Lever

Lever history

Autumn Stanley argues that the digging stick can be considered the first lever, which would position prehistoric women as the inventors of lever technology. The next earliest known cultural evidence of the application of the lever mechanism dates back to the ancient Egypt c. 5000 BC, when it was used in a simple balance scale. In ancient Egypt c. 4400 BC, a foot pedal was used for the earliest horizontal frame loom. In Mesopotamia (modern Iraq) c. 3000 BC, the shadouf, a crane-like device that uses a lever mechanism, was invented. In ancient Egypt, workmen used the lever to move and uplift obelisks weighing more than 100 tons. This is evident from the recesses in the large blocks and the handling bosses that could not be used for any purpose other than for levers.

The earliest remaining writings regarding levers date from the third century BC and were provided, by common belief, by the Greek mathematician Archimedes, who famously stated "Give me a lever (long enough and a fulcrum on which to place it), and I shall move the world". (The Greek usually attributed to Archimedes does not include details about length of lever or fulcrum, i.e., δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω .) That statement has given rise to the phrase "an Archimedean lever" being adopted for use in many instances, not just regarding mechanics, including abstract concepts about the successful effect of a human behavior or action intended to achieve results that could not have occurred without it.

Force and levers

A lever is a beam connected to ground by a hinge, or pivot, called a fulcrum. The ideal lever does not dissipate or store energy, which means there is no friction in the hinge or bending in the beam. In this case, the power into the lever equals the power out, and the ratio of output to input force is given by the ratio of the distances from the fulcrum to the points of application of these forces. This is known as the law of the lever.

The mechanical advantage of a lever can be determined by considering the balance of moments or torque, T, about the fulcrum. If the distance traveled is greater, then the output force is lessened.

$${\displaystyle \begin{array}{cc}{T}_1& =F_{1}a,\\ T_2& =F_{2}b\end{array}}$$

where F₁ is the input force to the lever and F₂ 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, ${\displaystyle T_1=T_2}$. So, ${\displaystyle F_{1}a=F_{2}b}$.

The mechanical advantage of a lever is the ratio of output force to input force.

$${\displaystyle MA=\frac{F_2}{F_1}=\frac{a}{b}.}$$

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.

Types of levers

Levers are classified by the relative positions of the fulcrum, effort, and resistance (or load). It is common to call the input force "effort" and the output force "load" or "resistance". This allows the identification of three classes of levers by the relative locations of the fulcrum, the resistance and the effort:

• Class I – Fulcrum is located between the effort and the resistance: The effort is applied on one side of the fulcrum and the resistance (or load) on the other side. For example, a seesaw, a crowbar, a pair of scissors, a balance scale, a pair of pliers, and a claw hammer (pulling a nail). With the fulcrum in the middle, the lever's mechanical advantage may be greater than, less than, or even equal to 1.
• Class II – Resistance (or load) is located between the effort and the fulcrum: The effort is applied on one side of the resistance and the fulcrum is located on the other side, e.g. a wheelbarrow, a nutcracker, a bottle opener, a wrench, a pair of bellows, and the brake pedal of a car. Since the load arm is smaller than the effort arm, the lever's mechanical advantage is always greater than 1. It is also called a force multiplier lever.
• Class III – Effort is located between the resistance and the fulcrum: The resistance (or load) is applied on one side of the effort and the fulcrum is located on the other side, e.g. a hoe, a pair of tweezers, a hammer, a pair of tongs, a fishing rod, and the mandible of a human skull. Since the effort arm is smaller than the load arm, the lever's mechanical advantage is always less than 1. It is also called a speed multiplier lever.

These cases are described by the mnemonic fre 123 where the f fulcrum is between r and e for the 1st class lever, the r resistance is between f and e for the 2nd class lever, and the e effort is between f and r for the 3rd class lever.

Compound lever

A compound lever comprises several levers acting in series: the resistance from one lever in a system of levers acts as effort for the next, and thus the applied force is transferred from one lever to the next. Examples of compound levers include scales, nail clippers, and piano keys.

The malleus, incus, and stapes are small bones in the middle ear, connected as compound levers, that transfer sound waves from the eardrum to the oval window of the cochlea.

Law of the lever

The lever is a bar that pivots on a point or hinge (fulcrum) attached to a fixed point. The lever operates by applying forces at different distances from the fulcrum, or a pivot.

As the lever rotates around the fulcrum, points farther from this pivot move faster than points closer to the pivot. Therefore, a force applied to a point farther from the pivot must be less than the force located at a point closer in, because power is the product of force and velocity, and the power on both sides is equal.

If a and b are distances from the fulcrum to points A and B and the force F_A applied to A is the input and the force F_B applied at B is the output, the ratio of the velocities of points A and B is given by a/b, so the ratio of the output force to the input force, or mechanical advantage, is given by: $${\displaystyle MA=\frac{F_B}{F_A}=\frac{a}{b}.}$$

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.

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_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 in radians.

Let the coordinate vector of the point P that defines the fulcrum be r_P, and introduce the lengths

$${\displaystyle a=|{\mathbf{r}}_A-{\mathbf{r}}_P|,b=|{\mathbf{r}}_B-{\mathbf{r}}_P|,}$$

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

$${\displaystyle {\mathbf{r}}_A-{\mathbf{r}}_P=a{\mathbf{e}}_A,{\mathbf{r}}_B-{\mathbf{r}}_P=b{\mathbf{e}}_B.}$$

The velocity of the points A and B are obtained as

$${\displaystyle {\mathbf{v}}_A=\dot{\theta }a{\mathbf{e}}_A^{\perp },{\mathbf{v}}_B=\dot{\theta }b{\mathbf{e}}_B^{\perp },}$$

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

$${\displaystyle F_{\theta }={\mathbf{F}}_A\cdot \frac{\mathrm{\partial }{\mathbf{v}}_A}{\mathrm{\partial }\dot{\theta }}-{\mathbf{F}}_B\cdot \frac{\mathrm{\partial }{\mathbf{v}}_B}{\mathrm{\partial }\dot{\theta }}=a({\mathbf{F}}_A\cdot {\mathbf{e}}_A^{\perp })-b({\mathbf{F}}_B\cdot {\mathbf{e}}_B^{\perp })=a{F}_A-b{F}_B,}$$

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

$${\displaystyle F_{\theta }=a{F}_A-b{F}_B=0.}$$

Thus, the ratio of the output force F_B to the input force F_A is obtained as

$${\displaystyle MA=\frac{F_B}{F_A}=\frac{a}{b},}$$

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.