# Classical mechanics: Wikis

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Classical mechanics
$\mathbf{F} = \frac{\mathrm{d}}{\mathrm{d}t}(m \mathbf{v})$
Newton's Second Law
History of ...

In the fields of physics, classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the set of physical laws governing and mathematically describing the motions of bodies and aggregates of bodies geometrically distributed within a certain boundary under the action of a system of forces. The other sub-field is quantum mechanics.

Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. It produces very accurate results within these domains, and is one of the oldest and largest subjects in science, engineering and technology.

Besides this, many related specialties exist, dealing with gases, liquids, and solids, and so on. Classical mechanics is enhanced by special relativity for objects moving with high velocity, approaching the speed of light; general relativity is employed to handle gravitation at a deeper level; and quantum mechanics handles the wave-particle duality of atoms and molecules.

The term classical mechanics was coined in the early 20th century to describe the system of mathematical physics begun by Isaac Newton and many contemporary 17th century natural philosophers, building upon the earlier astronomical theories of Johannes Kepler, which in turn were based on the precise observations of Tycho Brahe and the studies of terrestrial projectile motion of Galileo, but before the development of quantum physics and relativity. Therefore, some sources exclude so-called "relativistic physics" from that category. However, a number of modern sources do include Einstein's mechanics, which in their view represents classical mechanics in its most developed and most accurate form. The initial stage in the development of classical mechanics is often referred to as Newtonian mechanics, and is associated with the physical concepts employed by and the mathematical methods invented by Newton himself, in parallel with Leibniz, and others. This is further described in the following sections. More abstract and general methods include Lagrangian mechanics and Hamiltonian mechanics. Much of the content of classical mechanics was created in the 18th and 19th centuries and extends considerably beyond (particularly in its use of analytical mathematics) the work of Newton.

## Description of the theory

The analysis of projectile motion is a part of classical mechanics.

The following introduces the basic concepts of classical mechanics. For simplicity, it often models real-world objects as point particles, objects with negligible size. The motion of a point particle is characterized by a small number of parameters: its position, mass, and the forces applied to it. Each of these parameters is discussed in turn.

In reality, the kind of objects which classical mechanics can describe always have a non-zero size. (The physics of very small particles, such as the electron, is more accurately described by quantum mechanics). Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional degrees of freedom—for example, a baseball can spin while it is moving. However, the results for point particles can be used to study such objects by treating them as composite objects, made up of a large number of interacting point particles. The center of mass of a composite object behaves like a point particle.

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### Position and its derivatives

 The SI derived "mechanical" (that is, not electromagnetic or thermal) units with kg, m and s Position m Angular position/Angle unitless (radian) velocity m s−1 Angular velocity s−1 acceleration m s−2 Angular acceleration s−2 jerk m s−3 "Angular jerk" s−3 specific energy m2 s−2 absorbed dose rate m2 s−3 moment of inertia kg m2 momentum kg m s−1 angular momentum kg m2 s−1 force kg m s−2 torque kg m2 s−2 energy kg m2 s−2 power kg m2 s−3 pressure and energy density kg m−1 s−2 surface tension kg s−2 Spring constant kg s−2 irradiance and energy flux kg s−3 kinematic viscosity m2 s−1 dynamic viscosity kg m−1 s-1 Density(mass density) kg m−3 Density(weight density) kg m−2 s-2 Number density m−3 Action kg m2 s−1

The position of a point particle is defined with respect to an arbitrary fixed reference point, O, in space, usually accompanied by a coordinate system, with the reference point located at the origin of the coordinate system. It is defined as the vector r from O to the particle. In general, the point particle need not be stationary relative to O, so r is a function of t, the time elapsed since an arbitrary initial time. In pre-Einstein relativity (known as Galilean relativity), time is considered an absolute, i.e., the time interval between any given pair of events is the same for all observers. In addition to relying on absolute time, classical mechanics assumes Euclidean geometry for the structure of space.[1]

#### Velocity and speed

The velocity, or the rate of change of position with time, is defined as the derivative of the position with respect to time or

$\mathbf{v} = {\mathrm{d}\mathbf{r} \over \mathrm{d}t}\,\!$.

In classical mechanics, velocities are directly additive and subtractive. For example, if one car traveling East at 60 km/h passes another car traveling East at 50 km/h, then from the perspective of the slower car, the faster car is traveling east at 60 − 50 = 10 km/h. Whereas, from the perspective of the faster car, the slower car is moving 10 km/h to the West. Velocities are directly additive as vector quantities; they must be dealt with using vector analysis.

