Lever (mechanical)

See also: Lever

In Mechanical, a lever is a rigid part, lengthened, generally in Liaison pivot or simple support compared to a fixed part, which makes it possible to transform a movement. This very simple mechanism makes it possible to use the action leverage, which reduces the Force to be applied but requires to prolong the duration of the effort.

One calls arm of lever the distance separating an end from the lever and his fulcrum; one also indicates by arm of lever the report/ratio of the two arms of lever, which gives the amplitude of the action leverage which results from it.

History

If one understands by '' mechanism '' the coherent installation several solids with an aim of transforming a movement, then the lever is simplest of all. What probably makes of it the first Mécanisme or mechanical device used by the Man and this certainly well before the discovery of the wheel. It is made only of two solids: a support (a stone) and a lever (a branch, a stick) which laid out judiciously make it possible to gear down the muscular force.

Well later, Archimedes understood and controls all the possibilities that the lever offers. This device is besides at the origin of the one of its most famous quotations:

Give me a support and a lever and I will raise the world.

Mechanical study

The connection between the lever and its support is generally unilateral (only one direction of application of effort is then possible) in the ammovible case dun lever, or constitutes an articulation (pivot, kneecap…).

It will be noted that this support is not located obligatorily between the two points of application of the forces, it can be external with these points; it is the case for the wheelbarrow.

In addition, the lever undergoes a load (what one wants to raise or push), and a driving force (that which one exerts by wishing it weakest possible).

Balance under three forces (relation between 3 vectors) implies that the lever acts in a plan. It is possible that this plan is not fixed, and turns in space. It is the case of an automobile lever of speeds, maintained by a connection kneecap with finger.

The study which follows presents a lever in the kinematic and static plan. All the vectors are seen full-scale. The lever is maintained by a support (specific or linear rectilinear).

Kinematic study

The vectors $\backslash \; vec\; \{V\_1\}$ and $\backslash \; vec\; \{V\_2\}$ have even direction but are opposite directions: . These two vectors can represent a displacement (m), a speed (m/s) or an acceleration (m/s ²).

The Théorème from Thalès gives us the relation:

$\backslash frac\; \{\backslash |\{\backslash vec\; \{V\_1\}\}\backslash |\}\; \{\backslash |\{\backslash vec\; \{V\_2\}\}\backslash |\}\; =\; \backslash \; frac\; \{V\_1\}\; \{V\_2\}\; =\; \backslash \; frac\; \{L\_1\}\; \{L\_2\}$

This relation can be written more usefully:

$V\_1\; =\; \backslash \; frac\; \{L\_1\}\; \{L\_2\}\; \backslash \; times\; V\_2$ or $V\_2\; =\; \backslash \; frac\; \{L\_2\}\; \{L\_1\}\; \backslash \; times\; V\_1$.

A lever thus makes it possible to transform a displacement, a speed or an acceleration according to the report/ratio of its arms of lever.

Example: a warlike use of the kinematic aspect of the lever is the Trébuchet. In this case, a mass attached at an end (L₁) is accelerated by terrestrial gravity, the lever increases and transmits this acceleration to the other end (L₂) in order to project a ball.

Static study

The Principe Fundamental of Statics (PFS) applied to the system {LEVER} at the point O gives us two vectorial equations:

• For the resultants:

$\backslash \; vec\; \{F\_1\}\; +\; \backslash \; vec\; \{F\_2\}\; +\; \backslash \; vec\; \{R\}\; =\; \backslash \; vec\; \{0\}$

Maybe, in projection on $\backslash \; vec\; \{there\}$: $-\; F\_1-F\_2+R=0\; \backslash ,$ (1)

• For the moment:

$\backslash \; mathcal\; \{M\}\; \_o\; (\backslash \; vec\; \{F\_1\})\; +\; \backslash \; mathcal\; \{M\}\; \_o\; (\backslash \; vec\; \{F\_2\})\; =\; \backslash \; vec\; \{0\}$

$\backslash \; Leftrightarrow\; \backslash \; vec\; \{F\_1\}\; \backslash \; wedge\; \backslash \; vec\; \{AO\}\; +\; \backslash \; vec\; \{F\_2\}\; \backslash \; wedge\; \backslash \; vec\; \{BO\}\; =\; \backslash \; vec\; \{0\}$

$\backslash \; Leftrightarrow\; -\; F\_1.\; \backslash \; vec\; \{there\}\; \backslash \; wedge\; L\_1.\; \backslash \; vec\; \{X\}\; -\; F\_2.\; \backslash \; vec\; \{there\}\; \backslash \; wedge\; (-\; L\_2\; \backslash \; vec\; \{X\})\; =\; \backslash \; vec\; \{0\}$

