ini berhubungan dengan teorema equipartition.

Seperti tercantum dalam artikel tentang kapasitas panas, gas diatomik harus memiliki 7derajat kebebasan, tetapi gas yang lebih ringan bertindak seolah-olah mereka hanya 5.

Dengan demikian energi kinetik per kelvin (gas ideal monoatomik) adalah:

per mol: 12,47 J per molekul: 20,7 j = 129 eV

Pada suhu standar (273,15 K), kita mendapatkan:

per mol: 3406 J

per molekul: 5,65 ZJ = 35,2 MeV[sunting] Jumlah tumbukan dengan dinding

Satu dapat menghitung jumlah atom atau molekul tumbukan dengan dinding sebuah wadah

per satuan luas per satuan waktu.Dengan asumsi gas ideal, turunan [2] hasil dalam persamaan untuk jumlah tumbukan per

satuan waktu per wilayah:

[sunting] RMS kecepatan molekul

Dari rumus energi kinetik dapat ditunjukkan bahwa

dengan v dalam m / s, T di kelvin, dan R adalah konstanta gas. Massa molar diberikan

sebagai kg / mol. Kecepatan yang paling mungkin adalah 81,6% dari kecepatan rms dan

kecepatan rata-rata 92,1% (distribusi kecepatan).


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Benda tegar

For analytical convenience, human body segments are considered as rigidbodies. A rigid body is similar to a system of particles in the sense that it iscomposed of particles. Therefore most of the equations for a system of particlesare usable in the dynamics of a rigid body. The main difference is of course thatthe rigid body is rigid. There is no migration of mass within a rigid body. As a

result, the relative positions among the particles composing a rigid body do notchange. This will further simplify the equations obtained from a system ofparticles.

The main strategy in analyzing the motion of a rigid body is to split the motioninto the linear motion of the CM and the angular motion of the body about its CM.This is because all particles composing the body show the same relative angularmotion about the CM. In other words, one can describe the motion of the rigidbody as a whole rather than those of the particles individually. The physicalcharacteristics of a rigid body can be described by its inertial properties: massand moment of inertia. For this reason, a major portion of discussion will be

dedicated to the inertial properties of the rigid body: moment of inertia & inertiatensor.

The pages included in this section are:

Moment of InertiaCalculation of the MOIInertia TensorPrincipal AxesTransformation of the Inertia Tensor

Angular Momentum

Kinetic Energy

Moment of Inertia of a Systems of Particles

Newton's first law of motion says "A body maintains the current state of motionunless acted upon by an external force." The measure of the inertia in the linearmotion is the mass of the system and its angular counterpart is the so-calledmoment of inertia. The moment of inertia of a body is not only related to itsmass but also the distribution of the mass throughout the body. So two bodies ofthe same mass may possess different moments of inertia.

A rigid body can be considered as a system of particles in which the relativepositions of the particles do not change. The moment of inertia of a single particle(I) can be expressed as

[1]

where m = the mass of the particle, and r= the shortest distance from the axis ofrotation to the particle (Figure 1).
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Figure 1

As shown in [1], moment of inertia is equal to mass times square of the distanceand it is also referred to as the second mass moment. Mass times distance, mr,is called as the first mass moment. This concept of first mass moment isnormally used in deriving the center of mass of a system of particles or a rigidbody. See Center of Mass-System of Particles for the details.

Expanding [1] for a system of particles:

[2]

Top

Moment of Inertia of a Rigid Body

Based on [2], one can obtain the moment of inertia of a rigid by shown in Figure2:

Figure 2

[3]

where ri= the position of particle i, and n = the unit vector of the axis of rotation.Note here that the axis of rotation passes through the local reference frame, theOXYZsystem. Let

[4]

and
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[5]

where cos, cos& cos= the three direction cosines of vectorn to theXYZsystem. Substituting [4] & [5] into [3] leads to

[6]

where

[7]

Ixx, Iyy& Izzare called the moments of inertia while Ixy, Iyx, Iyz, Izy, Izx, & Ixzare theproducts of inertia. For a rigid body, the relative position of the particles do notchange and one can write [7] as:

[8]

When the shape and the density distribution of the rigid body is precisely known,one can use [8] to compute the moments and products of inertia. (See BSPEquations for the MOI equations of the typical geometric shapes commonly usedin human body modeling.) Otherwise, it is difficult to compute them throughintegration. Rather, the moment of inertia must be measured directly from theobject. See Measuring MOIfor the details.

