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{{redirects|Ramp}} |
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{{About|the physical structure}} |
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[[Image:Chartres, Hôtel Montescot 08 rampe PMR.jpg|thumb|[[Wheelchair ramp]], Hotel India, Chartres, France]] |
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[[File:Piano inclinato inv 1041 IF 21341.jpg|thumb|Demonstration inclined plane used in education, [[Museo Galileo]], Florence.]] |
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An '''inclined plane''', also known as a '''ramp''', is a flat supporting surface tilted at an angle, with one end higher than the other, used as an nless on an inclined plane due to [[friction]], without sliding down. This angle is equal to the [[arctangent]] of the [[coefficient of static friction]] ''μ<sub>s</sub>'' between the surfaces.<ref name="Ambekar">{{cite book|url=https://books.google.com/books?id=N-f5f-CytUIC&pg=PA446&dq=%22inclined+plane%22+%22angle+of+repose%22&hl=en&sa=X&ei=jB9dT_TaLfDYiQLuury6Cw&ved=0CDoQ6AEwAQ#v=onepage&q=%22inclined%20plane%22%20%22angle%20of%20repose%22&f=false|title=Mechanism and Machine Theory|last=Ambekar|first=A. G.|publisher=PHI Learning|year=2007|isbn=978-81-203-3134-1|location=|pages=446|doi=|id=|quote=Angle of repose is the limiting angle of inclination of a plane when a body, placed on the inclined plane, just starts sliding down the plane.|authorlink=}}</ref> |
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Two other simple machines are often considered to be derived from the inclined plane.<ref name="Rosen">{{cite book |
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| last = Rosen |
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| first = Joe |
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| authorlink = |
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|author2=Lisa Quinn Gothard |
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| title = Encyclopedia of Physical Science, Volume 1 |
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| publisher = Infobase Publishing |
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| year = 2009 |
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| location = |
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| pages = 375 |
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| url = https://books.google.com/books?id=avyQ64LIJa0C&pg=PA375&dq=%22inclined+plane%22+wedge+screw&hl=en&sa=X&ei=gyFdT7WXJNDYiALNpa2iDQ&ved=0CGYQ6AEwCA#v=onepage&q=%22inclined%20plane%22%20wedge%20screw&f=false |
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| isbn = 978-0-8160-7011-4}}</ref> The [[wedge (mechanical device)|wedge]] can be considered a moving inclined plane or two inclined planes connected at the base.<ref name="Ortleb">{{cite book|url=https://books.google.com/books?id=GgDAaXgL0ssC&pg=PR2&lpg=PR2&dq=%22inclined+plane%22+prehistoric+ancient&source=bl&ots=kuHkcGVJjq&sig=OA2J6tzAKOBSkBm12b9RUkpYwEI&hl=en&sa=X&ei=x0ldT7jaOKPmiAK3g4WUCw&ved=0CEkQ6AEwBDgK#v=onepage&q=%22inclined%20plane%22%20prehistoric%20ancient&f=false|title=Machines and Work|last=Ortleb|first=Edward P.|author2=Richard Cadice|publisher=Lorenz Educational Press|year=1993|isbn=978-1-55863-060-4|location=|pages=iv|doi=|id=|authorlink=}}</ref> The [[screw (simple machine)|screw]] consists of a narrow inclined plane wrapped around a [[cylinder (geometry)|cylinder]].<ref name="Ortleb" /> |
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The term may also refer to a specific implementation; a straight ramp cut into a steep hillside for transporting goods up and down the hill. It may include cars on rails or pulled up by a cable system; a [[funicular]] or [[cable railway]], such as the [[Johnstown Inclined Plane]]. |
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==Uses== |
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Inclined planes are widely used in the form of ''loading ramps'' to load and unload goods on trucks, ships and planes.<ref name="Edinformatics">{{cite web|url=http://www.edinformatics.com/math_science/simple_machines/inclined_plane.htm|title=The Inclined Plane|last=|first=|authorlink=|year=1999|work=Math and science activity center|publisher=Edinformatics|doi=|accessdate=March 11, 2012}}</ref> [[Wheelchair ramp]]s are used to allow people in [[wheelchair]]s to get over vertical obstacles without exceeding their strength.<ref name="Tiner">{{cite book|url=https://books.google.com/books?id=qTCT5ajPmRIC&pg=PA38&lpg=PA38|title=Exploring the World of Physics: From Simple Machines to Nuclear Energy|last=Tiner|first=John Hudson|publisher=New Leaf Publishing Group|year=2006|isbn=978-0-89051-466-5|location=|pages=37–38|doi=|id=|authorlink=}}</ref> [[Escalator]]s and slanted [[conveyor belt]]s are also forms of inclined plane.<ref name="TeachEngineering">{{cite web|url=http://www.teachengineering.org/view_lesson.php?url=collection/cub_/lessons/cub_simple/cub_simple_lesson04.xml|title=Lesson 04:Slide Right on By Using an Inclined Plane|last=Reilly|first=Travis|authorlink=|date=November 24, 2011|work=Teach Engineering|publisher=College of Engineering, Univ. of Colorado at Boulder|format=|doi=|accessdate=September 8, 2012}}</ref> In a [[funicular]] or [[cable railway]] a railroad car is pulled up a steep inclined plane using cables. Inclined planes also allow heavy fragile objects, including humans, to be safely lowered down a vertical distance by using the [[normal force]] of the plane to reduce the [[gravitational force]]. Aircraft [[evacuation slide]]s allow people to rapidly and safely reach the ground from the height of a passenger [[airliner]]. |
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{{multiple image |
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| align = center |
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| footer = |
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| image1 =2008-08-11 Unloading a VW New Beetle 1.jpg |
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| caption1 = Using ramps to load a car on a truck |
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| width1 = 151 |
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| image2 = Bulgarian Excavator on KrAZ-truck.jpg |
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| caption2 = Loading a truck on a ship using a ramp |
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| width2 = 143 |
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| image3 = Emergency exit slide.jpg |
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| caption3 = Aircraft emergency [[evacuation slide]] |
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| width3 = 154 |
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| image4 = Slope for Wheelchairs in omnibus.jpg |
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| caption4 = [[Wheelchair ramp]] on Japanese bus |
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| width4 = 134 |
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| image5 =PenskeRamp.jpg |
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| caption5 = Loading ramp on a truck |
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| width5 = 271 |
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}} |
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Other inclined planes are built into permanent structures. Roads for vehicles and railroads have inclined planes in the form of gradual slopes, ramps, and [[causeway]]s to allow vehicles to surmount vertical obstacles such as hills without losing traction on the road surface.<ref name="Edinformatics" /><ref name="Tiner" /> Similarly, pedestrian paths and [[sidewalk]]s have gentle ramps to limit their slope, to ensure that pedestrians can keep traction.<ref name="Cole">{{cite book|url=https://books.google.com/books?id=RhuciGEQ1G8C&pg=PA178|title=Explore science, 2nd Ed.|last=Cole|first=Matthew|publisher=Pearson Education|year=2008|isbn=978-981-06-2002-8|location=|pages=178|doi=|id=|authorlink=}}</ref><ref name="Silverman">{{cite book|url=https://books.google.com/books?id=d3QKpqsbxFIC&pg=PA8&dq=%22inclined+plane%22+work&hl=en&sa=X&ei=IY9cT9yxAdLWiALiqK3dCw&ved=0CE8Q6AEwBA#v=onepage&q=%22inclined%20plane%22%20work&f=false|title=Simple Machines: Forces in Action, 4th Ed.|last=Silverman|first=Buffy|publisher=Heinemann-Raintree Classroom|year=2009|isbn=978-1-4329-2317-4|location=USA|pages=7|doi=|id=|authorlink=}}</ref> Inclined planes are also used as entertainment for people to slide down in a controlled way, in [[playground slide]]s, [[water slide]]s, [[ski slope]]s and [[skateboard park]]s. |