Mathematically, if the velocity of the first object in the previous discussion is denoted by the vector u = ud and the velocity of the second object by the vector v = ve, where u is the speed of the first object, v is the speed of the second object, and d and e are unit vectors in the directions of motion of each particle respectively, then the velocity of the first object as seen by the second object is

$\mathbf{u}' = \mathbf{u} - \mathbf{v} \, .$

Similarly,

$\mathbf{v'}= \mathbf{v} - \mathbf{u} \, .$

When both objects are moving in the same direction, this equation can be simplified to

$\mathbf{u}' = ( u - v ) \mathbf{d} \, .$

Or, by ignoring direction, the difference can be given in terms of speed only:

$u' = u - v \, .$

#### Acceleration

The acceleration, or rate of change of velocity, is the derivative of the velocity with respect to time (the second derivative of the position with respect to time) or

$\mathbf{a} = {\mathrm{d}\mathbf{v} \over \mathrm{d}t}$.

Acceleration can arise from a change with time of the magnitude of the velocity or of the direction of the velocity or both. If only the magnitude v of the velocity decreases, this is sometimes referred to as deceleration, but generally any change in the velocity with time, including deceleration, is simply referred to as acceleration.

#### Frames of reference

While the position and velocity and acceleration of a particle can be referred to any observer in any state of motion, classical mechanics assumes the existence of a special family of reference frames in terms of which the mechanical laws of nature take a comparatively simple form. These special reference frames are called inertial frames. They are characterized by the absence of acceleration of the observer and the requirement that all forces entering the observer's physical laws originate in identifiable sources (charges, gravitational bodies, and so forth). A non-inertial reference frame is one accelerating with respect to an inertial one, and in such a non-inertial frame a particle is subject to acceleration by fictitious forces that enter the equations of motion solely as a result of its accelerated motion, and do not originate in identifiable sources. These fictitious forces are in addition to the real forces recognized in an inertial frame. A key concept of inertial frames is the method for identifying them. For practical purposes, reference frames that are unaccelerated with respect to the distant stars are regarded as good approximations to inertial frames.

Consider two reference frames S and S' . For observers in each of the reference frames an event has space-time coordinates of (x,y,z,t) in frame S and (x′,y′,z′,t′) in frame S′. Assuming time is measured the same in all reference frames, and if we require x = x' when t = 0, then the relation between the space-time coordinates of the same event observed from the reference frames S′ and S, which are moving at a relative velocity of u in the x direction is:

x′ = xut
y′ = y
z′ = z
t′ = t

This set of formulas defines a group transformation known as the Galilean transformation (informally, the Galilean transform). This group is a limiting case of the Poincaré group used in special relativity. The limiting case applies when the velocity u is very small compared to c, the speed of light.

The transformations have the following consequences:

• v′ = vu (the velocity v′ of a particle from the perspective of S′ is slower by u than its velocity v from the perspective of S)
• a′ = a (the acceleration of a particle is the same in any inertial reference frame)
• F′ = F (the force on a particle is the same in any inertial reference frame)
• the speed of light is not a constant in classical mechanics, nor does the special position given to the speed of light in relativistic mechanics have a counterpart in classical mechanics.

For some problems, it is convenient to use rotating coordinates (reference frames). Thereby one can either keep a mapping to a convenient inertial frame, or introduce additionally a fictitious centrifugal force and Coriolis force.

### Forces; Newton's second law

Newton was the first to mathematically express the relationship between force and momentum. Some physicists interpret Newton's second law of motion as a definition of force and mass, while others consider it to be a fundamental postulate, a law of nature. Either interpretation has the same mathematical consequences, historically known as "Newton's Second Law":

$\mathbf{F} = {\mathrm{d}\mathbf{p} \over \mathrm{d}t} = {\mathrm{d}(m \mathbf{v}) \over \mathrm{d}t}$.

The quantity mv is called the (canonical) momentum. The net force on a particle is thus equal to rate change of momentum of the particle with time. Since the definition of acceleration is a = dv/dt, the second law can be written in the simplified and more familiar form:

$\mathbf{F} = m \mathbf{a} \, .$

So long as the force acting on a particle is known, Newton's second law is sufficient to describe the motion of a particle. Once independent relations for each force acting on a particle are available, they can be substituted into Newton's second law to obtain an ordinary differential equation, which is called the equation of motion.

As an example, assume that friction is the only force acting on the particle, and that it may be modeled as a function of the velocity of the particle, for example:

$\mathbf{F}_{\rm R} = - \lambda \mathbf{v} \, ,$

where λ is a positive constant. Then the equation of motion is

$- \lambda \mathbf{v} = m \mathbf{a} = m {\mathrm{d}\mathbf{v} \over \mathrm{d}t} \, .$

This can be integrated to obtain

$\mathbf{v} = \mathbf{v}_0 e^{- \lambda t / m}$

where v0 is the initial velocity. This means that the velocity of this particle decays exponentially to zero as time progresses. In this case, an equivalent viewpoint is that the kinetic energy of the particle is absorbed by friction (which converts it to heat energy in accordance with the conservation of energy), slowing it down. This expression can be further integrated to obtain the position r of the particle as a function of time.

