Alever is asimple machine consisting of abeam or rigid rod pivoted at a fixedhinge, orfulcrum. A lever is a rigid body capable ofrotating on a point on itself. On the basis of the locations of fulcrum, load, and effort, the lever is divided intothree types. It is one of the sixsimple machines identified by Renaissance scientists. A lever amplifies an input force to provide a greater output force, which is said to provideleverage, which ismechanical advantage gained in the system, equal to the ratio of the output force to the input force. As such, the lever is amechanical advantage device, trading off force against movement.

Lever
Levers can be used to exert a large force over a small distance at one end by exerting only a small force (effort) over a greater distance at the other.
Classification	Simple machine
Components	fulcrum or pivot, load and effort
Examples	see-saw, bottle opener, etc.

The word "lever" enteredEnglish around 1300 fromOld French:levier. This sprang from the stem of the verblever, meaning "to raise". The verb, in turn, goes back toLatin:levare,^[1] itself from the adjectivelevis, meaning "light" (as in "not heavy"). The word's primary origin is theProto-Indo-European stemleg^wh-, meaning "light", "easy", or "nimble", among other things. The PIE stem also gave rise to the English-language antonym of "heavy", "light".^[2]

Autumn Stanley argues that thedigging stick can be considered the first lever, which would position prehistoric women as the inventors of lever technology.^[3] The next earliest known cultural evidence of the application of the lever mechanism dates back to theancient Near Eastc. 5000 BC, when it was used in a simplebalance scale.^[4] Inancient Egyptc. 4400 BC, a foot pedal was used for the earliest horizontal frameloom.^[5] InMesopotamia (modern Iraq)c. 3000 BC, theshadouf, a crane-like device that uses a lever mechanism, was invented.^[4] Inancient 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 thehandling bosses that could not be used for any purpose other than for levers.^[6]

The earliest remaining writings regarding levers date from the third century BC and were provided, by common belief, by the Greek mathematicianArchimedes, who famously stated "Give me a lever long enough and a fulcrum on which to place it, and I shall move the world". 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.^[7]

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 thelaw of the lever.

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

 

where F₁ is the input force to the lever and F₂ is the output force. The distancesa andb 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 botha andb change (diminish) as the lever changes to any position away from the horizontal.

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:^[8]

• 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, aseesaw, acrowbar, a pair ofscissors, abalance scale, a pair ofpliers, and aclaw 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. awheelbarrow, anutcracker, abottle opener, awrench, and thebrakepedal 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 pair oftweezers, ahammer, a pair oftongs, afishing rod, and themandible 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 mnemonicfre 123 where thef fulcrum is betweenr ande for the 1st class lever, ther resistance is betweenf ande for the 2nd class lever, and thee effort is betweenf andr for the 3rd class lever.

Acompound 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.

Themalleus,incus, andstapes are small bones in themiddle ear, connected as compound levers, that transfer sound waves from theeardrum to theoval window of thecochlea.

The lever is a movable bar that pivots on a 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.^[9]

Ifa andb are distances from the fulcrum to pointsA andB and the forceF_A applied toA is the input and the forceF_B applied atB is the output, the ratio of the velocities of pointsA andB is given bya/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 thelaw of the lever, which was proven byArchimedes using geometric reasoning.^[10] It shows that if the distancea from the fulcrum to where the input force is applied (pointA) is greater than the distanceb from fulcrum to where the output force is applied (pointB), then the lever amplifies the input force. On the other hand, if the distancea from the fulcrum to the input force is less than the distanceb 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 ofvirtual work.

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 forceF_A at a pointA located by the coordinate vectorr_A on the bar. The lever then exerts an output forceF_B at the pointB located byr_B. The rotation of the lever about the fulcrumP is defined by the rotation angleθ in radians.

Let the coordinate vector of the pointP that defines the fulcrum ber_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 pointA and to the output pointB, respectively.

Now introduce the unit vectorse_A ande_B from the fulcrum to the pointA andB, 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 pointsA andB are obtained as

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

wheree_A^⊥ ande_B^⊥ are unit vectors perpendicular toe_A ande_B, respectively.

The angleθ is thegeneralized coordinate that defines the configuration of the lever, and thegeneralized force associated with this coordinate is given by

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

whereF_A andF_B are components of the forces that are perpendicular to the radial segmentsPA andPB. The principle ofvirtual 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 forceF_B to the input forceF_A is obtained as

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

which is themechanical advantage of the lever.

This equation shows that if the distancea from the fulcrum to the pointA where the input force is applied is greater than the distanceb from fulcrum to the pointB 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 pointA is less than from the fulcrum to the output pointB, then the lever reduces the magnitude of the input force.

• Lever at Diracdelta science and engineering encyclopedia
• A Simple Lever byStephen Wolfram,Wolfram Demonstrations Project.
• Levers: Simple Machines at EnchantedLearning.com