View source for Mechanical energy

{{Short description|Sum of potential and kinetic energy}}
[[File:Orbital motion.gif|thumb|250px|An example of a mechanical system: A satellite is orbiting the Earth influenced only by the conservative gravitational force; its mechanical energy is therefore conserved. The satellite's acceleration is represented by the green vector and its velocity is represented by the red vector. If the satellite's orbit is an ellipse the potential energy of the satellite, and its kinetic energy, both vary with time but their sum remains constant.]]
In  [[Outline of physical science|physical sciences]], '''mechanical energy''' is the sum of [[potential energy]] and [[kinetic energy]]. The principle of conservation of mechanical energy states that if an [[Isolated system|isolated]] [[Physical system|system]] is subject only to [[conservative force]]s, then the mechanical energy is constant. If an object moves in the opposite direction of a conservative net force, the potential energy will increase; and if the [[speed]] (not the [[velocity]]) of the object changes, the kinetic energy of the object also changes. In all real systems, however, [[nonconservative force]]s, such as [[frictional force]]s, will be present, but if they are of negligible [[Magnitude (mathematics)|magnitude]], the mechanical energy changes little and its conservation is a useful approximation. In [[elastic collision]]s, the kinetic energy is conserved, but in [[inelastic collision]]s some mechanical energy may be converted into [[thermal energy]]. The equivalence between lost mechanical energy  and an increase in [[temperature]] was discovered by [[James Prescott Joule]].

Many devices are used to convert mechanical energy to or from other forms of [[energy]], e.g. an [[electric motor]] converts [[electrical energy]] to mechanical energy, an [[electric generator]] converts mechanical energy into [[electrical energy]] and a [[heat engine]] converts [[heat]] to mechanical energy.

==General==
Energy is a [[Scalar (physics)|scalar]] quantity and the mechanical energy of a system is the sum of the potential energy (which is measured by the position of the parts of the system) and the kinetic energy (which is also called the energy of motion):<ref name="Access CL">{{cite encyclopedia | last = Wilczek | first = Frank | encyclopedia = AccessScience | title = Conservation laws (physics) | url = http://www.accessscience.com/abstract.aspx?id=757423&referURL=http%3a%2f%2fwww.accessscience.com%2fcontent.aspx%3fsearchStr%3dConservation%2blaws%2b%28physics%29%26id%3d757423 | access-date = 2011-08-26 | year = 2008 | publisher = McGraw-Hill Companies | archive-url = https://web.archive.org/web/20130719083020/http://www.accessscience.com/abstract.aspx?id=757423&referURL=http%3a%2f%2fwww.accessscience.com%2fcontent.aspx%3fsearchStr%3dConservation%2blaws%2b%28physics%29%26id%3d757423 | archive-date = 2013-07-19 | url-status = dead }}</ref><ref name="Britannica mechE">{{cite encyclopedia | encyclopedia = [[Encyclopædia Britannica|The New Encyclopædia Britannica: Micropædia: Ready Reference]] | title = mechanical energy | edition = 15th | year = 2003 | volume = 7}}</ref>

<math display="block">E_\text{mechanical}=U+K</math>

The potential energy, ''U'', depends on the position of an object subjected to gravity or some other [[conservative force]]. The gravitational potential energy of an object is equal to the weight ''W'' of the object multiplied by the height ''h'' of the object's center of gravity relative to an arbitrary datum:

<math display="block">U = W h</math>

The potential energy of an object can be defined as the object's ability to do [[Work (physics)|work]] and is increased as the object is moved in the opposite direction of the direction of the force.{{#tag:ref|It is important to note that when measuring mechanical energy, an object is considered as a whole, as it is stated by [[Isaac Newton]] in his ''[[Principia Mathematica Philosophiae Naturalis|Principia]]'': "The motion of a whole is the same as the sum of the motions of the parts; that is, the change in position of its parts from their places, and thus the place of a whole is the same as the sum of the places of the parts and therefore is internal and in the whole body."<ref name="Newton409">{{Harvnb|Newton|1999|p=409}}</ref>|group="nb"}}<ref name="Access CL"/> If ''F'' represents the conservative force and ''x'' the position, the potential energy of the force between the two positions ''x<sub>1</sub>'' and ''x<sub>2</sub>'' is defined as the negative integral of ''F'' from ''x<sub>1</sub>'' to ''x<sub>2</sub>'':<ref name="tamuk">{{cite web | url = http://physics.tamuk.edu/~suson/html/2325/PotentialEnergy.html | title = Potential Energy | access-date = 2011-08-25 | publisher = Texas A&M University–Kingsville | archive-url = https://web.archive.org/web/20120414201746/http://physics.tamuk.edu/~suson/html/2325/PotentialEnergy.html | archive-date = 2012-04-14 | url-status = dead }}</ref>

<math display="block">U = - \int_{x_1}^{x_2} \vec{F}\cdot d\vec{x}</math>

The kinetic energy, ''K'', depends on the speed of an object and is the ability of a moving object to do work on other objects when it collides with them.{{#tag:ref|In physics, [[speed]] is a scalar quantity and [[velocity]] is a [[Euclidean vector|vector]]. Velocity is speed with a direction and can therefore change without changing the speed of the object since speed is the numerical magnitude of a velocity.<ref name="Brodie129-131">{{Harvnb|Brodie|Brown|Heslop|Ireson|1998|pp=129–131}}</ref><ref name="AccessSp">{{cite encyclopedia | last = Rusk | first = Rogers D. | encyclopedia = AccessScience | title = Speed | url = http://accessscience.com/abstract.aspx?id=643200&referURL=http%3a%2f%2faccessscience.com%2fcontent.aspx%3fsearchStr%3dspeed%26id%3d643200 | access-date = 2011-08-28 | year = 2008 | publisher = McGraw-Hill Companies | archive-url = https://web.archive.org/web/20130719083334/http://accessscience.com/abstract.aspx?id=643200&referURL=http%3a%2f%2faccessscience.com%2fcontent.aspx%3fsearchStr%3dspeed%26id%3d643200 | archive-date = 2013-07-19 | url-status = dead }}</ref><ref name="AccessVe">{{cite encyclopedia | last = Rusk | first = Rogers D. | encyclopedia = AccessScience | title = Velocity | url = http://accessscience.com/abstract.aspx?id=729500&referURL=http%3a%2f%2faccessscience.com%2fcontent.aspx%3fsearchStr%3dvelocity%26id%3d729500 | access-date = 2011-08-28 | year = 2008 | publisher = McGraw-Hill Companies | archive-url = https://web.archive.org/web/20130719071304/http://accessscience.com/abstract.aspx?id=729500&referURL=http%3a%2f%2faccessscience.com%2fcontent.aspx%3fsearchStr%3dvelocity%26id%3d729500 | archive-date = 2013-07-19 | url-status = dead }}</ref>|group="nb"}}<ref name="Brodie101">{{Harvnb|Brodie|Brown|Heslop|Ireson|1998|p=101}}</ref> It is defined as one half the product of the object's mass with the square of its speed, and the total kinetic energy of a system of objects is the sum of the kinetic energies of the respective objects:<ref name="Access CL"/><ref name="Jain9">{{Harvnb|Jain|2009|p=9}}</ref>

<math display="block">K={1 \over 2}mv^2</math>

The principle of conservation of mechanical energy states that if a body or system is subjected only to [[conservative force]]s, the mechanical energy of that body or system remains constant.<ref name="Jain12">{{Harvnb|Jain|2009|p=12}}</ref> The difference between a conservative and a [[Non-Conservative Force|non-conservative force]] is that when a conservative force moves an object from one point to another, the work done by the conservative force is independent of the path. On the contrary, when a non-conservative force acts upon an object, the work done by the non-conservative force is dependent of the path.<ref name="MIT Physics">{{cite web | url = http://web.mit.edu/8.02t/www/materials/modules/ReviewD.pdf | title = Review D: Potential Energy and the Conservation of Mechanical Energy | access-date = 2011-08-03 | last = Department of Physics | format = PDF | publisher = [[Massachusetts Institute of Technology]]}}</ref><ref>Resnick, Robert and Halliday, David (1966), ''Physics'', Section 8-3 (Vol I and II, Combined edition), Wiley International Edition, Library of Congress Catalog Card No. 66-11527</ref>