Important forces include the gravitational force and the Lorentz force for electromagnetism. In addition, Newton's third law can sometimes be used to deduce the forces acting on a particle: if it is known that particle A exerts a force F on another particle B, it follows that B must exert an equal and opposite reaction force, −F, on A. The strong form of Newton's third law requires that F and −F act along the line connecting A and B, while the weak form does not. Illustrations of the weak form of Newton's third law are often found for magnetic forces.

### Work and energy

If a constant force F is applied to a particle that achieves a displacement Δr,[nb 1] the work done by the force is defined as the scalar product of the force and displacement vectors:

$W = \mathbf{F} \cdot \Delta \mathbf{r} \, .$

More generally, if the force varies as a function of position as the particle moves from r1 to r2 along a path C, the work done on the particle is given by the line integral

$W = \int_C \mathbf{F}(\mathbf{r}) \cdot \mathrm{d}\mathbf{r} \, .$

If the work done in moving the particle from r1 to r2 is the same no matter what path is taken, the force is said to be conservative. Gravity is a conservative force, as is the force due to an idealized spring, as given by Hooke's law. The force due to friction is non-conservative.

The kinetic energy Ek of a particle of mass m travelling at speed v is given by

$E_k = \tfrac{1}{2}mv^2 \, .$

For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the kinetic energies of the particles.

The work-energy theorem states that for a particle of constant mass m the total work W done on the particle from position r1 to r2 is equal to the change in kinetic energy Ek of the particle:

$W = \Delta E_k = E_{k,2} - E_{k,1} = \tfrac{1}{2}m\left(v_2^{\, 2} - v_1^{\, 2}\right) \, .$

Conservative forces can be expressed as the gradient of a scalar function, known as the potential energy and denoted Ep:

$\mathbf{F} = - \mathbf{\nabla} E_p \, .$

If all the forces acting on a particle are conservative, and Ep is the total potential energy (which is defined as a work of involved forces to rearrange mutual positions of bodies), obtained by summing the potential energies corresponding to each force

$\mathbf{F} \cdot \Delta \mathbf{r} = - \mathbf{\nabla} E_p \cdot \Delta \mathbf{s} = - \Delta E_p \Rightarrow - \Delta E_p = \Delta E_k \Rightarrow \Delta (E_k + E_p) = 0 \, .$

This result is known as conservation of energy and states that the total energy,

$\sum E = E_k + E_p \, .$

is constant in time. It is often useful, because many commonly encountered forces are conservative.

### Beyond Newton's Laws

Classical mechanics also includes descriptions of the complex motions of extended non-pointlike objects. Euler's laws provide extensions to Newton's laws in this area. The concepts of angular momentum rely on the same calculus used to describe one-dimensional motion. The Rocket equation extends the notion of rate of change of an object's momentum to include the effects of an object "losing mass".

There are two important alternative formulations of classical mechanics: Lagrangian mechanics and Hamiltonian mechanics. These, and other modern formulations, usually bypass the concept of "force", instead referring to other physical quantities, such as energy, for describing mechanical systems.

The expressions given above for momentum and kinetic energy are only valid when there is no significant electromagnetic contribution. In electromagnetism, Newton's second law for current-carrying wires breaks down unless one includes the electromagnetic field contribution to the momentum of the system as expressed by the Poynting vector divided by c2, where c is the speed of light in free space.

## History

Some Greek philosophers of antiquity, among them Aristotle, may have been the first to maintain the idea that "everything happens for a reason" and that theoretical principles can assist in the understanding of nature. While to a modern reader, many of these preserved ideas come forth as eminently reasonable, there is a conspicuous lack of both mathematical theory and controlled experiment, as we know it. These both turned out to be decisive factors in forming modern science, and they started out with classical mechanics.

An early experimental scientific method was introduced into mechanics in the 11th century by al-Biruni, who along with al-Khazini in the 12th century, unified statics and dynamics into the science of mechanics, and combined the fields of hydrostatics with dynamics to create the field of hydrodynamics.[2] Concepts related to Newton's laws of motion were also enunciated by several other Muslim physicists during the Middle Ages. Early versions of the law of inertia, known as Newton's first law of motion, and the concept relating to momentum, part of Newton's second law of motion, were described by Ibn al-Haytham (Alhacen)[3][4] and Avicenna.[5][6] The proportionality between force and acceleration, an important principle in classical mechanics, was first stated by Hibat Allah Abu'l-Barakat al-Baghdaadi,[7] and Ibn Bajjah also developed the concept of a reaction force.[8] Theories on gravity were developed by Ja'far Muhammad ibn Mūsā ibn Shākir,[9] Ibn al-Haytham,[10] and al-Khazini.[11] It is known that Galileo Galilei's mathematical treatment of acceleration and his concept of impetus[12] grew out of earlier medieval analyses of motion, especially those of Avicenna,[5] Ibn Bajjah,[13] and Jean Buridan.

