For the textbooks, see Classical Mechanics (Goldstein book) and Classical Mechanics (Kibble and Berkshire book).
Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.
If the present state of an object is known it is possible to predict by the laws of classical mechanics how it will move in the future (determinism) and how it has moved in the past (reversibility)
The earliest development of classical mechanics is often referred to as Newtonian mechanics. It consists of the physical concepts employed by and the mathematical methods invented by Isaac Newton and Gottfried Wilhelm Leibniz and others in the 17th century to describe the motion of bodies under the influence of a system of forces.
Later, more abstract methods were developed, leading to the reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics. These advances, made predominantly in the 18th and 19th centuries, extend substantially beyond Newton's work, particularly through their use of analytical mechanics. They are, with some modification, also used in all areas of modern physics.
Classical mechanics provides extremely accurate results when studying large objects that are not extremely heavy and speeds not approaching the speed of light. When the objects being examined have about the size of an atom diameter, it becomes necessary to introduce the other major sub-field of mechanics: quantum mechanics. To describe velocities that are not small compared to the speed of light, special relativity is needed. In case that objects become extremely heavy, General relativity becomes applicable. However, a number of modern sources do include relativistic mechanics into classical physics, which in their view represents classical mechanics in its most developed and accurate form.[note 1]
Description of the theory
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 that 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, e.g., 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 of a large number of collectively acting point particles. The center of mass of a composite object behaves like a point particle.
Classical mechanics uses common-sense notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as location in space and speed. Non-relativistic mechanics also assumes that forces act instantaneously (see also Action at a distance).
Position and its derivatives
Main article: Kinematics
The position of a point particle is defined in relation to a coordinate system centered on an arbitrary fixed reference point in space called the origin O. A simple coordinate system might describe the position of a particleP with a vector notated by an arrow labeled r that points from the origin O to point P. In general, the point particle does not need to be stationary relative to O. In cases where P is moving relative to O, r is defined as a function of t, time. In pre-Einstein relativity (known as Galilean relativity), time is considered an absolute, i.e., the time interval that is observed to elapse 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.
Velocity and speed
Main articles: 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:
In classical mechanics, velocities are directly additive and subtractive. For example, if one car travels east at 60 km/h and passes another car traveling in the same direction at 50 km/h, the slower car perceives the faster car as traveling east at 60 − 50 = 10 km/h. However, from the perspective of the faster car, the slower car is moving 10 km/h to the west, often denoted as -10 km/h where the sign implies opposite direction. 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 object respectively, then the velocity of the first object as seen by the second object is
Similarly, the first object sees the velocity of the second object as
When both objects are moving in the same direction, this equation can be simplified to
Or, by ignoring direction, the difference can be given in terms of speed only:
Main article: 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):
Acceleration represents the velocity's change over time. Velocity can change in either magnitude or direction, or both. Occasionally, a decrease in the magnitude of velocity "v" is referred to as deceleration, but generally any change in the velocity over time, including deceleration, is simply referred to as acceleration.
Frames of reference
Main articles: Inertial frame of reference and Galilean transformation
While the position, velocity and acceleration of a particle can be described with respect to any observer in any state of motion, classical mechanics assumes the existence of a special family of reference frames in which the mechanical laws of nature take a comparatively simple form. These special reference frames are called inertial frames.
An inertial frame is a frame of reference within which an object interacting with no forces (an idealized situation) appears either at rest or moving uniformly in a straight line. This is the fundamental definition of an inertial frame. These are characterized by the requirement that all forces entering the observer's physical laws[clarification needed] originate from identifiable sources caused by fields, such as electro-static field (caused by static electrical charges), electro-magnetic field (caused by moving charges), gravitational field (caused by mass), and so forth.
