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| [[Image:chain_hoist.svg|thumb|right|Example of a differential pulley]]
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| A '''differential pulley''', also called "Weston differential pulley", sometimes "chain hoist" or colloquially "chain fall", is used to manually lift very heavy objects like car engines. It is operated by pulling upon the slack section of a continuous chain that wraps around pulleys. The relative size of two connected pulleys determines the maximum weight that can be lifted by hand.
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| ==History==
| | website In tɦe foyer of the MGM Grand casino, where СomƄs heɑdlined a "platinum party" as part of a 20th anniѵersary celebration for Foxwoods.<br><br>website Growth, which averaged almost seven percеnt per annum, was brօadbaѕed with manufacturing, agriculture and services sectors posting rօbust growth. Instead of acкnowledging and appreсiating the growth peгformances of half a decade, some so cаlleɗ 'experts' began to criticise these vehemently and continued to do so սntil the [http://data.gov.uk/data/search?q=growth+momentum growth momentum] was halted іn 200708.<br><br>website<br>Furthermore, it features a rеmovable ID Һolder, ɑ comfortable leather handlе as well as shiny golden brass pieces. Ϲlosed by a Slock, it is accompanied by two crafted trunk latches which recall the timelesѕ craftsmаnship of Louіs [http://tinyurl.com/po55k38 cheap louis vuitton bags]. website<br>What seemed like a creative, high еnergy, glamourfi lleԁ occupation to the hopeful fгeshmanmay easily [http://tinyurl.com/po55k38 louis vuitton outlet] deteriorate into a monotonousdeskjob of phonecallѕ foг sourcing new [http://tinyurl.com/po55k38 cheap louis vuitton bags] [http://tinyurl.com/po55k38 cheap louis vuitton bags] handbags 2013 and production haggling. website [http://tinyurl.com/po55k38 cheap louis vuitton bags] Handbags |
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| The differential pulley was invented in 1854 by Thomas Aldridge Weston from [[King's Norton]], England.<ref>[http://oldengine.org/members/kmorris/Webpages/TangyeHistory3.html The Weston Pulley Block]</ref><ref name=HoistMagazine>Hoist Magazine - [http://www.hoistmagazine.com/features/the-history-makers/ The history makers]</ref>
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| The pulleys were manufactured in collaboration with [[Richard Tangye|Richard]] and George Tangye. According to Richard Tangye's autobiography, the Weston differential pulley evolved from the [[Chinese windlass]], with an endless chain replacing the finite length of rope. He claimed that many engineering firms conceded on the difficulty of efficiently disengaging the chain from the teeth as the pulleys turned, but his firm developed a "pitch" chain which solved the issue. Marketed as "Weston Differential Pulley Blocks with Patent Chain Guides", the pulley had good sales, namely, 3000 sets in 9 months. It was displayed in 5 sizes — from {{convert|10|long cwt|kg}} to {{convert|3|LT|kg}} — at the [[1862 International Exhibition]] in London and received a medal for "original application, practical utility and success".
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| An ironmonger challenged the Tangyes that the pulley had been in use for 30 years before Weston's patent but the judge, [[William Page Wood]] ruled in favour of the Tangyes because the engaging mechanism was substantially different from the one presented as evidence.<ref>Powerhouse Museum, Sydney, Australia - [http://www.powerhousemuseum.com/collection/database/?irn=12102 2003/45/1 Differential pulley block, large and small sheaves, 2 ton (2.032 tonnes) capacity, invented by T A Weston / made by Tangyes Ltd, Birmingham, England, 1860 - 1890]</ref>
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| The [[Yale Lock Company]] acquired the patent rights in 1876.<ref name=HoistMagazine />
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| ==Mode of operation==
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| A differential pulley can lift very large [[Mass|masses]] a short distance. It consists of two fixed [[pulley]]s of unequal radii that are attached to each other and rotate together, a single pulley bearing the load, and an endless rope looped around the pulleys. To avoid slippage, the rope is usually replaced by a chain, and the connected pulleys by [[sprocket]]s.
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| The two sections of chain carrying the single pulley exert opposing and unequal [[torque]]s on the connected pulleys, such that only the ''difference'' of these torques has to be compensated manually by pulling the loose part of the chain.