The first published causal explanation of the motions of planets was Johannes Kepler's Astronomia nova published in 1609. He concluded, based on Tycho Brahe's observations of the orbit of Mars, that the orbits were ellipses. This break with ancient thought was happening around the same time that Galilei was proposing abstract mathematical laws for the motion of objects. He may (or may not) have performed the famous experiment of dropping two cannon balls of different masses from the tower of Pisa, showing that they both hit the ground at the same time. The reality of this experiment is disputed, but, more importantly, he did carry out quantitative experiments by rolling balls on an inclined plane. His theory of accelerated motion derived from the results of such experiments, and forms a cornerstone of classical mechanics.

As foundation for his principles of natural philosophy, Newton proposed three laws of motion: the law of inertia, his second law of acceleration (mentioned above), and the law of action and reaction; and hence laid the foundations for classical mechanics. Both Newton's second and third laws were given proper scientific and mathematical treatment in Newton's Philosophiæ Naturalis Principia Mathematica, which distinguishes them from earlier attempts at explaining similar phenomena, which were either incomplete, incorrect, or given little accurate mathematical expression. Newton also enunciated the principles of conservation of momentum and angular momentum. In Mechanics, Newton was also the first to provide the first correct scientific and mathematical formulation of gravity in Newton's law of universal gravitation. The combination of Newton's laws of motion and gravitation provide the fullest and most accurate description of classical mechanics. He demonstrated that these laws apply to everyday objects as well as to celestial objects. In particular, he obtained a theoretical explanation of Kepler's laws of motion of the planets.

Newton previously invented the calculus, of mathematics, and used it to perform the mathematical calculations. For acceptability, his book, the Principia, was formulated entirely in terms of the long established geometric methods, which were soon to be eclipsed by his calculus. However it was Leibniz who developed the notation of the derivative and integral preferred[citation needed] today.

Newton, and most of his contemporaries, with the notable exception of Huygens, worked on the assumption that classical mechanics would be able to explain all phenomena, including light, in the form of geometric optics. Even when discovering the so-called Newton's rings (a wave interference phenomenon) his explanation remained with his own corpuscular theory of light.

After Newton, classical mechanics became a principal field of study in mathematics as well as physics.

Some difficulties were discovered in the late 19th century that could only be resolved by more modern physics. Some of these difficulties related to compatibility with electromagnetic theory, and the famous Michelson-Morley experiment. The resolution of these problems led to the special theory of relativity, often included in the term classical mechanics.

A second set of difficulties were related to thermodynamics. When combined with thermodynamics, classical mechanics leads to the Gibbs paradox of classical statistical mechanics, in which entropy is not a well-defined quantity. Black-body radiation was not explained without the introduction of quanta. As experiments reached the atomic level, classical mechanics failed to explain, even approximately, such basic things as the energy levels and sizes of atoms and the photo-electric effect. The effort at resolving these problems led to the development of quantum mechanics.

Since the end of the 20th century, the place of classical mechanics in physics has been no longer that of an independent theory. Emphasis has shifted to understanding the fundamental forces of nature as in the Standard model and its more modern extensions into a unified theory of everything.[14] Classical mechanics is a theory for the study of the motion of non-quantum mechanical, low-energy particles in weak gravitational fields.

In the 21st century classical mechanics has been extended into the complex domain and complex classical mechanics exhibits behaviours very similar to quantum mechanics.[15]

## Limits of validity

Domain of validity for Classical Mechanics

Many branches of classical mechanics are simplifications or approximations of more accurate forms; two of the most accurate being general relativity and relativistic statistical mechanics. Geometric optics is an approximation to the quantum theory of light, and does not have a superior "classical" form.

### The Newtonian approximation to special relativity

In special relativity, the momentum of a particle is given by

$\mathbf{p} = \frac{m \mathbf{v}}{ \sqrt{1-v^2/c^2}} \, ,$

where m is the particle's mass, v its velocity, and c is the speed of light.

If v is very small compared to c, v2/c2 is approximately zero, and so

$\mathbf{p} \approx m\mathbf{v} \, .$

Thus the Newtonian equation p = mv is an approximation of the relativistic equation for bodies moving with low speeds compared to the speed of light.

For example, the relativistic cyclotron frequency of a cyclotron, gyrotron, or high voltage magnetron is given by

$f=f_c\frac{m_0}{m_0+T/c^2} \, ,$

where fc is the classical frequency of an electron (or other charged particle) with kinetic energy T and (rest) mass m0 circling in a magnetic field. The (rest) mass of an electron is 511 keV. So the frequency correction is 1% for a magnetic vacuum tube with a 5.11 kV direct current accelerating voltage.

### The classical approximation to quantum mechanics

The ray approximation of classical mechanics breaks down when the de Broglie wavelength is not much smaller than other dimensions of the system. For non-relativistic particles, this wavelength is

$\lambda=\frac{h}{p}$

where h is Planck's constant and p is the momentum.

