# Mechanical work

Work (abbreviated W) is the energy transferred by a force to a moving object. Work is a scalar quantity, but it can be positive or negative.

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## Definition

Note: Readers not familiar with multivariate calculus or vectors, please see "Simpler formulae" below)

Work is defined as the following line integral

[itex]W = \int_{C} \vec F \cdot \vec{ds} \,\![itex]

where

C is the path or curve traversed by the object;
[itex] \vec F[itex] is the force vector;
[itex]\vec s[itex] is the position vector.

## Units

The SI derived unit of work is the joule (J), which is defined as the work done by a force of one newton acting over a distance of one metre. The dimensionally equivalent newton-meter (Nm) is sometimes used instead; however, it is also sometimes reserved for torque to distinguish its units from work or energy.

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 steady direction, and acted on by a force parallel to that direction, the work is given by the formula

[itex]W = \mathbf{F} s \,\![itex]

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:

[itex]W = \vec F \cdot \vec s = |\mathbf{F}| |s| \cos\phi \,\![itex]

where [itex]\phi[itex] is the angle between the force and the displacement vector.

This formula holds true even when the force acts at an angle to the direction of travel. 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, d, to express the infinitesimal work done by the force over an infinitesimal time, thus:

[itex]dW = \vec F \cdot \vec{ds} \,\![itex]

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

## Types of work

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.

Heat conduction from a warmer body to a colder one is not normally considered to be a form of mechanical work, because at the macroscopic level, there is no measurable force. At the atomic level, there are forces as the atoms collide, but they average to nearly zero in bulk.

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.

## Mechanical energy

In physics, mechanical energy is one of several forms of energy. It is distinguished by the property that it can be transferred from one system to another by forces known to Newtonian mechanics. This category includes kinetic energy, Ek, and gravitational potential energy, Ep.

### Conservation of mechanical energy

The conservation of mechanical energy is a principle which states that the total mechanical energy of a system in a gravitational field where the gravitation is the only force acting upon it, is constant. It is also the sum of the kinetic and potential energy. If an object with constant mass is in free fall, the total energy of position 1 will be equal position 2.

[itex](E_k + E_p)_1 = (E_k + E_p)_2 \,\![itex]

where

Ek is the kinetic energy, and
Ep is the potential energy.

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