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From Wikipedia

Mechanics

Mechanics (GreekΜηχανική) is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment. The discipline has its roots in several ancient civilizations (see History of classical mechanics and Timeline of classical mechanics). During the early modern period, scientists such as Galileo, Kepler, and especially Newton, laid the foundation for what is now known as classical mechanics.

The system of study of mechanics is shown in the table below:

Classical versus quantum

The major division of the mechanics discipline separates classical mechanics from quantum mechanics.

Historically, classical mechanics came first, while quantum mechanics is a comparatively recent invention. Classical mechanics originated with Isaac Newton's Laws of motion in Principia Mathematica, while quantum mechanics didn't appear until 1900. Both are commonly held to constitute the most certain knowledge that exists about physical nature. Classical mechanics has especially often been viewed as a model for other so-called exact sciences. Essential in this respect is the relentless use of mathematics in theories, as well as the decisive role played by experiment in generating and testing them.

Quantum mechanics is of a wider scope, as it encompasses classical mechanics as a sub-discipline which applies under certain restricted circumstances. According to the correspondence principle, there is no contradiction or conflict between the two subjects, each simply pertains to specific situations. The correspondence principle states that the behavior of systems described by quantum theories reproduces classical physics in the limit of large quantum numbers. Quantum mechanics has superseded classical mechanics at the foundational level and is indispensable for the explanation and prediction of processes at molecular and (sub)atomic level. However, for macroscopic processes classical mechanics is able to solve problems which are unmanageably difficult in quantum mechanics and hence remains useful and well used. Modern descriptions of such behavior begin with a careful definition of such quantities as displacement (distance moved), time, velocity, acceleration, mass, and force. Until about 400 years ago, however, motion was explained from a very different point of view. For example, following the ideas of Greek philosopher and scientist Aristotle, scientists reasoned that a cannonball falls down because its natural position is in the earth; the sun, the moon, and the stars travel in circles around the earth because it is the nature of heavenly objects to travel in perfect circles.

The Italian physicist and astronomer Galileo brought together the ideas of other great thinkers of his time and began to analyze motion in terms of distance traveled from some starting position and the time that it took. He showed that the speed of falling objects increases steadily during the time of their fall. This acceleration is the same for heavy objects as for light ones, provided air friction (air resistance) is discounted. The English mathematician and physicist Sir Isaac Newton improved this analysis by defining force and mass and relating these to acceleration. For objects traveling at speeds close to the speed of light, Newton’s laws were superseded by Albert Einstein’s theory of relativity. For atomic and subatomic particles, Newton’s laws were superseded by quantum theory. For everyday phenomena, however, Newton’s three laws of motion remain the cornerstone of dynamics, which is the study of what causes motion.

Einsteinian versus Newtonian

Analogous to the quantum versus classical reformation, Einstein's general and special theories of relativity have expanded the scope of mechanics beyond the mechanics of Newton and Galileo, and made fundamental corrections to them, that become significant and even dominant as speeds of material objects approach the speed of light, which cannot be exceeded. Relativistic corrections are also needed for quantum mechanics, although General relativity has not been integrated; the two theories remain incompatible, a hurdle which must be overcome in developing the Grand Unified Theory.

History

Antiquity

The main theory of mechanics in antiquity was Aristotelian mechanics. A later developer in this tradition was Hipparchus.

Medieval age

In the Middle Ages, Aristotle's theories were criticized and modified by a number of figures, beginning with John Philoponus in the 6th century. A central problem was that of projectile motion, which was discussed by Hipparchus and Philoponus. This led to the development of the theory of impetus by 14th century French Jean Buridan, which developed into the modern theories of inertia, velocity, acceleration and Algorithm and Algorithm characterizations.

An example: Algorithm specification of addition m+n

Choice of machine model:

There is no “best�, or “preferred� model. The Turing machine, while considered the standard, is notoriously awkward to use. And different problems seem to require different models to study them. Many researchers have observed these problems, for example:

“The principal purpose of this paper is to offer a theory which is closely related to Turing's but is more economical in the basic operations� (Wang (1954) p. 63)
“Certain features of Turing machines have induced later workers to propose alternative devices as embodiments of what is to be meant by effective computability.... a Turing machine has a certain opacity, its workings are known rather than seen. Further a Turing machine is inflexible ... a Turing machine is slow in (hypothetical) operation and, usually complicated. This makes it rather hard to design it, and even harder to investigate such matters as time or storage optimization or a comparison between efficiency of two algorithms.� (Melzak (1961) p. 281)
Shepherdson-Sturgis (1963) proposed their register-machine model because “these proofs [using Turing machines] are complicated and tedious to follow for two reasons: (1) A Turing machine has only one head... (2) It has only one tape....� They were in search of “a form of idealized computer which is sufficiently flexible for one to be able to convert an intuitive computational procedure into a program for such a machine� (p. 218).
“I would prefer something along the lines of the random access computers of Angluin and Valiant [as opposed to the pointer machine of Schönhage]� (Gurivich 1988 p. 6)
“Showing that a function is Turing computable directly...is rather laborious ... we introduce an ostensibly more flexible kind of idealized machine, an abacus machine...� (Boolos-Burgess-Jeffrey 2002 p.45).