A key concept of inertial frames is the method for identifying them. For practical purposes, reference frames that do not accelerate with respect to distant stars (an extremely distant point) are regarded as good approximations to inertial frames. Non-inertial reference frames accelerate in relation to an existing inertial frame. They form the basis for Einstein's relativity. Due to the relative motion, particles in the non-inertial frame appear to move in ways not explained by forces from existing fields in the reference frame. Hence, it appears that there are other forces that enter the equations of motion solely as a result of the relative acceleration. These forces are referred to as fictitious forces, inertia forces, or pseudo-forces.
Consider two reference framesS and . For observers in each of the reference frames an event has space-time coordinates of (x,y,z,t) in frame S and (,,,) in frame . Assuming time is measured the same in all reference frames, and if we require x = when t = 0, then the relation between the space-time coordinates of the same event observed from the reference frames and S, which are moving at a relative velocity of u in the x direction is:
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′ = v − u (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
Main articles: Force and Newton's laws of motion
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 a fundamental postulate, a law of nature. Either interpretation has the same mathematical consequences, historically known as "Newton's Second Law":
The quantity mv is called the (canonical) momentum. The net force on a particle is thus equal to the rate of change of the 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:
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:
where λ is a positive constant, the negative sign states that the force is opposite the sense of the velocity. Then the equation of motion is
This can be integrated to obtain
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), and the particle is slowing 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.[clarification needed]
Work and energy
Main articles: Work (physics), kinetic energy, and potential energy
If a constant force F is applied to a particle that makes a displacement Δr,[note 2] the work done by the force is defined as the scalar product of the force and displacement vectors:
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
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 energyEk of a particle of mass m travelling at speed v is given by
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 as it moves from position r1 to r2 is equal to the change in kinetic energyEk of the particle:
Conservative forces can be expressed as the gradient of a scalar function, known as the potential energy and denoted Ep:
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
The decrease in the potential energy is equal to the increase in the kinetic energy
This result is known as conservation of energy and states that the total energy,
is constant in time. It is often useful, because many commonly encountered forces are conservative.
Beyond Newton's laws
Classical mechanics also describes the more 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, speed and momentum, for describing mechanical systems in generalized coordinates.
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.
Limits of validity
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.
When both quantum mechanics and classical mechanics cannot apply, such as at the quantum level with many degrees of freedom, quantum field theory (QFT) is of use. QFT deals with small distances and large speeds with many degrees of freedom as well as the possibility of any change in the number of particles throughout the interaction. When treating large degrees of freedom at the macroscopic level, statistical mechanics becomes useful. Statistical mechanics describes the behavior of large (but countable) numbers of particles and their interactions as a whole at the macroscopic level. Statistical mechanics is mainly used in thermodynamics for systems that lie outside the bounds of the assumptions of classical thermodynamics. In the case of high velocity objects approaching the speed of light, classical mechanics is enhanced by special relativity. In case that objects become extremely heavy (i.e. their Schwarzschild radius is not negligibly small for a given application), deviations from Newtonian mechanics become apparent and can be quantified by using the Parameterized post-Newtonian formalism. In that case, General relativity (GR) becomes applicable. However, until now there is no theory of Quantum gravity unifying GR and QFT in the sense that it could be used when objects become extremely small and heavy.
The Newtonian approximation to special relativity
In special relativity, the momentum of a particle is given by
where m is the particle's rest 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
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
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
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 wavelength of 0.167 nm, which was long enough to exhibit a single diffractionside 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 transistorgates 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.
Main article: History of classical mechanics
See also: Timeline of classical mechanics
The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology,
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ENG101 Assignment Question no 1Word formation denotes the processes of creation of new lexical units. It also refers to the ways in which new words are made on the basis of other words or morphemes. There are a number of methods of word formation including affixes, compounding, blending, clipping, acronyms, back formation, conversion, etc. Read the given paragraph carefully and categorize the underlined words accordingly by filling the tables given below.
