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A lever is a
simple machine A simple machine is a mechanical device that changes the direction or magnitude of a force. In general, they can be defined as the simplest mechanisms that use mechanical advantage (also called leverage) to multiply force. Usually the term refer ...
consisting of a
beam Beam may refer to: Streams of particles or energy *Light beam, or beam of light, a directional projection of light energy **Laser beam *Particle beam, a stream of charged or neutral particles **Charged particle beam, a spatially localized grou ...
or rigid rod pivoted at a fixed
hinge A hinge is a mechanical bearing that connects two solid objects, typically allowing only a limited angle of rotation between them. Two objects connected by an ideal hinge rotate relative to each other about a fixed axis of rotation: all other ...
, or ''
fulcrum A fulcrum is the support about which a lever pivots. Fulcrum may also refer to: Companies and organizations * Fulcrum (Anglican think tank), a Church of England think tank * Fulcrum Press, a British publisher of poetry * Fulcrum Wheels, a bicy ...
''. 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. Also,
leverage Leverage or leveraged may refer to: *Leverage (mechanics), mechanical advantage achieved by using a lever * ''Leverage'' (album), a 2012 album by Lyriel *Leverage (dance), a type of dance connection *Leverage (finance), using given resources to ...
is mechanical advantage gained in a system. It is one of the six
simple machine A simple machine is a mechanical device that changes the direction or magnitude of a force. In general, they can be defined as the simplest mechanisms that use mechanical advantage (also called leverage) to multiply force. Usually the term refer ...
s identified by Renaissance scientists. A lever amplifies an input force to provide a greater output force, which is said to provide leverage. The ratio of the output force to the input force is the
mechanical advantage Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. The device trades off input forces against movement to obtain a desired amplification in the output force. The model for t ...
of the lever. As such, the lever is a
mechanical advantage device A simple machine that exhibits mechanical advantage is called a mechanical advantage device - e.g.: * Lever: The beam shown is in static equilibrium around the fulcrum. This is due to the moment created by vector force ''"A"'' counterclockwise ( ...
, trading off force against movement.


Etymology

The word "lever" entered
English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ide ...
around 1300 from
Old French Old French (, , ; Modern French: ) was the language spoken in most of the northern half of France from approximately the 8th to the 14th centuries. Rather than a unified language, Old French was a linkage of Romance dialects, mutually intelligib ...
, in which the word was ''levier''. This sprang from the stem of the verb ''lever'', meaning "to raise". The verb, in turn, goes back to the
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
''levare'', itself from the adjective ''levis'', meaning "light" (as in "not heavy"). The word's primary origin is the
Proto-Indo-European Proto-Indo-European (PIE) is the reconstructed common ancestor of the Indo-European language family. Its proposed features have been derived by linguistic reconstruction from documented Indo-European languages. No direct record of Proto-Indo-E ...
stem , meaning "light", "easy" or "nimble", among other things. The PIE stem also gave rise to the English word "light".


History

The earliest evidence of the lever mechanism dates back to the
ancient Near East The ancient Near East was the home of early civilizations within a region roughly corresponding to the modern Middle East: Mesopotamia (modern Iraq, southeast Turkey, southwest Iran and northeastern Syria), ancient Egypt, ancient Iran ( Elam, ...
circa 5000 BC, when it was first used in a simple
balance scale A scale or balance is a device used to measure weight or mass. These are also known as mass scales, weight scales, mass balances, and weight balances. The traditional scale consists of two plates or bowls suspended at equal distances from a ...
. In ancient Egypt circa 4400 BC, a foot pedal was used for the earliest horizontal frame
loom A loom is a device used to weave cloth and tapestry. The basic purpose of any loom is to hold the warp threads under tension to facilitate the interweaving of the weft threads. The precise shape of the loom and its mechanics may vary, but th ...
. In
Mesopotamia Mesopotamia ''Mesopotamíā''; ar, بِلَاد ٱلرَّافِدَيْن or ; syc, ܐܪܡ ܢܗܪ̈ܝܢ, or , ) is a historical region of Western Asia situated within the Tigris–Euphrates river system, in the northern part of the F ...
(modern Iraq) circa 3000 BC, the shadouf, a crane-like device that uses a lever mechanism, was invented. In
ancient Egypt technology Ancient Egyptian technology describes devices and technologies invented or used in Ancient Egypt. The Egyptians invented and used many simple machines, such as the ramp and the lever, to aid construction processes. They used rope trusses to stiff ...
, 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 which could not be used for any purpose other than for levers. The earliest remaining writings regarding levers date from the 3rd century BC and were provided by the Greek mathematician
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
, who famously stated "Give me a lever long enough and a fulcrum on which to place it, and I shall move the world."