About all that one can insist upon is that the algorithm-writer specify in exacting detail (i) the machine model to be used and (ii) its instruction set.

Atomization of the instruction set:

The Turing machine model is primitive, but not as primitive as it can be. As noted in the above quotes this is a source of concern when studying complexity and equivalence of algorithms. Although the observations quoted below concern the Random access machine model – a Turing-machine equivalent – the problem remains for any Turing-equivalent model:

“...there hardly exists such a thing as an ‘innocent’ extension of the standard RAM model in the uniform time measure; either one only has additive arithmetic, or one might as well include all multiplicative and/or bitwise Boolean instructions on small operands....� (van Emde Boas (1992) p. 26)
“Since, however, the computational power of a RAM model seems to depend rather sensitively on the scope of its instruction set, we nevertheless will have to go into detail...
“One important principle will be to admit only such instructions which can be said to be of an atomistic nature. We will describe two versions of the so-called successor RAM, with the successor function as the only arithmetic operation....the RAM0 version deserves special attention for its extreme simplicity; its instruction set consists of only a few one letter codes, without any (explicit) addressing.� (Schönhage (1980) p.494)

Example #1: The most general (and original) Turing machine – single-tape with left-end, multi-symbols, 5-tuple instruction format – can be atomized into the Turing machine of Boolos-Burgess-Jeffrey (2002) – single-tape with no ends, two "symbols" { B, | } (where B symbolizes "blank square" and | symbolizes "marked square"), and a 4-tuple instruction format. This model in turn can be further atomized into a Post-Turing machine– single-tape with no ends, two symbols { B, | }, and a 0- and 1-parameter instruction set ( e.g. { Left, Right, Mark, Erase, Jump-if-marked to instruction xxx, Jump-if-blank to instruction xxx, Halt } ).

Example #2: The RASP can be reduced to a RAM by moving its instructions off the tape and (perhaps with translation) into its finite-state machine “table� of instructions, the RAM stripped of its indirect instruction and reduced to a 2- and 3-operand “abacus� register machine; the abacus in turn can be reduced to the 1- and 2-operand Minsky (1967)/Shepherdson-Sturgis (1963) counter machine, which can be further atomized into the 0- and 1-operand instructions of Schönhage (and even a 0-operand Schönhage-like instruction set is possible).

Cost of atomization:

Atomization comes at a (usually severe) cost: while the resulting instructions may be “simpler�, atomization (usually) creates more instructions and the need for more computational steps. As shown in the following example the increase in computation steps may be significant (i.e. orders of magnitude – the following example is “tame�), and atomization may (but not always, as in the case of the Post-Turing model) reduce the usability and readability of “the machine code�. For more see Turing tarpit.

Example: The single register machine instruction "INC 3" – increment the contents of register #3, i.e. increase its count by 1 – can be atomized into the 0-parameter instruction set of Schönhage, but with the equivalent number of steps to accomplish the task increasing to 7; this number is directly related to the register number “n� i.e. 4+n):

More examples can be found at the pages Register machine and Random access machine where the addition of "convenience instructions" CLR h and COPY h1,h1 are shown to reduce the number of steps dramatically. Indirect addressing is the other significant example.

Precise specification of Turing-machine algorithm m+n

As described in Algorithm characterizations per the specifications of Boolos-Burgess-Jeffrey (2002) and Sipser (2006), and with a nod to the other characterizations we proceed to specify:

(i) Number format: unary strings of marked squares (a "marked square" signfied by the symbol 1) separated by single blanks (signified by the symbol B) e.g. “2,3� = B11B111B
(ii) Machine type: Turing machine: single-tape left-ended or no-ended, 2-symbol { B, 1 }, 4-tuple instruction format.
(iii) Head location: See more at “Implementation Description� below. A symbolic representation of the head's location in the tape's symbol string will put the current state to the right of the scanned symbol. Blank squares may be included in this protocol. The state's number will appear with brackets around it, or sub-scripted. The head is shown as

Mechanical equilibrium

A standard definition of static equilibrium is:

A system of particles is in static equilibrium when all the particles of the system are at rest and the total force on each particle is permanently zero.

This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next.

A standard definition of mechanical equilibrium for a particle is:

The necessary and sufficient conditions for a particle to be in mechanical equilibrium is that the net force acting upon the particle is zero.

The necessary conditions for mechanical equilibrium for a system of particles are:

(i)The vector sum of all external forces is zero;
(ii) The sum of the moments of all external forces about any line is zero.

As applied to a rigid body, the necessary and sufficient conditions become:

A rigid body is in mechanical equilibrium when the sum of all forces on all particles of the system is zero, and also the sum of all torques on all particles of the system is zero.

A rigid body in mechanical equilibrium is undergoing neither linear nor rotational acceleration; however it could be translating or rotating at a constant velocity.