==Conservation of mechanical energy==
[[File:Physicsworks.ogv|thumb|200px|alt="Lecture demonstrating conservation of mechanical energy"|MIT professor [[Walter Lewin]] demonstrating conservation of mechanical energy]]

According to the principle of conservation of mechanical energy, the mechanical energy of an [[isolated system]] remains constant in time, as long as the system is free of [[friction]] and other non-conservative forces. In any real situation, frictional forces and other non-conservative forces are present, but in many cases their effects on the system are so small that the principle of conservation of mechanical energy can be used as a fair [[approximation]]. Though energy cannot be created or destroyed, it can be [[Energy conversion|converted]] to another form of energy.<ref name="Access CL"/><ref name="Access CoE">{{cite encyclopedia | last = E. Roller | first = Duane |author2=Leo Nedelsky | encyclopedia = AccessScience | title = Conservation of energy | url = http://www.accessscience.com/abstract.aspx?id=157600&referURL=http%3a%2f%2fwww.accessscience.com%2fcontent.aspx%3fsearchStr%3dconservation%2bof%2benergy%26id%3d157600 | access-date = 2011-08-26 | year = 2008 | publisher = McGraw-Hill Companies}}</ref>

===Swinging pendulum===
[[File:Pendulum animation.gif|200px|thumb|A swinging pendulum with the velocity vector (green) and acceleration vector (blue). The magnitude of the velocity vector, the speed, of the pendulum is greatest in the vertical position and the pendulum is farthest from Earth in its extreme positions.]]
{{Main|Pendulum}}
In a [[mechanical system]] like a swinging [[pendulum]] subjected to the conservative [[gravitational force]] where frictional forces like air drag and friction at the pivot are negligible, energy passes back and forth between kinetic and potential energy but never leaves the system. The pendulum reaches greatest kinetic energy and least potential energy when in the vertical position, because it will have the greatest speed and be nearest the Earth at this point. On the other hand, it will have its least kinetic energy and greatest potential energy at the extreme positions of its swing, because it has zero speed and is farthest from Earth at these points. However, when taking the frictional forces into account, the system loses mechanical energy with each swing because of the negative work done on the pendulum by these non-conservative forces.<ref name="Britannica mechE"/>

===Irreversibilities===

{{Main|Irreversible process}}
That the loss of mechanical energy in a system always resulted in an increase of the system's temperature has been known for a long time, but it was the amateur physicist [[James Prescott Joule]] who first experimentally demonstrated how a certain amount of work done against friction resulted in a definite quantity of [[heat]] which should be conceived as the random motions of the particles that comprise matter.<ref>{{cite encyclopedia | encyclopedia = Scientists: Their Lives and Works | title = James Prescott Joule | year = 2006 | publisher = Gale}} as cited on {{cite web | url = http://www.gale.cengage.com/InContext/resources.htm | title = Student Resources in Context | access-date = 2011-08-28 | publisher = Gale}}</ref> This equivalence between mechanical energy and heat is especially important when considering colliding objects. In an [[elastic collision]], mechanical energy is conserved – the sum of the mechanical energies of the colliding objects is the same before and after the collision. After an [[inelastic collision]], however, the mechanical energy of the system will have changed. Usually, the mechanical energy before the collision is greater than the mechanical energy after the collision. In inelastic collisions, some of the mechanical energy of the colliding objects is transformed into kinetic energy of the constituent particles. This increase in kinetic energy of the constituent particles is perceived as an increase in temperature. The collision can be described by saying some of the mechanical energy of the colliding objects has been converted into an equal amount of heat. Thus, the total energy of the system remains unchanged though the mechanical energy of the system has reduced.<ref name="Access CL"/><ref name="Access Cp">{{cite encyclopedia | last = Schmidt | first = Paul W. | encyclopedia = AccessScience | title = Collision (physics) | url = http://accessscience.com/abstract.aspx?id=149000&referURL=http%3a%2f%2faccessscience.com%2fcontent.aspx%3fsearchStr%3dCollision%2b%28physics%29%26id%3d149000 | access-date = 2011-09-03 | year = 2008 | publisher = McGraw-Hill Companies}}</ref>