Again, this happens with electrons before it happens with heavier particles. For example, the electrons used by Clinton Davisson and Lester Germer in 1927, accelerated by 54 volts, had a wave length of 0.167 nm, which was long enough to exhibit a single diffraction side lobe when reflecting from the face of a nickel crystal with atomic spacing of 0.215 nm. With a larger vacuum chamber, it would seem relatively easy to increase the angular resolution from around a radian to a milliradian and see quantum diffraction from the periodic patterns of integrated circuit computer memory.

More practical examples of the failure of classical mechanics on an engineering scale are conduction by quantum tunneling in tunnel diodes and very narrow transistor gates in integrated circuits.

Classical mechanics is the same extreme high frequency approximation as geometric optics. It is more often accurate because it describes particles and bodies with rest mass. These have more momentum and therefore shorter De Broglie wavelengths than massless particles, such as light, with the same kinetic energies.

## Branches

Branches of mechanics

Classical mechanics was traditionally divided into three main branches:

• Statics, the study of equilibrium and its relation to forces
• Dynamics, the study of motion and its relation to forces
• Kinematics, dealing with the implications of observed motions without regard for circumstances causing them

Another division is based on the choice of mathematical formalism:

Alternatively, a division can be made by region of application:

## Notes

1. ^ The displacement Δr is the difference of the particle's initial and final positions: Δr = rfinalrinitial.
1. ^ MIT physics 8.01 lecture notes (page 12) (PDF)
2. ^ Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 614-642 [642], Routledge, London and New York
3. ^ Abdus Salam (1984), "Islam and Science". In C. H. Lai (1987), Ideals and Realities: Selected Essays of Abdus Salam, 2nd ed., World Scientific, Singapore, p. 179-213.
4. ^ Seyyed Hossein Nasr, "The achievements of Ibn Sina in the field of science and his contributions to its philosophy", Islam & Science, December 2003.
5. ^ a b Fernando Espinoza (2005). "An analysis of the historical development of ideas about motion and its implications for teaching", Physics Education 40 (2), p. 141.
6. ^ Seyyed Hossein Nasr, "Islamic Conception Of Intellectual Life", in Philip P. Wiener (ed.), Dictionary of the History of Ideas, Vol. 2, p. 65, Charles Scribner's Sons, New York, 1973-1974.
7. ^ Shlomo Pines (1970). "Abu'l-Barakāt al-Baghdādī, Hibat Allah". Dictionary of Scientific Biography. 1. New York: Charles Scribner's Sons. pp. 26–28. ISBN 0684101149.
(cf. Abel B. Franco (October 2003). "Avempace, Projectile Motion, and Impetus Theory", Journal of the History of Ideas 64 (4), p. 521-546 [528]
8. ^ Shlomo Pines (1964), "La dynamique d’Ibn Bajja", in Mélanges Alexandre Koyré, I, 442-468 [462, 468], Paris.
(cf. Abel B. Franco (October 2003). "Avempace, Projectile Motion, and Impetus Theory", Journal of the History of Ideas 64 (4), p. 521-546 [543].)
9. ^ Robert Briffault (1938). The Making of Humanity, p. 191.
10. ^ Nader El-Bizri (2006), "Ibn al-Haytham or Alhazen", in Josef W. Meri (2006), Medieval Islamic Civilization: An Encyclopaedia, Vol. II, p. 343-345, Routledge, New York, London.
11. ^ Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", in Roshdi Rashed, ed., Encyclopaedia of the History of Arabic Science, Vol. 2, p. 622. London and New York: Routledge.
12. ^ Galileo Galilei, Two New Sciences, trans. Stillman Drake, (Madison: Univ. of Wisconsin Pr., 1974), pp 217, 225, 296-7.
13. ^ Ernest A. Moody (1951). "Galileo and Avempace: The Dynamics of the Leaning Tower Experiment (I)", Journal of the History of Ideas 12 (2), p. 163-193.
14. ^ Page 2-10 of the Feynman Lectures on Physics says "For already in classical mechanics there was indeterminability from a practical point of view." The past tense here implies that classical physics is no longer fundamental.
15. ^ Complex Elliptic Pendulum, Carl M. Bender, Daniel W. Hook, Karta Kooner

## References

• Feynman, Richard (1996). Six Easy Pieces. Perseus Publishing. ISBN 0-201-40825-2.
• Feynman, Richard; Phillips, Richard (1998). Six Easy Pieces. Perseus Publishing. ISBN 0-201-32841-0.
• Feynman, Richard (1999). Lectures on Physics. Perseus Publishing. ISBN 0-7382-0092-1.
• Landau, L. D.; Lifshitz, E. M. (1972). Mechanics Course of Theoretical Physics , Vol. 1. Franklin Book Company. ISBN 0-08-016739-X.
• Eisberg, Robert Martin (1961). Fundamentals of Modern Physics. John Wiley and Sons.
• Fundamental university physics. Addison-Wesley.
• Structure and Interpretation of Classical Mechanics. MIT Press. 2001. ISBN 0-262-19455-4.
• An Introduction to Mechanics. McGraw-Hill. 1973. ISBN 0-07-035048-5.
• Classical Mechanics (3rd ed.). Addison Wesley. 2002. ISBN 0-201-65702-3.

