# PHYSICS PROJECT REPORT ON LEVER

A **lever** (/ˈliːvər/ or US: /ˈlɛvər/) 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 location of fulcrum, load and effort, the lever is divided into three types. 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.

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## Etymology[edit]

The word “lever” entered English about 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 (PIE) stem *legwh-*, meaning “light”, “easy” or “nimble”, among other things. The PIE stem also gave rise to the English word “light”.^{[1]}

## Early use[edit]

The earliest remaining writings regarding levers date from the 3rd century BCE and were provided by Archimedes. ‘Give me a place to stand, and I shall move the Earth with it’ is a remark of Archimedes who formally stated the correct mathematical principle of levers (quoted by Pappus of Alexandria).

It is assumed^{[by whom?]} that in ancient Egypt, constructors used the lever to move and uplift obelisks weighing more than 100 tons.

## Force and levers[edit]

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.

- {\displaystyle T_{1}=F_{1}a,\!} {\displaystyle T_{2}=F_{2}b\!}

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, {\displaystyle T_{1}=T_{2}\!} . So, {\displaystyle F_{1}a=F_{2}b\!}.

The mechanical advantage of the 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 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.

## Classes of levers[edit]

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:^{[2]}

**Class 1**: Fulcrum in the middle: 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 or a pair of scissors. Mechanical advantage may be greater than, less than, or equal to 1.**Class 2**: Resistance (or load) in the middle: the effort is applied on one side of the resistance and the fulcrum is located on the other side, for example, a wheelbarrow, a nutcracker, a bottle opener or the brake pedal of a car. Load arm is smaller than the effort arm. Mechanical advantage is always greater than 1. It is also called force multiplier lever.**Class 3**: Effort in the middle: 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 or the human mandible. The effort arm is smaller than the load arm. Mechanical advantage is always less than 1. It is also called speed multiplier lever.

These cases are described by the mnemonic *fre 123* where the *f*ulcrum is in the middle for the 1st class lever, the *r*esistance is in the middle for the 2nd class lever, and the *e*ffort is in the middle for the 3rd class lever.

## Law of the lever[edit]

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.

Assuming the lever does not dissipate or store energy, the power into the lever must equal the power out of the lever. 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.^{[3]}

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*applied at

_{B}*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

- {\displaystyle MA={\frac {F_{B}}{F_{A}}}={\frac {a}{b}}.}

This is the *law of the lever*, which was proven by Archimedes using geometric reasoning.^{[4]} 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[edit]

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

- {\displaystyle a=|\mathbf {r} _{A}-\mathbf {r} _{P}|,\quad b=|\mathbf {r} _{B}-\mathbf {r} _{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

- {\displaystyle \mathbf {r} _{A}-\mathbf {r} _{P}=a\mathbf {e} _{A},\quad \mathbf {r} _{B}-\mathbf {r} _{P}=b\mathbf {e} _{B}.}

The velocity of the points *A* and *B* are obtained as

- {\displaystyle \mathbf {v} _{A}={\dot {\theta }}a\mathbf {e} _{A}^{\perp },\quad \mathbf {v} _{B}={\dot {\theta }}b\mathbf {e} _{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 associated with this coordinate is given by

- {\displaystyle F_{\theta }=\mathbf {F} _{A}\cdot {\frac {\partial \mathbf {v} _{A}}{\partial {\dot {\theta }}}}-\mathbf {F} _{B}\cdot {\frac {\partial \mathbf {v} _{B}}{\partial {\dot {\theta }}}}=a(\mathbf {F} _{A}\cdot \mathbf {e} _{A}^{\perp })-b(\mathbf {F} _{B}\cdot \mathbf {e} _{B}^{\perp })=aF_{A}-bF_{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 states that at equilibrium the generalized force is zero, that is

- {\displaystyle F_{\theta }=aF_{A}-bF_{B}=0.\,\!}

Thus, the ratio of the output force *F*_{B} to the input force *F*_{A} is obtained as

- {\displaystyle MA={\frac {F_{B}}{F_{A}}}={\frac {a}{b}},}

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.