Top

Ellipsoid of Inertia

The moments and products of inertia shown in [7] and [8] are basically specific tothe local reference frame defined and reflect the mass distribution within the bodyin relation to the local reference frame. As shown in [6], the actual moment ofinertia of a rigid body about an axis of rotation is a function of not only the

moments and products of inertia for a given reference frame but also theorientation of the axis of rotation, , & . Thus, it would be more accurate to say
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that the moment of inertia of a rigid body reflects the mass distribution within thebody with respect to the axis of rotation.

As the axis of rotation changes, so does the moment of inertia. To show thispoint clearly, let

[9]

Substituting [9] into [6] yields

[10]

Interestingly, [10] suffices the general form of the ellipsoid with its center at theorigin of the reference frame. When Ixy= Iyz= Izx= 0, the ellipsoid defined by [10]definitely becomes symmetric about the three axes.

Since

[11]

the distance from the center of the ellipsoid to the surface is 1 divided by the

square root of the moment of inertia of the rigid body for a given orientation, ,

& . The ellipsoid defined by [10] is called the ellipsoid of inertia since itdescribes the moment of inertia of an object as a function of the orientation of the

axis of rotation.

Top

Calculation of the MOI

In computation of the moment of inertia, one can replace the summation shownin [2] ofInertia Tensorby an integration over the body:

[1]

where r= the perpendicular distance from the particle to the axis of rotation, anddm = the mass of the particle which is a function of the density.

Thin Rod

Let's now apply [1] to a thin uniform rod shown in Figure 1. The MOI of the rodabout the Yaxis is

Figure 1

[2]

since
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[3]

where= the density of the rod, m = the mass of the rod, and L = the length ofthe rod.

Circular Ring

The moment of inertia of the uniform circular ring shown in Figure 2 about the Zaxis (the symmetry axis) is

Figure 2

[4]

since

[5]

where dl= the length of the arc formed by d. [5] is also applicable to a circularcylinder.

Circular Disc

A uniform circular disc of radius Rcan be considered as a cascade of uniformcircular rings as shown in Figure 3. Thus, from [5], the moment of inertia aboutthe Zaxis (the symmetry axis) becomes

Figure 3


19/33

[6]

since

[7]

[6] can be used in computing the MOI of a circular bar about its longitudinal axisas well.

Sphere

A uniform sphere can be considered as a cascade of uniform circular discs asshown in Figure 4. From [6], the moment of inertia about the vertical axis (Zaxis)is

Figure 4

[8]

since

[9]

See BSP Equations for the moment-of-inertia equations for the geometric shapescommonly used in the human body modeling.

Top

Perpendicular-Axis Theorem

Now, let's go back to the uniform circular disc (Figure 3). The moment of inertiaof a uniform circular disc about its perpendicular axis (Zaxis) can be expressedas
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[10]

since

[11]

In other words, the MOI about the Zaxis is equal to sum of those about theXandYaxes. [10] is true for any rigid lamina: the MOI of any rigid lamina about an axisnormal to the lamina plane is equal to the sum of the MOIs about any two

perpendicular axes lying on the plane and passing through the normal axis. Thisis the so-called perpendicular-axis theorem.

Since the circular disc has symmetric shape,

[12]

and, from [6]:

[13]

One can directly obtain [13] from [11] or

[14]

[2] can be used in further developing [14] to obtain [13]. See BSP Equationsfor

this approach.

