Editing Virtual work (section)

== Law of the lever ==
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'''<sub>''A''</sub> at a point ''A'' located by the coordinate vector '''r'''<sub>''A''</sub> on the bar.  The lever then exerts an output force '''F'''<sub>''B''</sub> at the point ''B'' located by '''r'''<sub>''B''</sub>.   The rotation of the lever about the fulcrum ''P'' is defined by the rotation angle ''θ''.
[[Image:Archimedes lever (Small).jpg|thumb|right|300px|This is an engraving from ''Mechanics Magazine'' published in London in 1824.]]

Let the coordinate vector of the point ''P'' that defines the fulcrum be '''r'''<sub>''P''</sub>, and introduce the lengths 
<math display="block"> a = |\mathbf{r}_A -  \mathbf{r}_P|, \quad  b = |\mathbf{r}_B -  \mathbf{r}_P|, </math>
which are the distances from the fulcrum to the input point ''A'' and to the output point ''B'', respectively.

Now introduce the unit vectors '''e'''<sub>''A''</sub> and '''e'''<sub>''B''</sub> from the fulcrum to the point ''A'' and ''B'', so
<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>
This notation allows us to define the velocity of the points ''A'' and ''B'' as
<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>
where '''e'''<sub>''A''</sub><sup>⊥</sup> and '''e'''<sub>''B''</sub><sup>⊥</sup> are unit vectors perpendicular to '''e'''<sub>''A''</sub> and '''e'''<sub>''B''</sub>, respectively.

The angle ''θ'' is the generalized coordinate that defines the configuration of the lever, therefore using the formula above for forces applied to a one degree-of-freedom mechanism, the generalized force is given by
<math display="block"> Q =  \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).</math>

Now, denote as ''F''<sub>''A''</sub> and ''F''<sub>''B''</sub> the components of the forces that are perpendicular to the radial segments ''PA'' and ''PB''.   These forces are given by
<math display="block"> F_A = \mathbf{F}_A \cdot \mathbf{e}_A^\perp, \quad  F_B = \mathbf{F}_B \cdot \mathbf{e}_B^\perp.</math>
This notation and the principle of virtual work yield the formula for the generalized force as
<math display="block"> Q = a F_A - b F_B = 0. </math>

The ratio of the output force ''F''<sub>''B''</sub> to the input force ''F''<sub>''A''</sub> is the [[mechanical advantage]] of the lever, and is obtained from the principle of virtual work as
<math display="block"> MA = \frac{F_B}{F_A} = \frac{a}{b}.</math>

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.

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}}</ref>