The 52-year-old, well-poised and proud lady, who is fully dressed this morning by eight o’clock, with her hair fashionably coiffed and makeup perfectly applied, moved to UCLA in smog today for her joining as hostel warden. Her room was not ready so she started to wait in the lobby. She got a newspaper there and started to read it. There she read an article about the human rights of the employees of the nylon industry. She started to send a fax to the government as she has been actively involved in many human rights campaigns. After many hours of waiting patiently in the lobby, she smiled sweetly when told her room was ready. As she maneuvered her walker to the elevator, I provided a visual description of her tiny room, including the eyelet sheets that had been hung on her window. "I love it," she stated with the enthusiasm of an eight-year-old having just been presented with a new puppy.
Recommended :Eng101 short questions for midterm
ENG101 Assignment Question no 2:The main idea is the central or the most important idea in a paragraph or a passage. It states the purpose and sets the direction of the paragraph or passage. The main idea statement must be like the “umbrella” sentence. Read the given paragraphs carefully. Each paragraph is followed by four statements. Select the statement that best expresses the main idea.
1. The fact that electronic computers are now used for data processing has led the general public to believe that it is a mysterious, complicated science and that the computers are giant brains. Both of these ideas are false. A computer is basically just a high-speed adding machine that performs the functions it is told to. If the input data are varied even a little, the computer is unable to operate until it is programmed to accept the variations. The business operations it performs are impressive only because of the extremely high speed of manipulation, but most of these operations have been used for decades. Unlike man, the computer performs repetitive calculations without getting tired or bored.
- A computer is a high-speed adding machine.
- A computer is a mysterious giant brain.
- A computer is impressive because of its high speed.
- A computer is superior to man in many ways.
What is the main idea of the above passage?
- Ali dislikes violence.
- Ali likes to think.
- Ali enjoys Monopoly.
- Ali enjoys playing games.
- How far is it to the sun?
- It’s so far to the Sun that it's hard to comprehend.
- In actual distance, it’s approximately 93 million miles to the Sun.
- It takes a long time to get to the Sun, no matter how you travel.
- Across the country, many states have abolished the policy of “social promotion.
- A similar study of New York City children in the 1980s revealed that the children who repeated a grade were more likely to drop out upon reaching high school.
- Yet there is little indication that it does students any real good.
- In 1989, University of Georgia Professor Thomas Holmes surveyed sixty-three studies that compared the performance of kids who had repeated a grade with those who had received a social promotion.
- There are many types of literature.
- Novels, short stories, drama, fairy tales and fables are types of fiction.
- Fiction is a type of narrative writing that comes from the imagination of the author rather than from history or fact.
- Novels and short stories are types of fiction.
ENG101 Assignment No 1 Solution Spring 2017
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ENG101 Assignment Question No. 1 Solution idea
Blending - is one of the many ways new words are made in English. It refers to joining the beginning of one word and the end of another to make a new word with a new meaning.
Smog: from smoke and fog
Clipping - is one of the ways new words are created in English. It involves the shortening ... which is a clipped form of mathematics, is an example of this. Informal examples include 'bro' … from brother, Maths from mathematics
Compounds - Words are combined into compound structures in a variety of ways
Closed form, in which the words are melded together, such as firefly, secondhand
hyphenated form, such as daughter-in-law, master-at-arms, over-the-counter
open form, such as post office, real estate, middle class
well-poised, eight-year-old, makeup, eyelet sheets
affixation/affix is a word element of English grammar used to alter the meaning or form of a word and comes in the form of either a prefix or a suffix. Prefixes include examples like "un-" "self-" and "re-" while suffixes come in the form of ending elements like "-hood" "-ing" or "-ed."
Coinage: the act of creating a new word or phrase that other people begin to use
Nylon : 1938 coined as a generic by the du Pont Chemical Co. as distinct from known words and having no prior meaning or use, but with the suffix -on suggesting other textile fibers such as rayon
An acronym is a word or name formed as an abbreviation from the initial components in a phrase or a word, usually individual letters (as in NATO or laser)
UCLA: University of California , Los Angeles
This Solution idea is as per my knowledge. Please confirm before uploading. Check for makeup It can fit in compound or coinage