[ 0 => '{{Short description|Simple machine consisting of a beam pivoted at a fixed hinge}}', 1 => '{{About|the simple machine}}', 2 => '{{Infobox machine', 3 => '| name = Lever', 4 => '| image = Palanca-ejemplo.jpg', 5 => '| caption = Levers can be used to exert a large force over a small distance at one end by exerting only a small force (effort) over a greater distance at the other.', 6 => '| classification = [[Simple machine]]', 7 => '| industry = ', 8 => '| application =', 9 => '| dimensions ', 10 => '| weight = ', 11 => '| fuel_source = ', 12 => '| powered =', 13 => '| self-propelled =', 14 => '| wheels =', 15 => '| tracks =', 16 => '| legs =', 17 => '| aerofoils =', 18 => '| axles =', 19 => '| components = fulcrum or pivot, load and effort', 20 => '| invented = ', 21 => '| inventor = ', 22 => '| examples = see-saw, bottle opener, etc.', 23 => '}}', 24 => 'A '''lever''' is a [[simple machine]] consisting of a [[beam (structure)|beam]] or rigid rod pivoted at a fixed [[hinge]], or ''[[:wikt:fulcrum|fulcrum]]''. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is divided into [[Lever#Classes of levers|three types]]. It is one of the six [[simple machine]]s identified by Renaissance scientists. A lever amplifies an input force to provide a greater output force, which is said to provide '''leverage''', which is [[mechanical advantage]] gained in the system, equal to the ratio of the output force to the input force. As such, the lever is a [[mechanical advantage device]], trading off force against movement.', 25 => '', 26 => '== Etymology ==', 27 => 'The word "lever" entered [[English language|English]] around AD 1300 from {{lang-fr|label=[[Old French]]|link=yes|text=levier|i=yes}}. This sprang from the stem of the verb ''lever'', meaning "to raise". The verb, in turn, goes back to {{lang-la|levare}},<ref>{{cite EB1911 |wstitle=Lever |volume=16 |page=510}}</ref> itself from the adjective ''levis'', meaning "light" (as in "not heavy"). The word's primary origin is the [[Proto-Indo-European language|Proto-Indo-European]] stem {{lang|mis|legwh-}}, meaning "light", "easy" or "nimble", among other things. The PIE stem also gave rise to the English word "light".<ref>{{Cite web |url=http://www.etymonline.com/index.php?term=lever |title=Etymology of the word "lever" in the Online Etymological |access-date=2015-04-29 |archive-date=2015-05-12 |archive-url=https://web.archive.org/web/20150512104818/http://www.etymonline.com/index.php?term=lever |url-status=live }}</ref>', 28 => '', 29 => '== History ==', 30 => 'The earliest evidence of the lever mechanism dates back to the [[ancient Near East]] {{circa|5000 BC}}, when it was first used in a simple [[balance scale]].<ref name="Paipetis">{{cite book |last1=Paipetis |first1=S. A. |last2=Ceccarelli |first2=Marco |title=The Genius of Archimedes -- 23 Centuries of Influence on Mathematics, Science and Engineering: Proceedings of an International Conference held at Syracuse, Italy, June 8-10, 2010 |date=2010 |publisher=[[Springer Science & Business Media]] |isbn=9789048190911 |page=416}}</ref> In [[ancient Egypt]] {{circa|4400 BC}}, a foot pedal was used for the earliest horizontal frame [[loom]].<ref>{{cite book |last1=Bruno |first1=Leonard C. |last2=Olendorf |first2=Donna |title=Science and technology firsts |date=1997 |publisher=[[Gale Research]] |isbn=9780787602567 |page=[https://archive.org/details/sciencetechnolog0000brun/page/2 2] |url=https://archive.org/details/sciencetechnolog0000brun |url-access=registration |quote=4400 B.C. Earliest evidence of the use of a horizontal loom is its depiction on a pottery dish found in Egypt and dated to this time. These first true frame looms are equipped with foot pedals to lift the warp threads, leaving the weaver's hands free to pass and beat the weft thread.