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{{multiple image |
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| align = center |
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| footer = |
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| image1 =Vista general de Masada.jpg |
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| caption1 = Earth ramp ''(right)'' built by Romans in 72 AD to invade [[Masada]], Palestine |
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| width1 = 169 |
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| image2 = Planalto do Planalto Entrance.jpg |
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| caption2 = Pedestrian ramp, Palacio do Planalto, Brasilia |
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| width2 = 188 |
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| image3 = JohnstownIncline.jpg |
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| caption3 = Johnstown Inclined Plane, a [[funicular]] railroad. |
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| width3 = 172 |
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| image4 = Ledo Burma Roads Assam-Burma-China.gif |
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| caption4 = Burma Road, Assam, India, through Burma to China 1945 |
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| width4 = 104 |
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| image5 =Rollin1.jpg |
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| caption5 = Inclined planes in a skateboard park |
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| width5 = 192 |
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}} |
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==History== |
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{| class="toccolours" style="float: margin-left: 1em; margin-right: 1em; font-size: 100%; background:#c6dbf7; color:black; width: 40%; float: right;" cellspacing="3" |
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| style="text-align: center;" |'''Stevin's proof''' |
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|- |
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| style="text-align: left;" |[[Image:StevinEquilibrium.svg|center|150px]] In 1586, Flemish engineer [[Simon Stevin]] (Stevinus) derived the mechanical advantage of the inclined plane by an argument that used a string of beads.<ref name="Koetsier">{{cite conference |
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| first = Teun |
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| last = Koetsier |
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| authorlink = |
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| title = Simon Stevin and the rise of Archimedean mechanics in the Renaissance |
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| booktitle = 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 |
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| pages = 94–99 |
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| publisher = Springer |
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| year = 2010 |
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| location = |
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| url = https://books.google.com/books?id=65Pz4_XJrgwC&pg=PA95 |
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| doi = |
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| isbn = 978-90-481-9090-4 |
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| accessdate = }}</ref> He imagined two inclined planes of equal height but different slopes, placed back-to-back (above) as in a prism. A loop of string with beads at equal intervals is draped over the inclined planes, with part hanging down below. The beads resting on the planes act as loads on the planes, held up by the tension force in the string at point ''T''. Stevin's argument goes like this:<ref name="Koetsier" /><ref name="Devreese">{{cite book |
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| last = Devreese |
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| first = Jozef T. |
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| authorlink = |
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|author2=Guido Vanden Berghe |
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| title = 'Magic is no magic': The wonderful world of Simon Stevin |
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| publisher = WIT Press |
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| year = 2008 |
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| location = |
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| pages = 136–139 |
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| url = https://books.google.com/books?id=f59h2ooQGmcC&pg=PA136 |
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| doi = |
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| id = |
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| isbn = 978-1-84564-391-1}}</ref><ref name="Feynman">{{cite book |
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| last = Feynman |
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| first = Richard P. |
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| authorlink = |
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|author2=Robert B. Leighton |author3=Matthew Sands |
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| title = The Feynman Lectures on Physics, Vol. I |
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| publisher = California Inst. of Technology |
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| year = 1963 |
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| location = USA |
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| pages = 4.4 – 4.5 |
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| url = https://books.google.com/books?id=bDF-uoUmttUC&pg=SA4-PA4 |
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| doi = |
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| id = |
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| isbn = 978-0-465-02493-3}}</ref> |
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*The string must be stationary, in [[static equilibrium]]. If it was heavier on one side than the other, and began to slide right or left under its own weight, when each bead had moved to the position of the previous bead the string would be indistinguishable from its initial position and therefore would continue to be unbalanced and slide. This argument could be repeated indefinitely, resulting in a circular [[perpetual motion]], which is absurd. Therefore, it is stationary, with the forces on the two sides at point ''T'' (''above'') equal. |
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*The portion of the chain hanging below the inclined planes is symmetrical, with an equal number of beads on each side. It exerts an equal force on each side of the string. Therefore, this portion of the string can be cut off at the edges of the planes ''(points S and V)'', leaving only the beads resting on the inclined planes, and this remaining portion will still be in static equilibrium. |
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*Since the beads are at equal intervals on the string, the total number of beads supported by each plane, the total load, is proportional to the length of the plane. Since the input supporting force, the tension in the string, is the same for both, the mechanical advantage of each plane is proportional to its slant length |
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As pointed out by Dijksterhuis,<ref>E.J.Dijksterhuis: ''Simon Stevin'' 1943</ref> Stevin's argument is not completely tight. The forces exerted by the hanging part of the chain need not be symmetrical because the hanging part ''need not retain its shape'' when let go. Even if the chain is released with a zero angular momentum, motion including oscillations is possible unless the chain is initially in its equilibrium configuration, a supposition which would make the argument circular. |
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|} |
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Inclined planes have been used by people since prehistoric times to move heavy objects.<ref name="Conn">Therese McGuire, ''Light on Sacred Stones'', in {{cite book |
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| last = Conn |
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| first = Marie A. |
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| authorlink = |