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 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 div ...
.'' The mechanical advantage of a lever can be determined by considering the balance of moments or
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
, ''T'', about the fulcrum. If the distance traveled is greater, then the output force is lessened. \begin T_ &= F_a,\quad \\ T_ &= F_b\! \end 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, T_ = T_ \!. So, F_a = F_b \!. The mechanical advantage of the lever is the ratio of output force to input force. MA = \frac = \frac.\! 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 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.


Classification 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 ''the effort'' and the output force ''the load'' or ''the 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 between the effort and 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 seesaw (also known as a teeter-totter or teeterboard) is a long, narrow board supported by a single pivot point, most commonly located at the midpoint between both ends; as one end goes up, the other goes down. These are most commonly found a ...
, a crowbar or a
pair of scissors Scissors are hand-operated shearing tools. A pair of scissors consists of a pair of metal blades pivoted so that the sharpened edges slide against each other when the handles (bows) opposite to the pivot are closed. Scissors are used for cutti ...
, a
balance scale A scale or balance is a device used to measure weight or mass. These are also known as mass scales, weight scales, mass balances, and weight balances. The traditional scale consists of two plates or bowls suspended at equal distances from a ...
, a claw hammer. Mechanical advantage may be greater than, less than, or equal to 1. * Class II – Resistance (or load) between the effort and fulcrum: The effort is applied on one side of the resistance and the fulcrum is located on the other side, e.g. in a
wheelbarrow A wheelbarrow is a small hand-propelled vehicle, usually with just one wheel, designed to be pushed and guided by a single person using two handles at the rear, or by a sail to push the ancient wheelbarrow by wind. The term "wheelbarrow" is mad ...
, a
nutcracker A nutcracker is a tool designed to open nuts by cracking their shells. There are many designs, including levers, screws, and ratchets. The lever version is also used for cracking lobster and crab shells. A decorative version portrays a person w ...
, a
bottle opener A bottle opener is a device that enables the removal of metal bottle caps from glass bottles. More generally, it might be thought to include corkscrews used to remove cork or plastic stoppers from wine bottles. A metal bottle cap is affixed t ...
or the
brake A brake is a mechanical device that inhibits motion by absorbing energy from a moving system. It is used for slowing or stopping a moving vehicle, wheel, axle, or to prevent its motion, most often accomplished by means of friction. Background ...
pedal A pedal (from the Latin '' pes'' ''pedis'', "foot") is a lever designed to be operated by foot and may refer to: Computers and other equipment * Footmouse, a foot-operated computer mouse * In medical transcription, a pedal is used to control p ...
of a car. The load arm is smaller than the effort arm, and the mechanical advantage is always greater than 1. It is also called a force multiplier lever. * Class III – Effort between the fulcrum and resistance: The resistance (or load) is on one side of the effort and the fulcrum is located on the other side, for example, a pair of
tweezers Tweezers are small hand tools used for grasping objects too small to be easily handled with the human fingers. Tweezers are thumb-driven forceps most likely derived from tongs used to grab or hold hot objects since the dawn of recorded history. ...
, a
hammer A hammer is a tool, most often a hand tool, consisting of a weighted "head" fixed to a long handle that is swung to deliver an impact to a small area of an object. This can be, for example, to drive nails into wood, to shape metal (as w ...
, a pair of
tongs Tongs are a type of tool used to grip and lift objects instead of holding them directly with hands. There are many forms of tongs adapted to their specific use. The first pair of tongs belongs to the Egyptians. Tongs likely started off as b ...
, a
fishing rod A fishing rod is a long, thin rod used by angling, anglers to fishing, catch fish by manipulating a fishing line, line ending in a fish hook, hook (formerly known as an ''angle'', hence the term "angling"). At its most basic form, a fishing ...
, or the
mandible In anatomy, the mandible, lower jaw or jawbone is the largest, strongest and lowest bone in the human facial skeleton. It forms the lower jaw and holds the lower tooth, teeth in place. The mandible sits beneath the maxilla. It is the only movabl ...
of a human skull. The effort arm is smaller than the load arm. 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 The compound lever is a simple machine operating on the premise that 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. Almost all scales use som ...
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 The malleus, or hammer, is a hammer-shaped small bone or ossicle of the middle ear. It connects with the incus, and is attached to the inner surface of the eardrum. The word is Latin for 'hammer' or 'mallet'. It transmits the sound vibrations fro ...
'', ''
incus The ''incus'' (plural incudes) or anvil is a bone in the middle ear. The anvil-shaped small bone is one of three ossicles in the middle ear. The ''incus'' receives vibrations from the ''malleus'', to which it is connected laterally, and transmit ...
'' and ''
stapes The ''stapes'' or stirrup is a bone in the middle ear of humans and other animals which is involved in the conduction of sound vibrations to the inner ear. This bone is connected to the oval window by its annular ligament, which allows the foo ...
'' are small bones in the
middle ear The middle ear is the portion of the ear medial to the eardrum, and distal to the oval window of the cochlea (of the inner ear). The mammalian middle ear contains three ossicles, which transfer the vibrations of the eardrum into waves in the ...
, connected as compound levers, that transfer sound waves from the
eardrum In the anatomy of humans and various other tetrapods, the eardrum, also called the tympanic membrane or myringa, is a thin, cone-shaped membrane that separates the external ear The outer ear, external ear, or auris externa is the extern ...
to the
oval window The oval window (or ''fenestra vestibuli'' or ''fenestra ovalis'') is a membrane-covered opening from the middle ear to the cochlea of the inner ear. Vibrations that contact the tympanic membrane travel through the three ossicles and into the in ...
of the
cochlea The cochlea is the part of the inner ear involved in hearing. It is a spiral-shaped cavity in the bony labyrinth, in humans making 2.75 turns around its axis, the modiolus. A core component of the cochlea is the Organ of Corti, the sensory org ...
.