}}</ref> In [[Mesopotamia]] (modern Iraq) {{circa|3000 BC}}, the [[shadouf]], a crane-like device that uses a lever mechanism, was invented.<ref name="Paipetis"/> In [[Ancient Egyptian technology|ancient Egypt]], workmen used the lever to move and uplift obelisks weighing more than 100 tons. This is evident from the recesses in the large blocks and the [[Lifting boss|handling bosses]] which could not be used for any purpose other than for levers.<ref>{{cite book |last1=Clarke |first1=Somers |last2=Engelbach |first2=Reginald |title=Ancient Egyptian Construction and Architecture |date=1990 |publisher=[[Courier Corporation]] |isbn=9780486264851 |pages=86–90}}</ref> ', 31 => '', 32 => 'The earliest remaining writings regarding levers date from the 3rd century BC and were provided by the Greek mathematician [[Archimedes]], who famously stated "Give me a lever long enough and a fulcrum on which to place it, and I shall move the world."', 33 => '', 34 => 'Autumn Stanley argues that the digging stick can be considered the first lever, which would position prehistoric women as the inventors of lever technology.<ref>{{Cite book |last=Stanley |first=Autumn |title=Machina Ex Dea: Feminist Perspectives on Technology |publisher=Pergamon Press |year=1983 |editor-last=Rothschild |editor-first=Joan |chapter="Women Hold Up Two-Thirds of the Sky: Notes for a Revised History of Technology."}}</ref> ', 35 => '', 36 => '== Force and levers ==', 37 => '[[File:Lever Principle 3D.png|thumb|right|A lever in balance]]', 38 => 'A lever is a beam connected to ground by a hinge, or pivot, called a fulcrum. The ideal lever does not dissipate or store energy, which means there is no friction in the hinge or bending in the beam. In this case, the power into the lever equals the power out, and the ratio of output to input force is given by the ratio of the distances from the fulcrum to the points of application of these forces. This is known as the '''law of the lever'''.', 39 => '', 40 => 'The mechanical advantage of a lever can be determined by considering the balance of [[Moment (physics)|moments]] or [[torque]], ''T'', about the fulcrum. If the distance traveled is greater, then the output force is lessened.', 41 => '', 42 => '<math display="block">\begin{align}', 43 => 'T_{1} &= F_{1}a,\quad \\', 44 => 'T_{2} &= F_{2}b\!', 45 => '\end{align}</math>', 46 => '', 47 => 'where F₁ is the input force to the lever and F₂ is the output force. The distances ''a'' and ''b'' are the perpendicular distances between the forces and the fulcrum.', 48 => '', 49 => 'Since the moments of torque must be balanced, <math>T_{1} = T_{2} \!</math>. So, <math>F_{1}a = F_{2}b \!</math>.', 50 => '', 51 => 'The mechanical advantage of a lever is the ratio of output force to input force.', 52 => '', 53 => '<math display="block">MA = \frac{F_{2}}{F_{1}} = \frac{a}{b}.\!