Virtual Work - Law of The Lever

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 F_A at a point A located by the coordinate vector r_A on the bar. The lever then exerts an output force F_B at the point B located by r_B. 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 r_P, 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 e_A and e_B 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 e_A⊥ and e_B⊥ are unit vectors perpendicular to e_A and e_B, 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 F_A and F_B 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 F_B to the input force F_A 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.