Editing Mechanical advantage (section)

== Levers ==
{{main|Lever}}
{{duplicates|section=y|dupe=Law of the lever}}

The lever is a movable bar that pivots on a [[wikt:fulcrum|fulcrum]] attached to or positioned on or across a fixed point. The lever operates by applying forces at different distances from the fulcrum, or pivot.  The location of the fulcrum determines a lever's [[Lever#Classes of levers|class]].  Where a lever rotates continuously, it functions as a rotary second-class lever.   The motion of the lever's end-point describes a fixed orbit, where mechanical energy can be exchanged.  (see a hand-crank as an example.)

In modern times, this kind of rotary leverage is widely used;  see a (rotary) 2nd-class lever; see gears, pulleys or friction drive, used in a mechanical power transmission scheme.  It is common for mechanical advantage to be manipulated in a 'collapsed' form, via the use of more than one gear (a gearset).  In such a gearset, gears having smaller radii and less inherent mechanical advantage are used.  In order to make use of non-collapsed mechanical advantage, it is necessary to use a 'true length' rotary lever.   See, also, the incorporation of mechanical advantage into the design of certain types of electric motors; one design is an 'outrunner'.  

[[Image:Lever mechanical advantage.png|thumb|right|300px]]
As the lever pivots on the fulcrum, points farther from this pivot move faster than points closer to the pivot.  The [[Power (physics)#Mechanical power|power]] into and out of the lever is the same, so must come out the same when calculations are being done. Power is the product of force and velocity, so forces applied to points farther from the pivot must be less than when applied to points closer in.<ref name="Uicker"/>

If ''a'' and ''b'' are distances from the fulcrum to points ''A'' and ''B'' and if force ''F<sub>A</sub>'' applied to ''A'' is the input force and ''F<sub>B</sub>'' exerted at ''B'' is the output, the  ratio of the velocities of points ''A'' and ''B'' is given by {{sfrac|''a''|''b''}} so the ratio of the output force to the input force, or mechanical advantage, is given by

:<math display="block">\mathit{MA} = \frac{F_b}{F_a} = \frac{a}{b}.</math>

This is the ''law of the lever'', which [[Archimedes]] formulated 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}}</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.  If the distance from the fulcrum to the input force is less than from the fulcrum to the output force, then the lever reduces the input force.   To Archimedes, who recognized the profound implications and practicalities of the law of the lever, has been attributed the famous claim, "Give me a place to stand and with a lever I will move the whole world."<ref>[[John Tzetzes]] ''Book of Histories (Chiliades) 2'' p 129-130, 12th century AD, translation by Francis R. Walton</ref>

The use of velocity in the static analysis of a lever is an application of the principle of [[virtual work]].
<!-- A technical note:   While it is common for an elementary study of the lever to focus on the work done by the input and output forces as they move through a displacement, the theory considers only differential or ''virtual'' displacements.  The easiest way to define a virtual displacement is as a velocity over a ''virtual''  moment of time.  This leads to the consideration of power rather than work, which has the practical benefit that power is the primary consideration in the study of machines and mechanisms.  This approach to the static analysis of lever determines its mechanical advantage in exactly same way as for the drive train of a car, and for a robot arm.-->