</math>', 54 => '', 55 => 'This relationship shows that the mechanical advantage can be computed from ratio of the distances from the fulcrum to where the input and output forces are applied to the lever, assuming a weightless lever and no losses due to friction, flexibility or wear. This remains true even though the "horizontal" distance (perpendicular to the pull of gravity) of both ''a'' and ''b'' change (diminish) as the lever changes to any position away from the horizontal.', 56 => '', 57 => '== Classes of levers ==', 58 => '[[File:Lever (PSF).png|thumb|right|Three classes of levers]]', 59 => '[[File:Levers of the Human Body.svg|thumb|The three classifications of levers with examples of the human body]]', 60 => 'Levers are classified by the relative positions of the fulcrum, effort and resistance (or load). It is common to call the input force "effort" and the output force "load" or "resistance". This allows the identification of three classes of levers by the relative locations of the fulcrum, the resistance and the effort:<ref>{{cite book', 61 => ' |title=Physics in Biology and Medicine |edition=3rd', 62 => ' |first1=Paul', 63 => ' |last1=Davidovits', 64 => ' |publisher=Academic Press', 65 => ' |year=2008', 66 => ' |isbn=978-0-12-369411-9', 67 => ' |page=10', 68 => ' |chapter-url=https://books.google.com/books?id=e9hbt3xisb0C&pg=PA10', 69 => ' |chapter=Chapter 1', 70 => ' |access-date=2016-02-23', 71 => ' |archive-date=2014-01-03', 72 => ' |archive-url=https://web.archive.org/web/20140103072006/http://books.google.com/books?id=e9hbt3xisb0C&pg=PA10', 73 => ' |url-status=live', 74 => ' }}</ref>', 75 => '*{{anchor|Class 1 lever}} '''Class I''' – Fulcrum is located between the effort and the resistance: The effort is applied on one side of the fulcrum and the resistance (or load) on the other side. For example, a [[seesaw]], a [[Crowbar (tool)|crowbar]], a pair of [[scissors]], a [[balance scale]], a pair of [[pliers]], and a [[claw hammer]] (pulling a nail). With the fulcrum in the middle, the lever's mechanical advantage may be greater than, less than, or even equal to 1.', 76 => '*{{anchor|Class 2 lever}} '''Class II''' – Resistance (or load) is located between the effort and the fulcrum: The effort is applied on one side of the resistance and the fulcrum is located on the other side, e.g. a [[wheelbarrow]], a [[nutcracker]], a [[bottle opener]], a [[wrench]], and the [[brake]] [[Automobile pedal|pedal]] of a car. Since the load arm is smaller than the effort arm, the lever's mechanical advantage is always greater than 1. It is also called a force multiplier lever.', 77 => '*{{anchor|Class 3 lever}} '''Class III''' – Effort is located between the resistance and the fulcrum: The resistance (or load) is applied on one side of the effort and the fulcrum is located on the other side, e.g. a pair of [[tweezers]], a [[hammer]], a pair of [[tongs]], a [[fishing rod]], and the [[mandible]] of a human skull. Since the effort arm is smaller than the load arm, the lever's mechanical advantage is always less than 1. It is also called a speed multiplier lever.', 78 => '', 79 => 'These cases are described by the mnemonic ''fre 123'' where the ''f'' fulcrum is between ''r'' and ''e'' for the 1st class lever, the ''r'' resistance is between ''f'' and ''e'' for the 2nd class lever, and the ''e'' effort is between ''f'' and ''r'' for the 3rd class lever.', 80 => '', 81 => '== Compound lever ==', 82 => '{{Main|Compound lever}} ', 83 => 'A [[compound lever]] comprises several levers acting in series: the resistance from one lever in a system of levers acts as effort for the next, and thus the applied force is transferred from one lever to the next. Examples of compound levers include scales, nail clippers and piano keys.', 84 => '', 85 => 'The ''[[malleus]]'', ''[[incus]]'' and ''[[stapes]]'' are small bones in the [[middle ear]], connected as compound levers, that transfer sound waves from the [[eardrum]] to the [[oval window]] of the [[cochlea]].', 86 => '', 87 => '== Law of the lever {{anchor|Law}} ==', 88 => '{{See also|Mechanical advantage#Lever}}', 89 => 'The lever is a movable bar that pivots on a fulcrum attached to a fixed point. The lever operates by applying forces at different distances from the fulcrum, or a pivot.', 90 => '', 91 => 'As the lever rotates around the fulcrum, points farther from this pivot move faster than points closer to the pivot. Therefore, a force applied to a point farther from the pivot must be less than the force located at a point closer in, because power is the product of force and velocity.