Law of The Lever
A lever is modeled as a rigid bar connected 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.
Read more about this topic: Virtual Work
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