===Satellite===
{{Main|Vis-viva equation}}
[[File:R = geo Re 2012-10-08 1809.png|thumb|plot of kinetic energy <math>K</math>, gravitational potential energy, <math>U</math> and mechanical energy <math>E_\text{mechanical}</math> versus distance away from centre of earth, r at R= Re, R= 2*Re, R=3*Re and lastly R = geostationary radius]]
A satellite of mass <math> m</math> at a distance <math> r</math> from the centre of Earth possesses both kinetic energy, <math> K</math>, (by virtue of its motion) and gravitational potential energy, <math>U</math>, (by virtue of its position within the Earth's gravitational field; Earth's mass is <math>M</math>).
Hence, mechanical energy <math>E_\text{mechanical}</math> of the satellite-Earth system is given by
<math display="block">E_\text{mechanical} = U + K</math>
<math display="block">E_\text{mechanical} = - G \frac{M m}{r}\ + \frac{1}{2}\, m v^2</math>

If the satellite is in circular orbit, the energy conservation equation can be further simplified into
<math display="block">E_\text{mechanical} = - G \frac{M m}{2r} </math>
since in circular motion, Newton's 2nd Law of motion can be taken to be
<math display="block">G \frac{M m}{r^2}\ = \frac{m v^2}{r} </math>

==Conversion==

Today, many technological devices convert mechanical energy into other forms of energy or vice versa. These devices can be placed in these categories:
* An [[electric motor]] converts [[electrical energy]] into mechanical energy.<ref>{{cite encyclopedia | last = Kopicki | first = Ronald J. | editor = Kutler, Stanley I. | encyclopedia = Dictionary of American History | title = Electrification, Household | edition = 3rd | year = 2003 | publisher = Charles Scribner's Sons | volume = 3 | location = New York | pages = 179–183}} as cited on {{cite web | url = http://www.gale.cengage.com/InContext/resources.htm | title = Student Resources in Context | access-date = 2011-09-07 | publisher = Gale}}</ref><ref>{{cite encyclopedia |editor1=Lerner, K. Lee |editor2=Lerner, Brenda Wilmoth | encyclopedia = The Gale Encyclopedia of Science | title = Electric motor | edition = 4th | year = 2008 | publisher = Gale | location = Detroit}} as cited on {{cite web | url = http://www.gale.cengage.com/InContext/resources.htm | title = Student Resources in Context | access-date = 2011-09-07 | publisher = Gale}}</ref><ref>{{cite encyclopedia | encyclopedia = U*X*L Encyclopedia of Science | title = Electric motor | year = 2007 | publisher = U*X*L}} as cited on {{cite web | url = http://www.gale.cengage.com/InContext/resources.htm | title = Student Resources in Context | access-date = 2011-09-07 | publisher = Gale}}</ref>
* A [[Electric generator|generator]] converts mechanical energy into electrical energy.<ref>{{cite encyclopedia | encyclopedia = U*X*L Encyclopedia of Science | title = Generator | date = 2007-07-16  | publisher = U*X*L}} as cited on {{cite web | url = http://www.gale.cengage.com/InContext/resources.htm | title = Student Resources in Context | access-date = 2011-10-09 | publisher = Gale}}</ref>
* A [[hydroelectric powerplant]] converts the mechanical energy of water in a storage dam into electrical energy.<ref>{{cite web|title=Hydroelectric Power|url=http://www.waterencyclopedia.com/Ge-Hy/Hydroelectric-Power.html|publisher=Water Encyclopedia}} Retrieved 2013-08-23</ref>
* An [[internal combustion engine]] is a [[heat engine]] that obtains mechanical energy from [[chemical energy]] by burning [[fuel]]. From this mechanical energy, the internal combustion engine often generates electricity.<ref>{{cite encyclopedia |editor1=Lerner, K. Lee |editor2=Lerner, Brenda Wilmoth | encyclopedia = The Gale Encyclopedia of Science | title = Internal combustion engine | edition = 4th | year = 2008 | publisher = Gale | location = Detroit}} as cited on {{cite web | url = http://www.gale.cengage.com/InContext/resources.htm | title = Student Resources in Context | access-date = 2011-10-09 | publisher = Gale}}</ref>
* A [[steam engine]] converts the [[Heat|heat energy]] of steam into mechanical energy.<ref>{{cite encyclopedia | encyclopedia = U*X*L Encyclopedia of Science | title = Steam engine | date = 2007-07-16  | publisher = U*X*L}} as cited on {{cite web | url = http://www.gale.cengage.com/InContext/resources.htm | title = Student Resources in Context | access-date = 2011-10-09 | publisher = Gale}}</ref>
* A [[turbine]] converts the kinetic energy of a stream of gas or liquid into mechanical energy.<ref>{{cite encyclopedia |editor1=Lerner, K. Lee |editor2=Lerner, Brenda Wilmoth | encyclopedia = The Gale Encyclopedia of Science | title = Turbine | edition = 4th | year = 2008 | publisher = Gale | location = Detroit}} as cited on {{cite web | url = http://www.gale.cengage.com/InContext/resources.htm | title = Student Resources in Context | access-date = 2011-10-09 | publisher = Gale}}</ref>