<ref>{{cite book', 92 => ' | last1 = Uicker', 93 => ' | first1 = John', 94 => ' | last2 = Pennock', 95 => ' | first2 = Gordon', 96 => ' | last3 = Shigley', 97 => ' | first3 = Joseph', 98 => ' | title = Theory of Machines and Mechanisms', 99 => ' | publisher = Oxford University Press USA', 100 => ' | edition = 4th', 101 => ' | year = 2010', 102 => ' | isbn =978-0-19-537123-9 ', 103 => '}}</ref>', 104 => '', 105 => 'If ''a'' and ''b'' are distances from the fulcrum to points ''A'' and ''B'' and the force ''F_A'' applied to ''A'' is the input and the force ''F_B'' applied at ''B'' is the output, the ratio of the velocities of points ''A'' and ''B'' is given by ''a/b'', so we have the ratio of the output force to the input force, or mechanical advantage, is given by:', 106 => '<math display="block">MA = \frac{F_B}{F_A} = \frac{a}{b}.</math>', 107 => '', 108 => 'This is the ''law of the lever'', which was proven by [[Archimedes]] using geometric reasoning.<ref name="Usher1954">{{cite book|author=Usher, A. P.|author-link=Abbott Payson Usher|title=A History of Mechanical Inventions|url=https://books.google.com/books?id=Zt4Aw9wKjm8C&pg=PA94|page=94|access-date=7 April 2013|year=1929|publisher=Harvard University Press (reprinted by Dover Publications 1988)|isbn=978-0-486-14359-0|oclc=514178|archive-date=26 July 2020|archive-url=https://web.archive.org/web/20200726002155/https://books.google.com/books?id=Zt4Aw9wKjm8C&pg=PA94|url-status=live}}</ref> It shows that if the distance ''a'' from the fulcrum to where the input force is applied (point ''A'') is greater than the distance ''b'' from fulcrum to where the output force is applied (point ''B''), then the lever amplifies the input force. On the other hand, if the distance ''a'' from the fulcrum to the input force is less than the distance ''b'' from the fulcrum to the output force, then the lever reduces the input force.', 109 => '', 110 => 'The use of velocity in the static analysis of a lever is an application of the principle of [[virtual work#Law of the Lever|virtual work]].', 111 => '', 112 => '== Virtual work and the law of the lever ==', 113 => 'A lever is modeled as a rigid bar connected to a ground frame by a hinged joint called a fulcrum. The lever is operated by applying an input force '''F'''_''A'' at a point ''A'' located by the coordinate vector '''r'''_''A'' on the bar. The lever then exerts an output force '''F'''_''B'' at the point ''B'' located by '''r'''_''B''. The rotation of the lever about the fulcrum ''P'' is defined by the rotation angle ''θ'' in radians.', 114 => '[[File:Archimedes lever (Small).jpg|thumb|right|Archimedes lever, Engraving from ''Mechanics Magazine'', published in London in 1824]]', 115 => '', 116 => 'Let the coordinate vector of the point ''P'' that defines the fulcrum be '''r'''_''P'', and introduce the lengths', 117 => '', 118 => '<math display="block"> a = |\mathbf{r}_A - \mathbf{r}_P|, \quad b = |\mathbf{r}_B - \mathbf{r}_P|, </math>', 119 => '', 120 => 'which are the distances from the fulcrum to the input point ''A'' and to the output point ''B'', respectively.', 121 => '', 122 => 'Now introduce the unit vectors '''e'''_''A'' and '''e'''_''B'' from the fulcrum to the point ''A'' and ''B'', so', 123 => '', 124 => '<math display="block"> \mathbf{r}_A - \mathbf{r}_P = a\mathbf{e}_A, \quad \mathbf{r}_B - \mathbf{r}_P = b\mathbf{e}_B.