Top

Parallel-Axis Theorem

Now, let's compute the MOI of a uniform circular column. Circular bar can beregarded as a cascade of circular discs as shown in Figure 5. From [13]:

Figure 5
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[15]

since

[16]

where mdisc = mass of the circular disc, and Iy'y'(disc) = the MOI of the disc aboutthe Y'axis. [16] basically says that the MOI of a circular plate about the Yaxis is

equal to the sum of the MOI about the parallel axis on the disc (Y') and the massof the disc times square of the distance between the two axes. This is the so-called parallel-axis theorem.

The MOI of the circular column is therefore

[17]

since

[18]

Comparing [18] with [2], one can clearly see the difference in the MOI between athin rod and a thick rod (circular column). Similarly, [13] and [17] shows thedifference in the MOI between a circular disc and a thick circular plate (circularcolumn).

Top

Physical Pendulum & Direct Measurement

Unfortunately, the integration approach is possible only when the body has aknown geometric shape. In the mathematical human body models such asHanavan (1964) and Yeadon (1990), it is assumed that the body segments showa group of geometric shapes such as ellipsoid of revolution, elliptical solids, and

stadium solids. See BSP Equations for the details.

If the body has a irregular shape, the integration approach has not much use anda direct measurement must be attempted. Figure 6 shows a body with irregularshape which is rotating freely about an axis passing through its one end. The Xaxis is the axis of rotation, thus, the center of mass (CM) of the body moveswithin the YZplane.
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Figure 6

The torque produced by the weight of the body about theXaxis is then

[19]

where Tx= the torque about theXaxis, Ixx= MOI of the body about theXaxis, = angular acceleration, m = the mass of the body, g= the gravitationalacceleration (9.81 m/s2), and L = the distance between the axis of rotation to the

body's CM. For a small ,

[20]

and, from [19],

[21]

Solving [21] for, one obtains

[22]

where o = the amplitude, f= the frequency of the pendulum, = the phaseangle, T= the period of the pendulum. As shown in [22], the MOI of the bodyabout theXaxis, after all, can be computed from the period of a small pendulummotion of the body. The MOI about the parallel axis, which passes through theCM of the body, can be also computed based on the parallel-axis theorem:

[23]

See Chandler et al. (1975) for an example of this approach.

Angular Momentum of a Rigid Body

Angular momentum of a rigid body (Figure 1) can be obtained from the sum ofthe angular momentums of the particles forming the body:


23/33

[1]

Figure 1

where ri= the position vector of particle i, and = the angular velocity vector ofthe rigid body. Now, let

[2]

[3]

Note here that a local reference frame, theXYZsystem, is defined and fixed to

the body at O and ri& are described in this frame. Substituting [2] & [3] into [1]:

[4]

Top
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Inertia Tensor

Let

[5]

Substituting [5] into [4]:

[6]

[7]

where I = the inertia tensor. The angular momentum of a rigid body rotating aboutan axis passing through the origin of the local reference frame is in fact the

product of the inertia tensor of the object and the angular velocity. The diagonalelements in the inertia tensor shown in [7], Ixx, Iyy& Izz, are called the moments ofinertia while the rest of the elements are called the products of inertia. Alsosee Moment of Inertia & Ellipsoid of Inertia for more details of the moments andproducts of inertia. As shown in [7], the inertia tensor is symmetric.

The 3 x 3 matrix in [7] suffices the requirements of a tensor of the 2nd rank:

[8]

where i,j, k& l= 1 to 3, tij= an element of the orthogonal transformation matrix,and I'ij= an element of the transformed inertia tensor. This property is explainedin detail in Transformation of the Inertia Tensor. In fact, a scalar is a tensor ofzero rank:

[9]

where S = a scalar, S'= the transformed scalar. In addition, a vector is a tensorof the first rank since its transformation follows

[10]

[10] is identical to [2] shown in Transformation Matrix. A 2-dimensional symmetricmatrix is not necessarily a tensor of the 2nd rank. It must suffice [8] to be atensor.

Principal Moments of inertia

As shown in [6] in Inertia Tensor, the angular momentum of a rigid body withrespect to the origin of the local reference frame is expressed as
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[1]

If, by chance, all the off-diagonal terms of the inertia tensor shown in [1] becomezero, [1] can be further simplified to

[2]

This can happen when one aligns the axes of the local reference frame in such away that the mass of the body evenly distribute around the axes, thus, theproduct-of-inertia terms all vanish. The non-zero diagonal terms of the inertiatensor shown in [2] are called the principal moments of inertia of the object.

