Editing Lever (section)

== Virtual work and the 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'' in radians.
[[File:Archimedes lever (Small).jpg|thumb|right|Archimedes lever, 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>

The velocity of the points ''A'' and ''B'' are obtained 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, and the [[generalized force]] associated with this coordinate is given by

<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>

where ''F''<sub>''A''</sub> and ''F''<sub>''B''</sub> 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

<math display="block"> F_\theta = a F_A - b F_B = 0. \,\!</math>

[[File:Seesaw1902.jpg|thumb|Simple lever, fulcrum, and vertical posts]]
Thus, the ratio of the output force ''F''<sub>''B''</sub> to the input force ''F''<sub>''A''</sub> is obtained as

<math display="block"> MA = \frac{F_B}{F_A} = \frac{a}{b},</math>

which is the [[mechanical advantage]] of the lever.

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.