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|author2=Therese Benedict McGuire |
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| title = Not etched in stone: essays on ritual memory, soul, and society |
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| publisher = University Press of America |
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| year = 2007 |
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| location = |
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| pages = 23 |
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| url = https://books.google.com/books?id=kEPkDyvek3sC&pg=PA23 |
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| doi = |
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| id = |
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| isbn = 978-0-7618-3702-2}}</ref><ref name="Dutch">{{cite web |
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| last = Dutch |
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| first = Steven |
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| authorlink = |
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| title = Pre-Greek Accomplishments |
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| work = Legacy of the Ancient World |
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| publisher = Prof. Steve Dutch's page, Univ. of Wisconsin at Green Bay |
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| year = 1999 |
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| url = http://www.uwgb.edu/dutchs/westtech/xancient.htm |
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| doi = |
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| accessdate = March 13, 2012}}</ref> The sloping roads and [[causeway]]s built by ancient civilizations such as the Romans are examples of early inclined planes that have survived, and show that they understood the value of this device for moving things uphill. The heavy stones used in ancient stone structures such as [[Stonehenge]]<ref name="Moffett">{{cite book |
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| last = Moffett |
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| first = Marian |
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| authorlink = |
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|author2=Michael W. Fazio |author3=Lawrence Wodehouse |
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| title = A world history of architecture |
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| publisher = Laurence King Publishing |
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| year = 2003 |
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| location = |
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| pages = 9 |
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| url = https://books.google.com/books?id=IFMohetegAcC&pg=PT8 |
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| doi = |
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| id = |
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| isbn = 978-1-85669-371-4}}</ref> are believed to have been moved and set in place using inclined planes made of earth,<ref name="Peet">{{cite book |
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| last = Peet |
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| first = T. Eric |
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| authorlink = |
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| title = Rough Stone Monuments and Their Builders |
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| publisher = Echo Library |
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| year = 2006 |
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| location = |
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| pages = 11–12 |
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| url = https://books.google.com/books?id=2c15PS0uwHEC&printsec=frontcover&dq=ancient+prehistoric+%22inclined+plane%22&hl=en&sa=X&ei=yrhfT6xVhtiIApmJzNQE&ved=0CFoQ6AEwBjgo#v=snippet&q=slope&f=false |
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| doi = |
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| id = |
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| isbn = 978-1-4068-2203-8}}</ref> although it is hard to find evidence of such temporary building ramps. The [[Egyptian pyramids]] were constructed using inclined planes,<ref name="Thomas">{{cite web |
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| last = Thomas |
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| first = Burke |
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| authorlink = |
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| title = Transport and the Inclined Plane |
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| work = Construction of the Giza Pyramids |
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| publisher = world-mysteries.com |
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| year = 2005 |
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| url = http://www.world-mysteries.com/gw_tb_gp.htm |
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| doi = |
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| accessdate = March 10, 2012}}</ref><ref name="Isler">{{cite book |
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| last = Isler |
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| first = Martin |
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| authorlink = |
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| title = Sticks, stones, and shadows: building the Egyptian pyramids |
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| publisher = University of Oklahoma Press |
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| year = 2001 |
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| location = USA |
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| pages = 211–216 |
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| url = https://books.google.com/books?id=Ip-tqz1xGkoC&pg=PA325 |
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| doi = |
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| id = |
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| isbn = 978-0-8061-3342-3}}</ref><ref name="SpragueDeCamp">{{cite book |
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| last = Sprague de Camp |
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| first = L. |
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| authorlink = |
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| title = The Ancient Engineers |
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| publisher = Barnes & Noble |
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| year = 1990 |
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| location = USA |
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| pages = 43 |
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| url = https://books.google.com/books?id=cauMt9vJLs0C&printsec=frontcover&dq=%22sprague+de+camp%22+ancient+engineers&hl=en&sa=X&ei=LFJdT4CfCI_YiAL5773LCw&ved=0CDMQ6AEwAA#v=onepage&q=ramp&f=false |
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| doi = |
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| id = |
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| isbn = 978-0-88029-456-0}}</ref> [[Siege]] ramps enabled ancient armies to surmount fortress walls. The ancient Greeks constructed a paved ramp 6 km (3.7 miles) long, the [[Diolkos]], to drag ships overland across the [[Isthmus of Corinth]].<ref name="Silverman" /> |
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However the inclined plane was the last of the six classic [[simple machine]]s to be recognised as a machine. This is probably because it is a passive, motionless device (the load is the moving part),<ref name="Reuleaux" /> and also because it is found in nature in the form of slopes and hills. Although they understood its use in lifting heavy objects, the [[Ancient Greece|ancient Greek]] philosophers who defined the other five simple machines did not include the inclined plane as a machine.<ref>for example, the lists of simple machines left by Roman architect [[Marcus Vitruvius Pollio|Vitruvius]] (c. 80 – 15 BCE) and Greek philosopher [[Heron of Alexandria]] (c. 10 – 70 CE) consist of the five classical simple machines, excluding the inclined plane. – {{cite book |
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| last = Smith |
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| first = William |
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| authorlink = |