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| This leads to a [[mechanical advantage]]: the force needed to lift a load is only a fraction of the load's [[weight]]. At the same time, the distance the load is lifted is smaller than the length of chain pulled by the same factor. This factor (the mechanical advantage ''MA'') depends on the relative difference of the radii ''r'' and ''R'' of the connected pulleys:
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| :<math>M\!A = \frac{2 R}{R-r} = \frac{2}{1 - \frac{r}{R}}</math>
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| The effect on the forces and distances (see figure) is quantitatively:
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| :<math>F_\mathrm{Z}=\frac{F_\mathrm{L}}{M\!A}\quad,\quad h=\frac{s}{M\!A}\quad.</math>
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| The difference in radii can be made very small, making the mechanical advantage of this pulley system very large.<ref>Black, N. Henry, and Harvey N. Davis, ''Practical physics, fundamental principles and applications to daily life'', 2 ed., New York: McMillan, 1922, chapter II, section 34, p. [http://books.google.com/books?id=mbAXAAAAIAAJ&jtp=39 39].</ref><ref>United States Naval Education and Training Program Development Center, ''Basic machines and how they work'', New York: Dover Publications, 1994, p. [http://books.google.com/books?id=yDKzy4rKEg0C&pg=PT18 2-6].</ref>
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| In the extreme case of zero difference in radii, ''MA'' becomes infinite, thus no force (besides friction) is needed to move the chain, but moving the chain will no longer lift the load.
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| At the other extreme, when ''r'' is zero, the system becomes a simple [[Block_and_tackle#Example_Block_and_Tackle_Configurations|''gun tackle'']] with a mechanical advantage of 2.
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| [[File:Comparison_differential_pulley_windlass.svg|thumb|Comparison of a differential pulley (left) and a differential windlass or Chinese windlass (right). The rope of the windlass is depicted as spirals for clarity, but is more likely helices with axes perpendicular to the image.]] | |
| The same principle is used in a differential [[windlass]], where the connected pulleys are replaced by [[winch]]es.
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| ==Calculation of mechanical advantage==
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| In the above graphic, the four segments of the chain are labelled W, X, Y and Z. The magnitudes of their corresponding forces are ''F''<sub>W</sub>, ''F''<sub>X</sub>, ''F''<sub>Y</sub> and ''F''<sub>Z</sub>, respectively.
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| Assuming that the chain is massless, ''F''<sub>X</sub> = 0 because segment X is not supporting any weight.
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| Taking the system at equilibrium, ''F''<sub>W</sub> and ''F''<sub>Y</sub> are equal — if they were not, the lower pulley would freely turn until they were.
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| Next, the downward force acting on the lower pulley equal the upward forces acting on it, so
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| :''F''<sub>L</sub> = ''F''<sub>W</sub> + ''F''<sub>Y</sub>, or 2 ''F''<sub>W</sub> because ''F''<sub>W</sub> = ''F''<sub>Y</sub>.
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| Additionally, there is no net [[torque]] or moment around the compound pulley, so the clockwise torque is equal to the anticlockwise torque:
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| :''F''<sub>W</sub> ''R'' + ''F''<sub>X</sub> ''r'' = ''F''<sub>Y</sub> ''r'' + ''F''<sub>Z</sub> ''R'' . | |
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| Substituting ''F''<sub>X</sub> and ''F''<sub>Y</sub> from the above equations,
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| :''F''<sub>W</sub> ''R'' + 0 = ''F''<sub>W</sub> ''r'' + ''F''<sub>Z</sub> ''R'' .
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| Rearranging gives
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| :''F''<sub>W</sub> = ''F''<sub>Z</sub> · {{sfrac|''R''|''R'' − ''r''}} . | |
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| As ''F''<sub>W</sub> = {{sfrac|''F'' <sub>L</sub>|2}},
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| :{{sfrac|''F'' <sub>L</sub>|2}} = ''F''<sub>Z</sub> · {{sfrac|''R''|''R'' − ''r''}} .
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| Finally, the mechanical advantage, {{sfrac|''F'' <sub>L</sub>|''F'' <sub>Z</sub>}} = {{sfrac|2 ''R''|''R'' − ''r''}} or {{sfrac|2|1 − {{sfrac|''r''|''R''}} }} .
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| ==See also==
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| * [[Block and tackle]]
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| * [[Hoist (device)|Hoist]]
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| ==References==
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| {{reflist}}
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| [[Category:Mechanics]]
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| [[Category:Simple machines]]
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| [[Category:Mechanisms]]
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