## Further reading

• Thornton, Stephen T.; Marion, Jerry B. (2003). Classical Dynamics of Particles and Systems (5th ed.). Brooks Cole. ISBN 0-534-40896-6.

# Study guide

Up to date as of January 14, 2010

## Newton's Laws

Classical Mechanics is the study of the motion of particles and systems of particles and of the forces which induce changes in motion. Quantum mechanics and Relativity, developed in the 20th century, were advancements, which corrected classical mechanical errors in motion. However, since classical mechanics is easier to understand and widely applicable to most everyday physical situations it is still quite useful today.

Classical Mechanics were described in detail by Sir Isaac Newton in his two volume Philosophiae Naturalis Principia Mathematica in the 17th century, sometimes referred to as Principia or Principia Mathematica. We will begin our study of classical mechanics with Newton's three laws of motion.

1) An object at rest will remain at rest unless acted upon by an external force. An object in motion will move at a constant velocity unless acted upon by an external force.

This first law was a major advancement in Physics, as it was previously assumed that an object in motion on earth would always return to a state of rest unless acted on by a force. We now understand that the force of friction must act upon the object to return it to rest.

2) Force = the change in momentum of a particle (or system of particles) with respect to time.

This law is often portraited in the simple equation,

F = ma

which is true when the mass is constant. A derivation of this simpler equation is displayed below.

3) Every action has an equal and opposite action.

This can be displayed by pushing on the wall or firing a gun. When I apply a force to the wall, the wall in turn applies an equal and opposite force on me. When the propellant in a gun is ignited the resulting chamber pressure propells the bullet forward and an equal force pushes the gun backward, providing the kickback force.

## Introduction

Classical mechanics is the study of everyday forces which we come in contact with. The goal of classical physics is to be able to describe perfectly any and all kinds of motion. From this it would follow that if you knew only part of the motion of an object, then you could determine each other part of the motion.

Essentially, motion boils down to where an object is, how fast it is going, and how fast it is changing how fast it is going.

In strict physical terms, these quantities are:

Position: the location of an object (generally from its centre of gravity) in relation to all other objects in a system.

Velocity: the rate at which the position changes.

Acceleration: The rate at which the object's velocity changes.

The units used to describe these quantities are:

• meter (m): the standard unit for distance in one direction. A meter is about 39 inches, or slightly more the three feet.
• second (s): the standard unit for time. a second is the time that elapses during 9,192,631,770 vibrations of the cesium 133 atom.

These relate to each other as follows:

Position = x (in meters)

Velocity = change in position (distance d) divided by change in time (t)

v = (change in d)/t

Acceleration = change in velocity (v) divided by change in time

a = (change in v)/t

for example, if I were to walk in a straight line 10 m, and it took me 5 s to do this, my velocity would be:

v = (change in d)/t
v = 10 m/5 s
v = 2 m/s

my velocity was 2 m/s

In a similar fashion, if you knew my initial velocity to be 2 m/s and after a 20 s stretch, my velocity was 42 m/s, my acceleration would be

a = (change in v)/t
a = (42 m/s - 2 m/s)/20 s
a = (40 m/s)/20 s
a = 2 m/s²

my acceleration was 2 m/s²

In physics, there is a distinction between average velocity and instantaneous velocity. Instantaneous velocity is the velocity at a specific point in time, and can be only derived with advanced calculus which may be covered in an advanced section. Average velocity, however, is what we will be dealing with. Average velocity involves the difference between two points in space and the difference in time associated with them. This is different from average speed. Average speed is the distance traveled divided by the time expended. Speed is based on the path an object takes, while velocity is based on the displacement of an object, i.e. the amount of distance it actually covered. for example, if I were to move 10 m to the left, and then 10 m to the right, over the course of 1 s, my path would be 20 m long, but my displacement would be 0 m, because the starting point and ending point were the same. Therefore, my average speed would be 20 m/s, while my velocity would be 0 m/s. The difference is important, because in physics, speed is never used, only velocity. The reason for this is that a physicist is considered an outside observer, and should be able to describe the event perfectly without actually witnessing what occured. If only two data points are taken, then velocity is the only way to calculate things because no data has been taken in the middle, and no data about the path can be known.

Long ago, in the 1600's, a man named Isaac Newton established three laws relating to motion, which brought physics out of what was described above to a new level.