Law of the lever

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. If ''a'' and ''b'' are distances from the fulcrum to points ''A'' and ''B'' and the force ''FA'' applied to ''A'' is the input and the force ''FB'' applied at ''B'' is the output, the ratio of the velocities of points ''A'' and ''B'' is given by ''a/b'', so we have the ratio of the output force to the input force, or mechanical advantage, is given by: MA = \frac = \frac. This is the ''law of the lever'', which was proven by
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
using geometric reasoning. 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 In mechanics, virtual work arises in the application of the ''principle of least action'' to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different for d ...
.


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 a = , \mathbf_A - \mathbf_P, , \quad b = , \mathbf_B - \mathbf_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_A - \mathbf_P = a\mathbf_A, \quad \mathbf_B - \mathbf_P = b\mathbf_B. The velocity of the points ''A'' and ''B'' are obtained as \mathbf_A = \dot a \mathbf_A^\perp, \quad \mathbf_B = \dot b \mathbf_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 In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
that defines the configuration of the lever, and the
generalized force Generalized forces find use in Lagrangian mechanics, where they play a role conjugate to generalized coordinates. They are obtained from the applied forces, Fi, i=1,..., n, acting on a system that has its configuration defined in terms of generali ...
associated with this coordinate is given by F_\theta = \mathbf_A \cdot \frac - \mathbf_B \cdot \frac= a(\mathbf_A \cdot \mathbf_A^\perp) - b(\mathbf_B \cdot \mathbf_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 In mechanics, virtual work arises in the application of the ''principle of least action'' to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different for d ...
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 = \frac, which is the
mechanical advantage Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. The device trades off input forces against movement to obtain a desired amplification in the output force. The model for t ...
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.


See also

* * Balance lever coupling * * * *


References


External links


Lever
at Diracdelta science and engineering encyclopedia *
A Simple Lever
' by
Stephen Wolfram Stephen Wolfram (; born 29 August 1959) is a British-American computer scientist, physicist, and businessman. He is known for his work in computer science, mathematics, and theoretical physics. In 2012, he was named a fellow of the American Ma ...
,
Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
.
Levers: Simple Machines
at EnchantedLearning.com {{Authority control Mechanisms (engineering) Simple machines Ancient inventions Egyptian inventions