==Distinction from other types==

The classification of energy into different types often follows the boundaries of the fields of study in the natural sciences.

* [[Chemical energy]] is the kind of [[potential energy]] "stored" in [[chemical bonds]] and is studied in [[chemistry]].<ref>{{cite encyclopedia | last = Atkins | first = Peter W. | encyclopedia = AccessScience | title = Chemical energy | url = http://accessscience.com/abstract.aspx?id=127100&referURL=http%3a%2f%2faccessscience.com%2fcontent.aspx%3fsearchStr%3dchemical%2benergy%26id%3d127100 | access-date = 2011-10-17 | year = 2008 | publisher = McGraw-Hill Companies | archive-url = https://web.archive.org/web/20130719080323/http://accessscience.com/abstract.aspx?id=127100&referURL=http%3a%2f%2faccessscience.com%2fcontent.aspx%3fsearchStr%3dchemical%2benergy%26id%3d127100 | archive-date = 2013-07-19 | url-status = dead }}</ref>
* [[Nuclear potential energy|Nuclear energy]] is energy stored in interactions between the particles in the [[atomic nucleus]] and is studied in [[nuclear physics]].<ref>{{cite encyclopedia | last = Duckworth | first = Henry E. | author2 = Wilkinson, D. H. | encyclopedia = AccessScience | title = Nuclear binding energy | url = http://accessscience.com/abstract.aspx?id=457950&referURL=http%3a%2f%2faccessscience.com%2fcontent.aspx%3fsearchStr%3dnuclear%2benergy%26id%3d457950 | access-date = 2011-10-17 | year = 2008 | publisher = McGraw-Hill Companies | archive-url = https://web.archive.org/web/20130719064114/http://accessscience.com/abstract.aspx?id=457950&referURL=http%3a%2f%2faccessscience.com%2fcontent.aspx%3fsearchStr%3dnuclear%2benergy%26id%3d457950 | archive-date = 2013-07-19 | url-status = dead }}</ref>
* [[Electromagnetic energy]] is in the form of electric charges, magnetic fields, and [[photons]]. It is studied in [[electromagnetism]].<ref>{{cite encyclopedia | last = Hartwig | first = William H. | encyclopedia = AccessScience | title = Electrical energy measurement | url = http://accessscience.com/abstract.aspx?id=218350&referURL=http%3a%2f%2faccessscience.com%2fcontent.aspx%3fsearchStr%3dElectrical%2benergy%2bmeasurement%26id%3d218350 | access-date = 2011-10-17 | year = 2008 | publisher = McGraw-Hill Companies | archive-url = https://web.archive.org/web/20130719095247/http://accessscience.com/abstract.aspx?id=218350&referURL=http%3a%2f%2faccessscience.com%2fcontent.aspx%3fsearchStr%3dElectrical%2benergy%2bmeasurement%26id%3d218350 | archive-date = 2013-07-19 | url-status = dead }}</ref><ref>{{cite encyclopedia | last = Smythe | first = William R. | encyclopedia = AccessScience | title = Electromagnetic radiation | url = http://accessscience.com/abstract.aspx?id=222700&referURL=http%3a%2f%2faccessscience.com%2fcontent.aspx%3fsearchStr%3dElectromagnetic%2bradiation%26id%3d222700 | access-date = 2011-10-17 | year = 2008 | publisher = McGraw-Hill Companies | archive-url = https://web.archive.org/web/20130719082043/http://accessscience.com/abstract.aspx?id=222700&referURL=http%3a%2f%2faccessscience.com%2fcontent.aspx%3fsearchStr%3dElectromagnetic%2bradiation%26id%3d222700 | archive-date = 2013-07-19 | url-status = dead }}</ref>
* Various forms of energy in [[quantum mechanics]]; e.g., the [[energy level]]s of [[electron]]s in an atom.<ref>{{cite encyclopedia | last = Gerjuoy | first = Edward | encyclopedia = AccessScience | title = Quantum mechanics | url = http://accessscience.com/abstract.aspx?id=562900&referURL=http%3a%2f%2faccessscience.com%2fcontent.aspx%3fsearchStr%3dQuantum%2bmechanics%26id%3d562900 | access-date = 2011-10-17 | year = 2008 | publisher = McGraw-Hill Companies | archive-url = https://web.archive.org/web/20130719081651/http://accessscience.com/abstract.aspx?id=562900&referURL=http%3a%2f%2faccessscience.com%2fcontent.aspx%3fsearchStr%3dQuantum%2bmechanics%26id%3d562900 | archive-date = 2013-07-19 | url-status = dead }}</ref><ref>{{cite encyclopedia | last = March-Russell | first = John | encyclopedia = AccessScience | title = Energy level (quantum mechanics) | url = http://accessscience.com/abstract.aspx?id=232800&referURL=http%3a%2f%2faccessscience.com%2fcontent.aspx%3fsearchStr%3dEnergy%2blevel%2b%28quantum%2bmechanics%29%26id%3d232800 | access-date = 2011-10-17 | year = 2008 | publisher = McGraw-Hill Companies | archive-url = https://web.archive.org/web/20130719062552/http://accessscience.com/abstract.aspx?id=232800&referURL=http%3a%2f%2faccessscience.com%2fcontent.aspx%3fsearchStr%3dEnergy%2blevel%2b%28quantum%2bmechanics%29%26id%3d232800 | archive-date = 2013-07-19 | url-status = dead }}</ref>

==References==
'''Notes'''
{{reflist|group="nb"}}

'''Citations'''
{{reflist|30em}}

'''Bibliography'''
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* {{cite book | last1 = Jain | first1 = Mahesh C. | title = Textbook of Engineering Physics, Part I | publisher = PHI Learning Pvt. Ltd. | year = 2009 | location = New Delhi | url = https://books.google.com/books?id=DqZlU3RJTywC | access-date = 2011-08-25 | isbn = 978-81-203-3862-3}}
* {{cite book | last1 = Newton | first1 = Isaac | author-link1 = Isaac Newton | title = The Principia: mathematical principles of natural philosophy |editor1= I. Bernard Cohen |editor2=Anne Miller Whitman | publisher = University of California Press | year = 1999 | location = United States of America | isbn = 978-0-520-08816-0}}

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