His first law was that each object will stay in motion unless acted upon by an unbalanced outside force, and likewise for an object at rest. This established a thing called momentum, which is a property of matter based upon an object's mass and velocity. An object, like a train moving at 10 m/s, will have more momentum than, say, a pea at the same velocity. This means that, in the same amount of time, the train will require more energy to stop than the pea. However, velocity also plays a role, as a 20 m/s train will be harder to stop than a 10 m/s train.

His second law stated that force equals mass multiplied by acceleration. Derivation of F = ma from Newton's original definition (Force is the rate of change of momentum)

$F=\frac{dp}{dt} = \frac{d(mv)}{dt} =m\frac{dv}{dt} + \frac{dm}{dt}v = ma + \frac{dm}{dt}v$

Then, if mass is kept constant, $\frac{dm}{dt}=0$, and $F=ma \,$.

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# Wikibooks

Up to date as of January 23, 2010
(Redirected to Classical Mechanics article)

### From Wikibooks, the open-content textbooks collection

Classical mechanics is the study of the motion of bodies based upon Isaac Newton's famous laws of mechanics. There are no new physical concepts in classical mechanics that are not already extant in other areas of physics. What classical mechanics does is mathematically reformulate Newtonian physics to address a huge range of problems ranging from molecular dynamics to the motion of celestial bodies.

As one of the oldest branches of physics, it has long ago been displaced in many fields of study by newer theories (the foremost of these being quantum mechanics and relativity), but classical mechanics is far from being obsolete. Classical mechanics is very useful for analyzing problems in which quantum and relativistic effects are negligible, and its principles and mathematics are the foundation upon which numerous branches of modern physics are founded (including quantum mechanics and relativity). And finally, it is fascinating field of study unto itself—or at least some people think so. Maybe you'll be a fan of classical mechanics too, after having studied it.

## Prerequisites

The reader should be comfortable with Newton's laws and with basic physics concepts such as mass, moments of inertia, length, force and time (q.v. basic concepts). In addition, math is the crucial tool of physics, familiarity with geometry, algebra, and calculus is a must. In particular, the reader should be comfortable with multivariable calculus (if you do not know the difference between '∂f/∂x' and 'df/dx', then it's time to spend some quality time with a math textbook).

That said, mathematics is tool for physics, and only a tool. As much as it is important for the study of physics, physics is more than a mere exercise in math. It is also about finding different ways to look at the physical world, and developing intuition about how to predict natural phenomena ("how things work"). Readers need not have understood everything that was ever taught to them in a math course.

## Suggested books to study

There is an extraordinarily large number of textbooks in theoretical mechanics, because it is a fairly old and well-studied subject. You need any textbook on classical mechanics that you can understand and that talks about "Lagrangians" early on. (Books that only talk about accelerations, forces, and torques may be quite advanced but they do not cover the subject of theoretical mechanics.)

• H. Goldstein. Classical mechanics. - Has everything standard in it and quite a few advanced topics. An old classic.
• L.N. Hand, J.D. Finch. Analytical mechanics (Cambridge, 1998). - A fresher, more didactic exposition of mechanics. Standard material.
• L. Landau, E. Lifshitz. Mechanics. -- A short, clean, concise treatment of mechanics.
• V.I. Arnold. Mathematical methods of classical mechanics. -- Has almost nothing standard in it but is excellent for a more mathematically minded student.

## Contents

### Part 1: Core material

• Small oscillations
• Symmetries and conservation laws
• Mechanics of rigid bodies
• Special Relativity

### Part 2: Optional material

• Virial theorem
• Scattering theory
• Anharmonic oscillations

# Simple English

Mechanics is a part of physics. It says what happens when forces act on things. There are two parts of mechanics. The two parts are classical mechanics and quantum mechanics. Classical mechanics is used most of the time. It is good to say what happens to most of the things we can see. Some of the time, for example when the things are too small, classical mechanics is not good. Then we need to use quantum mechanics.

## Newton's Three Laws

Newton's three laws of motion are important to classical mechanics. Isaac Newton made them.

The first law says that, if there is no external force (meaning there is no motion, gravity, slope, or any sort of power), things that are stopped will stay stopped or un-moving, and things that are moving will keep moving. Before, people thought that things stopped if there was no force present. Often, people say, Objects that are stopped tend to stay stopped, and objects that are moving tend to stay moving, unless acted upon by an outside force, such as gravity, friction, etc....

The second law says how a force moves a thing. The force on a object equals the rate of change of the speed momentum.

The third law says that if one thing puts a force on another thing, the second thing also puts a force on the first thing. For example, if you jump forward off a boat, the boat moves backward. Often, people say, For every action there is an equal and opposite reaction.

## Kinematic Equations

In physics, kinematics is the part of classical mechanics that explains the movement of objects without looking at what causes the movement or what the movement affects.

### 1-Dimensional Kinematics

1-Dimensional (1D) Kinematics are used only when an object moves in one direction: either side to side (left to right) or up and down. There are equations with can be used to solve problems that have movement in only 1 dimension or direction. These equations come from the definitions of velocity, acceleration and distance.