However, this definition is of little use in continuum mechanics, for which the idea of a particle is foreign. In addition, this definition gives no information as to one of the most important and interesting aspects of equilibrium states – their stability.

An alternative definition of equilibrium that applies to conservative systems and often proves more useful is:

A system is in mechanical equilibrium if its position in configuration space is a point at which the gradient with respect to the generalized coordinates of the potential energy is zero.

Because of the fundamental relationship between force and energy, this definition is equivalent to the first definition. However, the definition involving energy can be readily extended to yield information about the stability of the equilibrium state.

For example, from elementary calculus, we know that a necessary condition for a local minimumor a maximum of a differentiable function is a vanishing first derivative (that is, the first derivative is becoming zero). To determine whether a point is a minimum or maximum, one may be able to use the second derivative test. The consequences to the stability of the equilibrium state are as follows:

  • Second derivative< 0 : The potential energy is at a local maximum, which means that the system is in an unstable equilibrium state. If the system is displaced an arbitrarily small distance from the equilibrium state, the forces of the system cause it to move even farther away.
  • Second derivative > 0 : The potential energy is at a local minimum. This is a stable equilibrium. The response to a small perturbation is forces that tend to restore the equilibrium. If more than one stable equilibrium state is possible for a system, any equilibria whose potential energy is higher than the absolute minimum represent metastable states.
  • Second derivative = 0 or does not exist: The second derivative test fails, and one must typically resort to using the first derivative test. Both of the previous results are still possible, as is a third: this could be a region in which the energy does not vary, in which case the equilibrium is called neutral or indifferent or marginally stable. To lowest order, if the system is displaced a small amount, it will stay in the new state.

In more than one dimension, it is possible to get different results in different directions, for example stability with respect to displacements in the x-direction but instability in the y-direction, a case known as a saddle point. Without further qualification, an equilibrium is stable only if it is stable in all directions.

The special case of mechanical equilibrium of a stationary object is static equilibrium. A paperweight on a desk would be in static equilibrium. The minimal number of static equilibria of homogeneous, convex bodies (when resting under gravity on a horizontal surface) is of special interest. In the planar case, the minimal number is 4, while in three dimensions one can build an object with just one stable and one unstable balance point, this is called Gomboc. A child sliding down a slide at constant speed would be in mechanical equilibrium, but not in static equilibrium.

An example of mechanical equilibrium will be a person trying to press a spring, he can push it up to a point after which it reaches a state where the force trying to compress it and the resistive force from the spring are equal, so the person can not further press it, at this state the system will be in mechanical equilibrium. When the pressing force is removed the spring attains its original state.



From Yahoo Answers

Question:I need five different examples

Answers:wind moving a mill water moving turbines steam moving turbines i c engine rotating the crank your feet rotating the pedal of whatever

Question:

Answers:kenetic, anything thats moving. car, bus, train, sprinter, lorry potential, anything that can fall. ball, plane, bird, rain, leaf electrical, anything that has electricity. light, computer, etc

Question:Is it impossible to, for example, confront the news of having to write a 5 page essay in 3 days without using defense mechanisms? Sublimation could get the essay done, but is it impossible to face this kind of instruction without feeling anxiety, or is the feeling necessary?

Answers:Uh, I guess that it depends on what the paper's on and your attitude toward writing. I, for instance, enjoy writing papers and usually start 3-5 page papers the night before they're due and do a good job. I don't use any defense mechanisms, just procrastination if that counts. I say just work on the essay gradually, and eventually you'll get it done. I mean, if you don't have a ton of other things to do in a situation such as the one you provided, you could easily write the paper with ample time and little or no defense mechanisms.

Question:I need at least 5 different energy exchanges. like a car has Mechanical energy from the engine and electromagnetic from the radio. i can't use that one though cos it was my teachers example(;

Answers:(1) Wind turns the propellers of a windmill, generating mechanical energy that is converted into electrical energy sent via high-tension wires to an electric utility plant. (2) Sunlight shines on a solar panel, which converts the electromagnetic waves of visible light into electrical energy to be used by appliances in a residential home. (3) Biochemical energy from glucose is used in oxidative phosphorylation to generate ATP (adenosine phosphate), which serves as an energy resource for synthesis of proteins in a cell. (4) ATP is converted to ADP (adenosine diphosphate), releasing energy that is converted into mechanical energy to be used by the muscles of a athlete (a soccer player or marathon runner). (5) Electrical energy is used by a high definition television set (HDTV) to convert electromagnetic waves from a geosynchronous satellite into pixelated images and sounds that can be viewed for entertainment or educational use.

From Youtube

The Periodic Tables of Physics - Relationship Examples from Mechanics (5/9) :This is part 5 of 9 in the Periodic Tables of Physics presentation series. When the physical elements are arranged on a grid that reflects their time and space relationships, a periodic table of physical elements is formed. This presentation provides several examples from mechanics of how important physical relationships are embodied in the rules by which the grid is constructed. This presentation has been incorporated into a written document that is available online at: www-personal.umich.edu

Sn1 Mechanism and Examples :Overview of the Sn1 substitution reaction and the different products we can form.