Lever

A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge or fulcrum. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load, and effort, the lever is divided into three types. It is one of the six simple machines identified by Renaissance scientists. A lever amplifies an input force to provide a greater output force, which is said to provide leverage, which is mechanical advantage gained in the system, equal to the ratio of the output force to the input force. As such, the lever is a mechanical advantage device, trading off force against movement.

A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge or fulcrum. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load, and effort, the lever is divided into three types. It is one of the six simple machines identified by Renaissance scientists. A lever amplifies an input force to provide a greater output force, which is said to provide leverage, which is mechanical advantage gained in the system, equal to the ratio of the output force to the input force. As such, the lever is a mechanical advantage device, trading off force against movement. \begin{align} T_{1} &= F_{1}a,\quad \\ T_{2} &= F_{2}b\! \end{align} where F 1 is the input force to the lever and F 2 is the output force. The distances a and b are the perpendicular distances between the forces and the fulcrum. Since the moments of torque must be balanced, T_{1} = T_{2} \! . So, F_{1}a = F_{2}b \! . The mechanical advantage of a lever is the ratio of output force to input force. MA = \frac{F_{2}}{F_{1}} = \frac{a}{b}.\! This relationship shows that the mechanical advantage can be computed from ratio of the distances from the fulcrum to where the input and output forces are applied to the lever, assuming a weightless lever and no losses due to friction, flexibility, or wear. This remains true even though the "horizontal" distance (perpendicular to the pull of gravity) of both a and b change (diminish) as the lever changes to any position away from the horizontal. MA = \frac{F_B}{F_A} = \frac{a}{b}. This is the law of the lever, as discussed by Archimedes. It shows that if the distance a from the fulcrum to where the input force is applied (point A) is greater than the distance b from fulcrum to where the output force is applied (point B), then the lever amplifies the input force. On the other hand, if the distance a from the fulcrum to the input force is less than the distance b from the fulcrum to the output force, then the lever reduces the input force. The use of velocity in the static analysis of a lever is an application of the principle of virtual work. a = |\mathbf{r}_A - \mathbf{r}_P|, \quad b = |\mathbf{r}_B - \mathbf{r}_P|, 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 \mathbf{r}_A - \mathbf{r}_P = a\mathbf{e}_A, \quad \mathbf{r}_B - \mathbf{r}_P = b\mathbf{e}_B. The velocity of the points A and B are obtained as \mathbf{v}_A = \dot{\theta} a \mathbf{e}_A^\perp, \quad \mathbf{v}_B = \dot{\theta} b \mathbf{e}_B^\perp, 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, and the generalized force associated with this coordinate is given by 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 , where F A and F B 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 F_\theta = a F_A - b F_B = 0. \,\! Thus, the ratio of the output force F B to the input force F A is obtained as MA = \frac{F_B}{F_A} = \frac{a}{b}, 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.

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