$\backslash Leftrightarrow\; F\_1.L\_1.\backslash \; vec\; \{Z\}\; -\; F\_2.L\_2\; \backslash \; vec\; \{Z\}\; =\; \backslash \; vec\; \{0\}\; \backslash ,$

What gives finally in projection on $\backslash \; vec\; \{Z\}$: $F\_1.L\_1-F\_2.L\_2=0\; \backslash ,$

that one will write more usefully: $\backslash \; frac\; \{F\_1\}\; \{F\_2\}\; =\; \backslash \; frac\; \{L\_2\}\; \{L\_1\}$ or $F\_1\; =\; \backslash \; frac\; \{L\_2\}\; \{L\_1\}\; \backslash \; times\; F\_2$ or $F\_2\; =\; \backslash \; frac\; \{L\_1\}\; \{L\_2\}\; \backslash \; times\; F\_1$.

Remarks on the results:

• the report/ratio of the arms of lever is reversed compared to the relation on speeds.
• the equation (1) makes it possible to calculate the effort that the sudden support.

The report/ratio of the forces is thus inversely proportional to the report/ratio of the arms of lever.

Examples: The Shears and the crowbar (also called Grip monseigneur) use the static aspect of the lever. A small effort applied by the user to the large arm of lever makes it possible to obtain a very great effort on the level of the small arm of lever and thus makes it possible to cut a branch or to tear off a nail.

In the same way, the interest of the Wheelbarrow in the transport of loads, rests on this principle.

Energy aspect

One of the great principle of the Physique is the conservation of energy. Let us check that the lever respects this principle.

In has, the power applied is $\backslash \; mathcal\; \{P\}\; \_A\; =\; F\_1\; \backslash \; times\; V\_1$.

The power transmitted out of B is $\backslash \; mathcal\; \{P\}\; \_B\; =\; F\_2\; \backslash \; times\; V\_2$.

However we saw that $V\_1\; =\; \backslash \; frac\; \{L\_1\}\; \{L\_2\}\; \backslash \; times\; V\_2$ and that $F\_1\; =\; \backslash \; frac\; \{L\_2\}\; \{L\_1\}\; \backslash \; times\; F\_2$.

There is thus $\backslash \; mathcal\; \{P\}\; \_A\; =\; F\_1\; \backslash \; times\; V\_1\; =\; \backslash \; frac\; \{L\_2\}\; \{L\_1\}\; \backslash \; times\; \backslash \; frac\; \{L\_1\}\; \{L\_2\}\; \backslash \; times\; F\_2\; \backslash \; times\; V\_2\; =\; F\_2\; \backslash \; times\; V\_2\; =\; \backslash \; mathcal\; \{P\}\; \_B$.

Thus the power and thus energy are completely transmitted point has at the point B.

In practice, a small portion of the power is degraded in the form of heat and/or of sound vibrations on the level of the connection with the support. To hold account of it is necessary to know the Rendement this connection.

Note: starting from the principle of conservation of energy (here in the form of the work of the forces) one finds the properties of the lever, in particular the fact that the report/ratio of the forces at the ends is equal to the inverse proportion lengths of the arms, which makes the happiness of the burglars users of the “foot of hind”.

Virtual work

We can introduce here the Principe of virtual work. Indeed, considering that displacements of the structure are so small that its geometry is not changed, we arrive at the same conclusions (ouf! Once more mathematics appeared exact). The infinitesimal sizes are preceded by the sign $d$ and we place item 1 has some and 2 out of B.

In 1, virtual work is worth $D\; \backslash \; mathcal\; \{W\}\; \_1\; =\; F\_1\; \backslash \; times\; dx\_2$.

Virtual work in 2 is $D\; \backslash \; mathcal\; \{W\}\; \_2\; =\; F\_2\; \backslash \; times\; dx\_1$.

As the structure does not work at rest, we must thus equalize this two virtual work $F\_1\; \backslash \; times\; dx\_1\; =\; F\_2\; \backslash \; times\; dx\_2$.

However displacements of items 1 and 2 are bound by geometry of structure because if we consider that the beams remain perfectly rigid, consequently, $L\_1\; \backslash \; times\; dx\_1\; =\; L\_2\; \backslash \; times\; dx\_2$.

There is thus $L\_1\; \backslash \; times\; F\_1\; =\; F\_2\; \backslash \; times\; L\_2$.

Of course, this grooved development brings the same conclusions as a reasoning simple to include/understand but what would you have made if, simply and without imagining cruelest, the structure had not been rigid?

See also:

• simple Machine

Simple: To raise Zh-yue: 槓桿

Random links:

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