page_title: Virtual work
section_title: Law of the lever
hierarchical_section_title: Virtual work / Law of the lever
caption_attribution_description: This an engraving from Mechanics Magazine published in London in 1824.
context_page_description: Virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different for different displacements. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action. This displacement is therefore the displacement followed by the particle according to the principle of least action. The work of a force on a particle along a virtual displacement is known as the virtual work.
  Historically, virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies, but they have also been developed for the study of the mechanics of deformable bodies.
context_section_description: 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  FA at a point A located by the coordinate vector rA on the bar.  The lever then exerts an output force FB at the point B located by rB.   The rotation of the lever about the fulcrum P is defined by the rotation angle θ.
  Let the coordinate vector of the point P that defines the fulcrum be rP, and introduce the lengths 
  which are the distances from the fulcrum to the input point A and to the output point B, respectively.
  Now introduce the unit vectors eA and eB from the fulcrum to the point A and B, so
  This notation allows us to define the velocity of the points A and B as
  where eA⊥ and eB⊥ are unit vectors perpendicular to eA and eB, 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
  Now, denote as FA and FB the components of the forces that are perpendicular to the radial segments PA and PB.   These forces are given by
  This notation and the principle of virtual work yield the formula for the generalized force as
  The ratio of the output force FB to the input force FA is the mechanical advantage of the lever, and is obtained from the principle of virtual work as
  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.
caption: This is an engraving from Mechanics Magazine published in London in 1824.
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