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A lever is a simple machine 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 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 is mechanical advantage gained in a system. 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. The ratio of the output force to the input force is the mechanical advantage 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 around 1300 from Old French, 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 ''levare'', itself from the adjective ''levis'', meaning "light" (as in "not heavy"). The word's primary origin is the Proto-Indo-European 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 circa 5000 BC, when it was first used in a simple balance scale. 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 (modern Iraq) circa 3000 BC, the
shadouf A shadoof or shaduf (from the Arabic word , ''šādūf'') is an irrigation tool. It is highly efficient, and has been known since 3000 BCE. Names It is also called a lift, well pole, well sweep, or simply a sweep in the US.Knight, Edward Henry ...
, 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.'' 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. \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
crowbar A crowbar, also called a wrecking bar, pry bar or prybar, pinch-bar, or occasionally a prise bar or prisebar, colloquially, in Britain and Australia sometimes called a jemmy or jimmy (also called jemmy bar), gooseneck, or pig foot, is a tool ...
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
claw hammer A claw hammer is a hammer primarily used in carpentry for driving nails into or pulling them from wood. Historically, a claw hammer has been associated with woodworking, but is also used in general applications. It is not suitable for heavy h ...
. 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 nutcracker, a bottle opener 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 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, 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, 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 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'', ''
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'' are small bones in the middle ear, 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 of the cochlea.


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 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 generaliz ...
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 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 The balance lever coupling, also known as rocking lever coupling or compensating coupling, is a type of central buffer coupling that has found widespread use, especially in narrow-gauge railways. In Switzerland this type of coupling is called a c ...
* * * *


References


External links


Lever
at Diracdelta science and engineering encyclopedia *
A Simple Lever
' by Stephen Wolfram, Wolfram Demonstrations Project.
Levers: Simple Machines
at EnchantedLearning.com {{Authority control Mechanisms (engineering) Simple machines Ancient inventions Egyptian inventions