</math>', 125 => '', 126 => 'The velocity of the points ''A'' and ''B'' are obtained as', 127 => '', 128 => '<math display="block"> \mathbf{v}_A = \dot{\theta} a \mathbf{e}_A^\perp, \quad \mathbf{v}_B = \dot{\theta} b \mathbf{e}_B^\perp,</math>', 129 => '', 130 => 'where '''e'''_''A''^⊥ and '''e'''_''B''^⊥ are unit vectors perpendicular to '''e'''_''A'' and '''e'''_''B'', respectively.', 131 => '', 132 => 'The angle ''θ'' is the [[generalized coordinate]] that defines the configuration of the lever, and the [[generalized force]] associated with this coordinate is given by', 133 => '', 134 => '<math display="block"> F_\theta = \mathbf{F}_A \cdot \frac{\partial\mathbf{v}_A}{\partial\dot{\theta}} - \mathbf{F}_B \cdot \frac{\partial\mathbf{v}_B}{\partial\dot{\theta}}= a(\mathbf{F}_A \cdot \mathbf{e}_A^\perp) - b(\mathbf{F}_B \cdot \mathbf{e}_B^\perp) = a F_A - b F_B ,</math>', 135 => '', 136 => 'where ''F''_''A'' and ''F''_''B'' are components of the forces that are perpendicular to the radial segments ''PA'' and ''PB''. The principle of [[virtual work]] states that at equilibrium the generalized force is zero, that is', 137 => '', 138 => '<math display="block"> F_\theta = a F_A - b F_B = 0. \,\!</math>', 139 => '', 140 => '[[File:Seesaw1902.jpg|thumb|Simple lever, fulcrum and vertical posts]]', 141 => 'Thus, the ratio of the output force ''F''_''B'' to the input force ''F''_''A'' is obtained as', 142 => '', 143 => '<math display="block"> MA = \frac{F_B}{F_A} = \frac{a}{b},</math>', 144 => '', 145 => 'which is the [[mechanical advantage]] of the lever.', 146 => '', 147 => 'This equation shows that if the distance ''a'' from the fulcrum to the point ''A'' where the input force is applied is greater than the distance ''b'' from fulcrum to the point ''B'' where the output force is applied, then the lever amplifies the input force. If the opposite is true that the distance from the fulcrum to the input point ''A'' is less than from the fulcrum to the output point ''B'', then the lever reduces the magnitude of the input force.', 148 => '', 149 => '== See also ==', 150 => '{{div col}}', 151 => '* {{annotated link|Applied mechanics}}', 152 => '* [[Balance lever coupling]]', 153 => '* [[Bascule (disambiguation)|bascule]]', 154 => '* {{annotated link|Linkage (mechanical)}}', 155 => '* {{annotated link|Mechanism (engineering)}}', 156 => '* {{annotated link|On the Equilibrium of Planes}}', 157 => '* {{annotated link|Simple machine}}', 158 => '', 159 => '{{div col end}}', 160 => '', 161 => '== References ==', 162 => '{{Reflist}}', 163 => '', 164 => '== External links ==', 165 => '{{Commons category|Levers}}', 166 => '{{Wiktionary}}', 167 => '*[https://web.archive.org/web/20070114204336/http://www.diracdelta.co.uk/science/source/l/e/lever/source.html Lever] at Diracdelta science and engineering encyclopedia', 168 => '* ''[http://demonstrations.wolfram.com/ASimpleLever/ A Simple Lever]'' by [[Stephen Wolfram]], [[Wolfram Demonstrations Project]].', 169 => '* [http://www.enchantedlearning.com/physics/machines/Levers.shtml Levers: Simple Machines] at EnchantedLearning.com', 170 => '', 171 => '{{Simple machines}}', 172 => '{{Authority control}}', 173 => '', 174 => '[[Category:Mechanisms (engineering)]]', 175 => '[[Category:Simple machines]]', 176 => '[[Category:Ancient inventions]]', 177 => '[[Category:Egyptian inventions]]' ]