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| title = Dictionary of Greek and Roman antiquities |
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| publisher = Walton and Maberly; John Murray |
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| year = 1848 |
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| location = London |
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| pages = 722 |
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| url = https://books.google.com/books?id=zfIrAAAAYAAJ&pg=PA722&dq=%22inclined+plane%22+%22mechanical+powers%22+greek&hl=en&sa=X&ei=TapdT8aaDufbiALkrKW1Ag&ved=0CDQQ6AEwAA#v=onepage&q=%22inclined%20plane%22%20%22mechanical%20powers%22%20greek&f=false |
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| doi = |
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| id = |
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| isbn = }}, {{cite book |
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|last=Usher |
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|first=Abbott Payson |
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|authorlink= |
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|title=A History of Mechanical Inventions |
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|publisher=Courier Dover Publications |
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|year=1988 |
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|location=USA |
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|pages=98, 120 |
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|url=https://books.google.com/books?id=xuDDqqa8FlwC&pg=PA196#v=snippet&q=wedge%20and%20screw&f=false |
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|doi= |
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|isbn=978-0-486-25593-4 |
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|postscript=.}}</ref> This view persisted among a few later scientists; as late as 1826 [[Karl Christian von Langsdorf|Karl von Langsdorf]] wrote that an inclined plane "''...is no more a machine than is the slope of a mountain.''<ref name="Reuleaux">Karl von Langsdorf (1826) ''Machinenkunde'', quoted in {{cite book |
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| last = Reuleaux |
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| first = Franz |
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| authorlink = |
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| title = The kinematics of machinery: Outlines of a theory of machines |
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| publisher = MacMillan |
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| year = 1876 |
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| location = |
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| pages = 604 |
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| url = https://books.google.com/books?id=WUZVAAAAMAAJ&pg=PA604&dq=#v=onepage&q&f=false |
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| doi = |
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| id = |
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| isbn = }}</ref> The problem of calculating the force required to push a weight up an inclined plane (its mechanical advantage) was attempted by Greek philosophers [[Heron of Alexandria]] (c. 10 - 60 CE) and [[Pappus of Alexandria]] (c. 290 - 350 CE), but they got it wrong.<ref>{{cite book |
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| last = Heath |
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| first = Thomas Little |
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| authorlink = |
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| title = A History of Greek Mathematics, Vol. 2 |
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| publisher = The Clarendon Press |
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| year = 1921 |
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| location = UK |
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| pages = 349, 433–434 |
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| url = https://books.google.com/books?id=7DDQAAAAMAAJ&pg=PA351 |
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| doi = |
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| id = |
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| isbn = }}</ref><ref name="Laird">Egidio Festa and Sophie Roux, ''The enigma of the inclined plane'' in {{cite book |
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| last = Laird |
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| first = Walter Roy |
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| authorlink = |
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|author2=Sophie Roux |
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| title = Mechanics and natural philosophy before the scientific revolution |
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| publisher = Springer |
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| year = 2008 |
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| location = USA |
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| pages = 195–221 |
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| url = https://books.google.com/books?id=z3pRa83qz2IC&pg=PA209&dq=stevin+#v=onepage&q=stevin%20&f=false |
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| doi = |
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| id = |
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| isbn = 978-1-4020-5966-7}}</ref><ref name="Meli">{{cite book |
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| last = Meli |
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| first = Domenico Bertoloni |
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| authorlink = |
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| title = Thinking With Objects: The Transformation of Mechanics in the Seventeenth Century |
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| publisher = JHU Press |
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| year = 2006 |
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| location = |
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| pages = 35–39 |
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| url = https://books.google.com/books?id=I6QreZN02joC&printsec=frontcover&dq=%22inclined+plane%22+stevin+jordanus+galileo&hl=en&sa=X&ei=JIhdT4PjHsPaiQKXkMWHCw&ved=0CG8Q6AEwCA#v=onepage&q=inclined%20plane&f=false |
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| doi = |
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| id = |
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| isbn = 978-0-8018-8426-9}}</ref> |
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It wasn't until the [[Renaissance]] that the inclined plane was classed with the other simple machines. The first correct analysis of the inclined plane appeared in the work of enigmatic 13th century author [[Jordanus de Nemore]],<ref name="Boyer">{{cite book |
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| last = Boyer |
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| first = Carl B. |
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| authorlink = |
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|author2=Uta C. Merzbach|author2-link= Uta Merzbach |
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| title = A History of Mathematics, 3rd Ed. |
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| publisher = John Wiley and Sons |
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| year = 2010 |
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| location = |
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| pages = |
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| url = https://books.google.com/books?id=BokVHiuIk9UC&pg=PT243&dq=%22inclined+plane%22+stevin+jordanus+galileo&hl=en&sa=X&ei=JIhdT4PjHsPaiQKXkMWHCw&ved=0CFMQ6AEwBA#v=onepage&q=%22inclined%20plane%22%20stevin%20jordanus%20galileo&f=false |
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| doi = |
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| id = |
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| isbn = 978-0-470-63056-3}}</ref><ref name="Usher">{{cite book |
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| last = Usher |
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| first = Abbott Payson |
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| authorlink = |
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| title = A History of Mechanical Inventions |
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| publisher = Courier Dover Publications |
