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 r ...
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. Also,
leverage 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 r ...
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 of the lever. As such, the lever is a
mechanical advantage device, 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 ...
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 intel ...
, 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 languages, 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 ...
''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 ...
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.
In
ancient Egypt circa 4400 BC, a foot pedal was used for the earliest horizontal frame
loom. 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 ...
(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, 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
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.
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,
. So,
.
The mechanical advantage of the lever is the ratio of output force to input force.
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. 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 ma ...
, a
nutcracker, a
bottle opener or the
brake 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
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 histo ...
, a
hammer, a pair of
tongs, a
fishing rod, 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 teeth in place. The mandible sits beneath the maxilla. It is the only movable bone ...
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'' and ''
stapes'' are small bones in the
middle ear, connected as compound levers, that transfer sound waves from the
eardrum 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 o ...
.
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 ''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 we have the ratio of the output force to the input force, or mechanical advantage, is given by:
This is the ''law of the lever'', which was proven by
Archimedes 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.
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
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
The velocity of the points ''A'' and ''B'' are obtained as
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
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
Thus, the ratio of the output force ''F''
''B'' to the input force ''F''
''A'' is obtained as
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
Leverat Diracdelta science and engineering encyclopedia
*
A Simple Lever' by
Stephen Wolfram,
Wolfram Demonstrations Project.
Levers: Simple Machinesat EnchantedLearning.com
{{Authority control
Mechanisms (engineering)
Simple machines
Ancient inventions
Egyptian inventions