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Principal Axes

As shown in [1], there is no guarantee that the angular momentum vector has thesame direction to that of the angular velocity vector. This causes a problem: if thedirection of the angular momentum keeps changing, it develops a torque whicheventually forces the axis of rotation to move. This is the main reason thatcauses wearing and vibration in machinery with rotating parts.

But in some special cases, the following condition may hold so that the angularmomentum and velocity vectors show the same direction:

[3]

where I= the equivalent scalar moment of inertia of the body about the axis ofrotation. Any axis of rotation of the body which suffices [3] is called a principal

axis. There are a group of principal axes (theoretically 3) in a three-dimensionalbody. For example, there are three perpendicular principal axes for the systemshown in Figure 1.

Figure 1

[3] basically says that the inertia tensor can be replaced with a single scalarmoment of inertia when the axis of rotation is a principal axis.

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Diagonalization of the Inertia Tensor

From [3]:
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[4]

Or[4] can be simplified to

[5]

where 1 = the identity matrix. Ishown in [5] is called an eigenvalue while is theeigenvector. [5] is the eigenvalue equation.

In order for[5] to have a nontrivial solution the determinant of the coefficientsshould vanish:

[6]

[6] leads to the secular equation which is basically cubic, thus provides threeroots (eigenvalues): I1, I2 & I3. Each root corresponds to a moment of inertia abouta principal axis. In fact the three roots are the principal moments of inertia of therigid body introduced in [2]:

[7]

Once the eigenvalues are known, the principal axes can be computed. Let

[8]

where n = the unit vector of the principal axis, thus,

[9]

From [4] & [8]:

[10]

For each eigenvalue, one can compute the corresponding nx, ny& nz from [9] &

[10]. One must pay attention to the direction of the eigenvector in this process.


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In motion analysis, the principal moments of inertia can be obtained from theinertial properties of the body segments. I1, I2 & I3 of each segment are generallyknown. The data are available in the form of the radius-of-gyration ratios (ratio ofthe radius of gyration to the segment length), regression equations, and scalingcoefficients. One can also compute the principal moments of inertia of the body

segments through modeling using some geometric shapes. See IndividualizedBSP Estimation for the details.

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Angular Momentum of a Rigid Body

As shown in [6] in Inertis Tensor, the angular momentum of a rigid body rotatingabout an axis passing through the origin of the local reference frame (frameA) is

[1]

Now, let's transforms this angular momentum vector to another reference frame(frame B):

[2]

Combination of[1] & [2] leads to

[3]

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Rotation

[3] shows that the angular momentum of the object described in frame B is equalto I' times the angular velocity described in frame B. The axis transformationdoes not alter the nature of any mechanical quantity, but describes it in a differentperspective. Since both the angular momentum & the angular velocity vectors arealready described in frame B, I' must be the inertia tensor of the object describedin frame B:

[4]

This type of transformation is called the similarity transformation.

Now, letA be a local reference frame (frame L) and B be the global (inertial)frame. Then, the inertia tensor of a body segment described in the globalcoordinates is

[5]

In motion analysis, one can compute the transformation matrix from the globalframe to a segmental reference frame based on the marker data, while the inertia
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tensor is typically first described in the corresponding segmental reference frame.[5] can be used to compute the inertia tensor described in the global frame.

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Tensor of the 2nd Rank

From [4]:

[6]

where i,j, k, & m = 1 to 3. Or simplifying [6], one can obtain

[7]

[7] is identical to [8] inInertia Tensor. [7] is the requirement of a tensor of the 2ndrank. A scalar is a tensor or zero rank, while a vector is a tensor of the 1st rank.

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Translation

Figure 1 shows two reference frames: the OXYZsystem & the O'X'Y'Z'system.TheXYZsystem is the local reference frame fixed to the body with its origin atthe body's center of mas

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