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| year = 1988 |
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| location = |
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| pages = 106 |
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| url = https://books.google.com/books?id=xuDDqqa8FlwC&pg=PA106#v=snippet&q=inclined%20plane&f=false |
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| doi = |
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| id = |
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| isbn = 978-0-486-25593-4}}</ref> however his solution was apparently not communicated to other philosophers of the time.<ref name="Laird" /> [[Girolamo Cardano]] (1570) proposed the incorrect solution that the input force is proportional to the angle of the plane.<ref name="Koetsier" /> Then at the end of the 16th century, three correct solutions were published within ten years, by Michael Varro (1584), [[Simon Stevin]] (1586), and Galileo Galilee (1592).<ref name="Laird" /> Although it was not the first, the derivation of Flemish engineer [[Simon Stevin]]<ref name="Meli" /> is the most well-known, because of its originality and use of a string of beads (see box).<ref name="Feynman" /><ref name="Boyer" /> In 1600, Italian scientist [[Galileo Galilei]] included the inclined plane in his analysis of simple machines in ''Le Meccaniche'' ("On Mechanics"), showing its underlying similarity to the other machines as a force amplifier.<ref name="Machamer">{{cite book |
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| last = Machamer |
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| first = Peter K. |
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| authorlink = |
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| title = The Cambridge Companion to Galileo |
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| publisher = Cambridge University Press |
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| year = 1998 |
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| location = London |
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| pages = 47–48 |
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| url = https://books.google.com/books?id=1wEFPLoqTeAC&pg=PA48&dq=%22inclined+plane%22+galileo+Meccaniche&hl=en&sa=X&ei=FphdT9abKYmpiQLZ26jSCw&ved=0CDkQ6AEwAQ#v=onepage&q=%22inclined%20plane%22%20galileo%20Meccaniche&f=false |
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| doi = |
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| id = |
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| isbn = 978-0-521-58841-6}}</ref> |
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The first elementary rules of sliding [[friction]] on an inclined plane were discovered by [[Leonardo da Vinci]] (1452-1519), but remained unpublished in his notebooks.<ref name="Armstrong">{{cite book |
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| last = Armstrong-Hélouvry |
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| first = Brian |
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| authorlink = |
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| title = Control of machines with friction |
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| publisher = Springer |
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| year = 1991 |
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| location = USA |
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| pages = 10 |
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| url = https://books.google.com/books?id=0zk_zI3xACgC&pg=PA10&dq=friction+leonardo+da+vinci+amontons+coulomb&hl=en&ei=b8GMTcP6EanE0QG9sKywCw&sa=X&oi=book_result&ct=result&resnum=2&ved=0CC8Q6AEwAQ#v=onepage&q=friction%20leonardo%20da%20vinci%20amontons%20coulomb&f=false |
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| doi = |
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| id = |
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| isbn = 978-0-7923-9133-3}}</ref> They were rediscovered by [[Guillaume Amontons]] (1699) and were further developed by [[Charles-Augustin de Coulomb]] (1785).<ref name="Armstrong" /> [[Leonhard Euler]] (1750) showed that the [[Tangent (trigonometric function)|tangent]] of the [[angle of repose]] on an inclined plane is equal to the [[coefficient of friction]].<ref name="Meyer">{{cite book |
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| last = Meyer |
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| first = Ernst |
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| authorlink = |
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| title = Nanoscience: friction and rheology on the nanometer scale |
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| publisher = World Scientific |
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| year = 2002 |
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| location = |
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| pages = 7 |
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| url = https://books.google.com/books?id=Rhi7odTe2BEC&pg=PA7&dq=%22Leonhard+euler%22+angle+%22inclined+plane%22&hl=en&sa=X&ei=YK9fT4CtJO3XiAKH_NTUBA&ved=0CDMQ6AEwAA#v=onepage&q=%22Leonhard%20euler%22%20angle%20%22inclined%20plane%22&f=false |
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| doi = |
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| id = |
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| isbn = 978-981-238-062-3}}</ref> |
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==Terminology== |
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===Slope=== |
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The [[mechanical advantage]] of an inclined plane depends on its ''[[slope]]'', meaning its [[Grade (slope)|gradient]] or steepness. The smaller the slope, the larger the mechanical advantage, and the smaller the force needed to raise a given weight. A plane's slope ''s'' is equal to the difference in height between its two ends, or "''rise''", divided by its horizontal length, or "''run''".<ref name="Tiner" /><ref name="Handley">{{cite book |
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| last = Handley |
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| first = Brett |
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| authorlink = |
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|author2=David M. Marshall |author3=Craig Coon |
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| title = Principles of Engineering |
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| publisher = Cengage Learning |
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| year = 2011 |
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| location = |
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| pages = 71–73 |
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| url = https://books.google.com/books?id=3YBeXkp-AacC&pg=PT91&lpg=PT91&dq=%22inclined+plane%22+slope+angle+%22mechanical+advantage%22&source=bl&ots=fv4F2vb0uT&sig=gJBSYxtVOdAU-RovppyExeG9tTA&hl=en&sa=X&ei=3TJMUNT3KM3biwKf74HYAg&ved=0CDEQ6AEwAA#v=onepage&q=%22inclined%20plane%22%20slope%20angle%20%22mechanical%20advantage%22&f=false |
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| doi = |
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| id = |
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| isbn = 978-1-4354-2836-2}}</ref> It can also be expressed by the angle the plane makes with the horizontal, ''θ''. |
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[[Image:Inclined plane terminology.svg|thumb|The inclined plane's geometry is based on a [[right triangle]].<ref name="Handley" /> The horizontal length is sometimes called ''Run'', the vertical change in height ''Rise''.]] |
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:<math>\theta = \tan^{-1} \bigg( \frac {\text{Rise}}{\text{Run}} \bigg) \,</math> |
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===Mechanical advantage=== |
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The [[mechanical advantage]] ''MA'' of a simple machine is defined as the ratio of the output force exerted on the load to the input force applied. For the inclined plane the output load force is just the gravitational force of the load object on the plane, its weight '''''F<sub>w</sub>'''''. The input force is the force '''''F<sub>i</sub>''''' exerted on the object, parallel to the plane, to move it up the plane. The mechanical advantage is |
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:<math>\mathrm{MA} = \frac {F_w}{F_i}. \,</math> |