1. The first 1D kinematic equation deals with acceleration and velocity. If acceleration and velocity do not change. (Does not need include distance)
Equation: $V_f=v_i+at$
Vf is the final velocity.
vi is the starting or initial velocity
a is the acceleration
t is time - how long the object was accelerated for.
2. The second 1D kinematic equation finds the distance moved, by using the average velocity and the time. (Does not need include acceleration)
Equation: $x=\left(\left(V_f+V_i\right)/2\right)t$
x is the distance moved.
Vf is the final velocity.
vi is the starting or initial velocity
t is time
3. The third 1D kinematic equation finds the distance travelled, while the object is accelerating. It deals with velocity, acceleration, time and distance. (Does not need include final velocity)
Equation: $X_f=x_i+v_it+\left(1/2\right)at^2$
$X_f$ is the final distance moved
xi is the starting or initial distance
vi is the starting or initial velocity
a is the acceleration
t is time
4. The fourth 1D kinematic equation finds the final velocity by using the initial velocity, acceleration and distance travelled. (Does not need include time)
Equation: $V_f^2=v_i^2+2ax$
Vf is the final velocity
vi is the starting or initial velocity
a is the acceleration
x is the distance moved

### 2-Dimensional Kinematics

2-Dimensional kinematics is used when motion happens in both the x-direction (left to right) and the y-direction (up and down). There are also equations for this type of kinematics. However, there are different equations for the x-direction and different equations for the y-direction. Galileo proved that the velocity in the x-direction does not change through the whole run. However, the y-direction is affected by the force of gravity, so the y-velocity does change during the run.

#### X-Direction Equations

Left and Right movement
1. The first x-direction equation is the only one that is needed to solve problems, because the velocity in the x-direction stays the same.
Equation: $X=V_x*t$
X is the distance moved in the x-direction
Vx is the velocity in the x-direction
t is time

#### Y-Direction Equations

Up and Down movement. Affected by gravity or other external acceleration
1. The first y-direction equation is almost the same as the first 1-Dimensional kinematic equation except it deals with the changing y-velocity. It deals with a freely falling body while its being affected by gravity. (Distance is not needed)
Equation: $V_fy=v_iy-gt$
Vfy is the final y-velocity
viy is the starting or initial y-velocity
g is the acceleration because of gravity which is 9.8 $m/s^2$ or 32 $ft/s^2$
t is time
2. The second y-direction equation is used when the object is being affected by a separate acceleration, not by gravity. In this case, the y-component of the acceleration vector is needed. (Distance is not needed)
Equation: $V_fy=v_iy+a_yt$
Vfy is the final y-velocity
viy is the starting or initial y-velocity
ay is the y-component of the acceleration vector
t is the time
3. The third y-direction equation finds the distance moved in the y-direction by using the average y-velocity and the time. (Does not need acceleration of gravity or external acceration)
Equation: $X_y=\left(\left(V_fy+V_iy\right)/2\right)t$
Xy is the distance moved in the y-direction
Vfy is the final y-velocity
viy is the starting or initial y-velocity
t is the time
4. The fourth y-direction equation deals with the distance moved in the y-direction while being affected by gravity. (Does not need final y-velocity)
Equation: $X_fy=X_iy+v_iy-\left(1/2\right)gt^2$
$X_fy$ is the final distance moved in the y-direction
xiy is the starting or initial distance in the y-direction
viy is the starting or initial velocity in the y-direction
g is the acceleration of gravity which is 9.8 $m/s^2$ or 32 $ft/s^2$
t is time
5. The fifth y-direction equation deals with the distance moved in the y-direction while being affected by a different acceleration other than gravity. (Does not need final y-velocity)
Equation: $X_fy=X_iy+v_iy+\left(1/2\right)a_yt^2$
$X_fy$ is the final distance moved in the y-direction
xiy is the starting or initial distance in the y-direction
viy is the starting or initial velocity in the y-direction
ay is the y-component of the acceleration vector
t is time
6. The sixth y-direction equation finds the final y-velocity while it is being affected by gravity over a certain distance. (Does not need time)
Equation: $V_fy^2=V_iy^2-2gx_y$
Vfy is the final velocity in the y-direction
Viy is the starting or initial velocity in the y-direction
g is the acceleration of gravity which is 9.8 $m/s^2$ or 32 $ft/s^2$
xy is the total distance moved in the y-direction
7. The seventh y-direction equation finds the final y-velocity while it is being affected by an acceleration other than gravity over a certain distance. (Does not need time)
Equation: $V_fy^2=V_iy^2+2a_yx_y$
Vfy is the final velocity in the y-direction
Viy is the starting or initial velocity in the y-direction
ay is the y-component of the acceleration vector
xy is the total distance moved in the y-direction

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