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.
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.
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 stemlegwh-, meaning "light", "easy", or "nimble", among other things. The PIE stem also gave rise to the English-language antonym of "heavy", "light".[2]
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". (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.[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,. So,.
The mechanical advantage of a 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 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.
Three classes of leversThe three classifications of levers with examples of the human body
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, a pair ofbellows, 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. ahoe, 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.
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 forceFA applied toA is the input and the forceFB 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:
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 forceFA at a pointA located by the coordinate vectorrA on the bar. The lever then exerts an output forceFB at the pointB located byrB. The rotation of the lever about the fulcrumP is defined by the rotation angleθ in radians.
Archimedes lever, Engraving fromMechanics Magazine, published in London in 1824
Let the coordinate vector of the pointP that defines the fulcrum berP, and introduce the lengths
which are the distances from the fulcrum to the input pointA and to the output pointB, respectively.
Now introduce the unit vectorseA andeB from the fulcrum to the pointA andB, so
The velocity of the pointsA andB are obtained as
whereeA⊥ andeB⊥ are unit vectors perpendicular toeA andeB, respectively.
whereFA andFB 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
Simple lever, fulcrum, and vertical posts
Thus, the ratio of the output forceFB to the input forceFA is obtained as
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.
^Stanley, Autumn (1983). ""Women Hold Up Two-Thirds of the Sky: Notes for a Revised History of Technology."". In Rothschild, Joan (ed.).Machina Ex Dea: Feminist Perspectives on Technology. Pergamon Press.
^abPaipetis, S. A.; Ceccarelli, Marco (2010).The Genius of Archimedes -- 23 Centuries of Influence on Mathematics, Science and Engineering: Proceedings of an International Conference held at Syracuse, Italy, June 8-10, 2010.Springer Science & Business Media. p. 416.ISBN9789048190911.
^Bruno, Leonard C.; Olendorf, Donna (1997).Science and technology firsts.Gale Research. p. 2.ISBN9780787602567.4400 B.C. Earliest evidence of the use of a horizontal loom is its depiction on a pottery dish found in Egypt and dated to this time. These first true frame looms are equipped with foot pedals to lift the warp threads, leaving the weaver's hands free to pass and beat the weft thread.
^Davidovits, Paul (2008)."Chapter 1".Physics in Biology and Medicine (3rd ed.). Academic Press. p. 10.ISBN978-0-12-369411-9.Archived from the original on 2014-01-03. Retrieved2016-02-23.
^Uicker, John; Pennock, Gordon; Shigley, Joseph (2010).Theory of Machines and Mechanisms (4th ed.). Oxford University Press USA.ISBN978-0-19-537123-9.
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