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The MA of an ideal inclined plane without friction is sometimes called ''ideal mechanical advantage'' (IMA) while the MA when friction is included is called the ''actual mechanical advantage'' (AMA).<ref name="Dennis">{{cite book |
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| last = Dennis |
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| first = Johnnie T. |
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| authorlink = |
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| title = The Complete Idiot's Guide to Physics |
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| publisher = Penguin |
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| year = 2003 |
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| location = |
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| pages = 116–117 |
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| url = https://books.google.com/books?id=y-cB37zUU90C&pg=PA117&lpg=PA117&dq=%22ideal+mechanical+advantage%22+actual+mechanical+advantage%22+%22inclined+plane%22&source=bl&ots=b9X5-cV-Hw&sig=ydicT0UVbTCHOAltVKE4RPY3S4s&hl=en&sa=X&ei=rydMUMuFEcGUiALx84G4BQ&ved=0CGEQ6AEwCDgK#v=onepage&q=%22ideal%20mechanical%20advantage%22%20actual%20mechanical%20advantage%22%20%22inclined%20plane%22&f=false |
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| doi = |
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| id = |
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| isbn = 978-1-59257-081-2}}</ref> |
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==Frictionless inclined plane== |
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[[File:Schiefe-ebene hg.jpg|right|thumb|Instrumented inclined plane used for physics education, around 1900. The lefthand weight provides the load force '''''F<sub>w</sub>'''''. The righthand weight provides the input force '''''F<sub>i</sub>''''' pulling the roller up the plane.]] |
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If there is no [[friction]] between the object being moved and the plane, the device is called an ''ideal inclined plane''. This condition might be approached if the object is rolling, like a [[barrel]], or supported on wheels or [[caster]]s. Due to [[conservation of energy]], for a frictionless inclined plane the [[work (physics)|work]] done on the load lifting it, '''''W<sub>out</sub>''''', is equal to the work done by the input force, '''''W<sub>in</sub>'''''<ref name="Hyperphysics">{{cite web |
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| last = Nave |
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| first = Carl R. |
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| authorlink = |
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| title = The Incline |
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| work = Hyperphysics |
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| publisher = Dept. of Physics and Astronomy, Georgia State Univ. |
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| year = 2010 |
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| url = http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/incline.html |
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| doi = |
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| accessdate = September 8, 2012}}</ref><ref name="Westminster">{{cite web |
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| last = Martin |
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| first = Lori |
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| authorlink = |
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| title = Lab Mech14:The Inclined Plane - A Simple Machine |
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| work = Science in Motion |
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| publisher = Westminster College |
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| year = 2010 |
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| url = http://www.westminster.edu/acad/sim/documents/STHEINCLINEDPLANE.pdf |
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| format = |
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| doi = |
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| accessdate = September 8, 2012}}</ref><ref name="Pearson">{{cite book |
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| last = Pearson |
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| first = |
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| authorlink = |
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| title = Physics class 10 - The IIT Foundation Series |
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| publisher = Pearson Education India |
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| year = 2009 |
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| location = New Delhi |
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| pages = 69 |
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| url = https://books.google.com/books?id=4Gm_PiLjwBcC&pg=PA69&lpg=PA69&dq=%22inclined+plane%22++%22mechanical+advantage%22+slope+angle&source=bl&ots=jHFSqdndUs&sig=A8riHRznWLbjOcvIPoA-_Gm7usI&hl=en&sa=X&ei=uy9MULfnCIOdiQLw2IHQBA&ved=0CEkQ6AEwBA#v=onepage&q=%22inclined%20plane%22%20%20%22mechanical%20advantage%22%20slope%20angle&f=false |
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| doi = |
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| id = |
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| isbn = 978-81-317-2843-7}}</ref> |
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:<math>W_{out} = W_{in} \,</math> |
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Work is defined as the force multiplied by the displacement an object moves. The work done on the load is just equal to its weight multiplied by the vertical displacement it rises, which is the "rise" of the inclined plane |
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:<math>W_{out} = F_w \cdot \text{Rise} \,</math> |
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The input work is equal to the force '''''F<sub>i</sub>''''' on the object times the diagonal length of the inclined plane. |
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:<math>W_{in} = F_i \cdot \text{Length} \,</math> |
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Substituting these values into the conservation of energy equation above and rearranging |
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:<math>\text{MA} = \frac{F_w}{F_i} = \frac {\text{Length}}{\text{Rise}} \,</math> |
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To express the mechanical advantage by the angle ''θ'' of the plane,<ref name="Westminster" /> it can be seen from the diagram ''(above)'' that |
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:<math>\sin \theta = \frac {\text{Rise}}{\text{Length}} \,</math> |
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So |
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:<math>\text{MA} = \frac{F_w}{F_i} = \frac {1}{\sin \theta} \,</math> |
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So the mechanical advantage of a frictionless inclined plane is equal to the reciprocal of the sine of the slope angle. The input force '''''F<sub>i</sub>''''' from this equation is the force needed to hold the load motionless on the inclined plane, or push it up at a constant velocity. If the input force is greater than this, the load will accelerate up the plane; if the force is less, it will accelerate down the plane. |
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==Inclined plane with friction== |
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Where there is [[friction]] between the plane and the load, as for example with a heavy box being slid up a ramp, some of the work applied by the input force is dissipated as heat by friction, '''''W'''''<sub>fric</sub>, so less work is done on the load. |
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:<math>W_\text{in} = W_\text{fric} + W_\text{out} \,</math> |
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Therefore, more input force is required, and the mechanical advantage is lower, than if friction were not present. |
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With friction, the load will only move if the net force parallel to the surface is greater than the frictional force '''''F<sub>f</sub>''''' opposing it.<ref name="Ambekar" /><ref name="Bansal">{{cite book |
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| last = Bansal |
