Mechanical work

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Work (abbreviated W) is the energy transferred in applying force over a distance.

Work can be calculated from the formula:

<math>W = \int\mathbf{F}\cdot d\mathbf{s}<math>

where

F is the force.

s is the position.

(Readers not familiar with vectors and calculus please see "Simpler Formulae" below.)

Work is a scalar quantity, but it can be positive or negative.

Not all forces do work. For instance, a centripetal force in uniform circular motion does not transfer energy; the speed of the object undergoing the motion remains constant. This fact is confirmed by the formula: if the vectors of force and displacement are perpendicular, their dot product is zero.

Forms of work that are not evidently mechanical, such as electrical work, can be considered as special cases of this principle; for instance, in the case of electricity, work is done on charged particles moving through a medium.

Units

The SI derived unit of work is the Joule, which is defined as the work done by a force of one newton acting over a distance of one metre in the direction of the force.

Non-SI units of work include the erg, the foot-pound, and the foot-poundal.

Simpler formulae

In the simplest case, that of a body moving in a constant direction, acted on by a force parallel to the direction of motion, the work is given by the formula

where F is the force and s is the distance traveled by the object. The work is taken to be negative when the force opposes the motion. More generally, the force and distance are taken to be vector quantities, and combined using the dot product:

<math>W = \mathbf{F}\cdot\mathbf{s}<math>

This formula holds true even when the force acts at an angle to the distance traveled. To further generalize the formula to situations in which the force and the object's direction of motion changes over time, it is necessary to use differentials to express the infinitesimal work done by the force over an infinitesimal time, thus:

<math>dW = \mathbf{F}\cdot d\mathbf{s}<math>

The integration of both sides of this equation yields the most general formula, as given above.

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