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| first = R.K |
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| authorlink = |
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| title = Engineering Mechanics and Strength of Materials |
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| publisher = Laxmi Publications |
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| year = 2005 |
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| location = |
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| pages = 165–167 |
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| url = https://books.google.com/books?id=PajxWzB6jusC&pg=PA165&dq=inclined+plane+friction&hl=en |
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| doi = |
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| id = |
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| isbn = 978-81-7008-094-7}}</ref><ref name="Gujral">This derives slightly more general equations which cover force applied at any angle: {{cite book |
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| last = Gujral |
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| first = I.S. |
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| authorlink = |
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| title = Engineering Mechanics |
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| publisher = Firewall Media |
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| year = 2008 |
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| location = |
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| pages = 275–277 |
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| url = https://books.google.com/books?id=JM0OG-XUyu0C&pg=PA275&dq=%22inclined+plane%22+%22angle+of+repose%22+coefficient+of+friction&hl=en&sa=X&ei=oURrT8S9GqXYiQKw3f2IBQ&ved=0CFIQ6AEwBDgU#v=onepage&q=%22inclined%20plane%22%20%22angle%20of%20repose%22%20coefficient%20of%20friction&f=false |
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| doi = |
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| id = |
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| isbn = 978-81-318-0295-3}}</ref> The maximum friction force is given by |
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:<math>F_f = \mu F_n \,</math> |
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where '''''F<sub>n</sub>''''' is the [[normal force]] between the load and the plane, directed normal to the surface, and ''μ'' is the [[coefficient of static friction]] between the two surfaces, which varies with the material. When no input force is applied, if the inclination angle ''θ'' of the plane is less than some maximum value ''φ'' the component of gravitational force parallel to the plane will be too small to overcome friction, and the load will remain motionless. This angle is called the [[angle of repose]] and depends on the composition of the surfaces, but is independent of the load weight. It is shown below that the [[tangent (trigonometry)|tangent]] of the angle of repose ''φ'' is equal to ''μ'' |
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:<math>\phi = \tan^{-1} \mu \,</math> |
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With friction, there is always some range of input force '''''F<sub>i</sub>''''' for which the load is stationary, neither sliding up or down the plane, whereas with a frictionless inclined plane there is only one particular value of input force for which the load is stationary. |
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===Analysis=== |
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[[Image:Free body1.3.svg|right|thumb|Key: ''F''<sub>n</sub> = ''N'' = [[Normal force]] that is perpendicular to the plane, ''F''<sub>i</sub> = ''f'' = input force, ''F''<sub>w</sub> = ''mg'' = weight of the load, where m = [[mass]], g = [[gravity]] ]] |
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A load resting on an inclined plane, when considered as a [[Free body diagram|free body]] has three forces acting on it:<ref name="Ambekar" /><ref name="Bansal" /><ref name="Gujral" /> |
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*The applied force, '''''F<sub>i</sub>''''' exerted on the load to move it, which acts parallel to the inclined plane. |
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*The weight of the load, '''''F<sub>w</sub>''''', which acts vertically downwards |
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*The force of the plane on the load. This can be resolved into two components: |
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**The normal force '''''F<sub>n</sub>''''' of the inclined plane on the load, supporting it. This is directed perpendicular ([[surface normal|normal]]) to the surface. |
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**The frictional force, '''''F<sub>f</sub>''''' of the plane on the load acts parallel to the surface, and is always in a direction opposite to the motion of the object. It is equal to the normal force multiplied by the [[coefficient of static friction]] μ between the two surfaces. |
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Using [[Newton's second law of motion]] the load will be stationary or in steady motion if the sum of the forces on it is zero. Since the direction of the frictional force is opposite for the case of uphill and downhill motion, these two cases must be considered separately: |
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*'''Uphill motion:''' The total force on the load is toward the uphill side, so the frictional force is directed down the plane, opposing the input force. |
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{| class="wikitable" width=100%; |
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|- |
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| |
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{{show |
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|Derivation of mechanical advantage for uphill motion |
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| |
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The equilibrium equations for forces parallel and perpendicular to the plane are |
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:<math>\sum F_{\|} = F_i - F_f - F_w \sin \theta = 0 \,</math> |
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:<math>\sum F_{\perp} = F_n - F_w \cos \theta = 0 \,</math> |
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:Substituting <math>F_f = \mu F_n \,</math> into first equation |
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:<math>F_i - \mu F_n - F_w \sin \theta = 0 \,</math> |
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:Solving second equation to get <math>F_n = F_w \cos \theta \,</math> and substituting into the above equation |
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:<math>F_i - \mu F_w \cos \theta - F_w \sin \theta = 0 \, </math> |
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:<math>\frac {F_w}{F_i} = \frac {1} {\sin \theta + \mu \cos \theta} \,</math> |
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:Defining <math>\mu = \tan \phi \,</math> |
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:<math>\frac {F_w}{F_i} = \frac {1} {\sin \theta + \tan \phi \cos \theta} = \dfrac{1}{\sin \theta + \dfrac {\sin \phi}{\cos \phi} \cos \theta} = \frac {\cos \phi} {\sin \theta \cos \phi + \cos \theta \sin \phi } \,</math> |
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:Using a sum-of-angles [[trigonometric identity]] on the denominator, |
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}} |
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|} |
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:The mechanical advantage is |
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:<math>\mathrm{MA} = \frac {F_w}{F_i} = \frac {\cos \phi} { \sin (\theta + \phi ) } \, </math> |
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:where <math>\phi = \tan^{-1} \mu \,</math>. This is the condition for ''impending motion'' up the inclined plane. If the applied force '''''F<sub>i</sub>''''' is greater than given by this equation, the load will move up the plane. |
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*'''Downhill motion:''' The total force on the load is toward the downhill side, so the frictional force is directed up the plane. |
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{| class="wikitable" width=100%; |
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|- |
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| |
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{{show |
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|Derivation of mechanical advantage for downhill motion |
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| |
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The equilibrium equations are |
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:<math>\sum F_{\|} = F_i + F_f - F_w \sin \theta = 0 \,</math> |
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:<math>\sum F_{\perp} = F_n - F_w \cos \theta = 0 \,</math> |
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:Substituting <math>F_f = \mu F_n \,</math> into first equation |
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:<math>F_i + \mu F_n - F_w \sin \theta = 0 \,</math> |
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:Solving second equation to get <math>F_n = F_w \cos \theta \,</math> and substituting into the above equation |
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:<math>F_i + \mu F_w \cos \theta - F_w \sin \theta = 0 \, </math> |
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:<math>\frac {F_w}{F_i} = \frac {1} {\sin \theta - \mu \cos \theta} \,</math> |
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:Substituting in <math>\mu = \tan \phi \,</math> and simplifying as above |
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:<math>\frac {F_w}{F_i} = \frac {\cos \phi} {\sin \theta \cos \phi - \cos \theta \sin \phi } \,</math> |
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:Using another [[trigonometric identity]] on the denominator, |
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}} |
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|} |
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:The mechanical advantage is |
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:<math>\mathrm{MA} = \frac {F_w}{F_i} = \frac {\cos \phi} { \sin (\theta - \phi ) } \,</math> |
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:This is the condition for impending motion down the plane; if the applied force '''''F<sub>i</sub>''''' is less than given in this equation, the load will slide down the plane. There are three cases: |
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:#<math>\theta < \phi\,</math>: The mechanical advantage is negative. In the absence of applied force the load will remain motionless, and requires some negative (downhill) applied force to slide down. |
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:#<math>\theta = \phi\,</math>: The '[[angle of repose]]'. The mechanical advantage is infinite. With no applied force, load will not slide, but the slightest negative (downhill) force will cause it to slide. |
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:#<math>\theta > \phi\,</math>: The mechanical advantage is positive. In the absence of applied force the load will slide down the plane, and requires some positive (uphill) force to hold it motionless |
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==Mechanical advantage using power== |
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[[Image:Free body.svg|right|thumb|Key: N = [[Normal force]] that is perpendicular to the plane, W=mg, where m = [[mass]], g = [[gravity]], and θ ([[theta]]) = Angle of inclination of the plane]] |
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The [[mechanical advantage]] of an inclined plane is the ratio of the weight of the load on the ramp to the force required to pull it up the ramp. If energy is not dissipated or stored in the movement of the load, then this mechanical advantage can be computed from the dimensions of the ramp. |
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In order to show this, let the position '''r''' of a rail car on along the ramp with an angle, ''θ'', above the horizontal be given by |
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:<math>\mathbf{r} = R (\cos\theta, \sin\theta),</math> |
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where ''R'' is the distance along the ramp. The velocity of the car up the ramp is now |
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:<math>\mathbf{v} = V (\cos\theta, \sin\theta).</math> |
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Because there are no losses, the power used by force ''F'' to move the load up the ramp equals the power out, which is the vertical lift of the weight ''W'' of the load. |
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The input power pulling the car up the ramp is given by |
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:<math>P_{\mathrm{in}} = F V,\!</math> |
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and the power out is |
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:<math>P_{\mathrm{out}} = \mathbf{W}\cdot\mathbf{v} = (0, W)\cdot V (\cos\theta, \sin\theta) = WV\sin\theta.</math> |
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Equate the power in to the power out to obtain the mechanical advantage as |
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:<math> \mathrm{MA} = \frac{W}{F} = \frac{1}{\sin\theta}.</math> |
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The mechanical advantage of an inclined can also be calculated from the ratio of length of the ramp ''L'' to its height ''H,'' because the sine of the angle of the ramp is given by |
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:<math> \sin\theta = \frac{H}{L},</math> |
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therefore, |
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:<math> \mathrm{MA} = \frac{W}{F} = \frac{L}{H}.</math> |
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[[File:Plan incliné machine stationnaire Liverpool Minard.jpg|right|thumb|400px|Layout of the cable drive system for the Liverpool Minard inclined plane.]] |
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Example: If the height of a ramp is H = 1 meter and its length is L = 5 meters, then the mechanical advantage is |
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:<math> \mathrm{MA} = \frac{W}{F} = 5,</math> |
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which means that a 20 lb force will lift a 100 lb load. |
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The Liverpool Minard inclined plane has the dimensions 1804 meters by 37.50 meters, which provides a mechanical advantage of |
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:<math> \mathrm{MA} = \frac{W}{F} = 1804/37.50 = 48.1,</math> |
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so a 100 lb tension force on the cable will lift a 4810 lb load. The grade of this incline is 2%, which means the angle θ is small enough that sinθ=tanθ. |
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==See also== |
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{{Div col|colwidth=30em}} |
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* [[Canal inclined plane]] |
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* [[Frictionless plane]] |
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* [[Grade (slope)]] |
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* [[Inclined plane railroad]] |
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* [[Mechanical advantage]] |
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* [[Ramp function]] |
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* [[Schiefe Ebene]] |
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* [[Simple machine]] |
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* [[Stairs]] |
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{{div col end}} |
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==References== |
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{{Reflist}} |
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==External links== |
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{{commons category|Inclined planes}} |
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*[http://www.phy.hk/wiki/englishhtm/Incline.htm An interactive simulation of Physics inclined plane] |
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<!-- This listing is better found in the article on simple machines {{Simple machines}}--> |
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<!-- Inclined planes are not architectural elements link rooms --> |
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{{Simple machines}} |
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{{Authority control}} |
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[[Category:Simple machines]] |
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