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. Work is associated with a change in energy, but not all changes in energy can be readily analysed in terms of work.

In addition, not all forces do work. For instance, a centripetal force in uniform circular motion does not transfer energy; the kinetic energy of the object undergoing the motion remains constant.

Definition

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

Work is defined as the following line integral:

$W = \int_{C} \vec F \cdot d\vec{s} \,\!$

where:

C is the path or curve traversed by the object;

$\vec F$ is the force vector;

$\vec s$ is the position vector.

This formula readily explains how a nonzero force can do zero work. The simplest case is where the force is always perpendicular to the direction of motion, making the integrand always zero (viz. circular motion). However, even if the integrand sometimes takes nonzero values, it can still integrate to zero if it is sometimes negative and sometimes positive.

The possibility of a nonzero force doing zero work exemplifies the difference between work and a related quantity: impulse (the integral of force over time). Impulse measures change in a body's momentum, a vector quantity sensitive to direction, whereas work considers only the magnitude of the velocity. For instance, as an object in uniform circular motion traverses half of a revolution, its centripetal force does no work, but it transfers a nonzero impulse.

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 meter. The dimensionally equivalent newton-meter (N·m) 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, the foot-poundal, and the liter-atmosphere.

Simpler formulae

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

$W = F s \,\!$

where

F is the force and

s is the distance travelled 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:

$W = \vec F \cdot \vec \mathbf{s} = |\mathbf{F}| |\mathbf{s}| \cos\phi \,\!$

where $φ$ is the angle between the force and the displacement vector. This formula holds true even when the object changes its direction of travel throughout the motion.

To further generalize the formula to situations in which the force changes over time, it is necessary to use differentials to express the infinitesimal work done by the force over an infinitesimal displacement, thus:

$dW = \vec F \cdot d\vec{s} \,\!$

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 in fact represent special cases of this principle. For instance, in the case of "electrical work", an electric field does work on charged particles as they move through a medium.

One mechanism of heat conduction is collisions between fast-moving atoms in a warm body with slow-moving atoms in a cold body. Although colliding atoms do work on each other, it averages to nearly zero in bulk, so conduction is not considered to be mechanical work.

PV work

Chemical thermodynamics studies PV work, which occurs when the volume of a fluid changes. PV work is represented by the following differential equation:

$dW = -P dV \,$

where:

W = work done on the system
P = external pressure
V = volume

Therefore, we have:

$W=-\int_{V_i}^{V_f} P\,dV$

Like all work functions, PV work is path-dependent. (The path in question is a curve in the Euclidean space specified by the fluid's pressure and volume, and infinitely many such curves are possible.) From a thermodynamic perspective, this fact implies that PV work is not a state function. This means that the differential $d W$ is an inexact differential; to be more rigorous, it should be written đW (with a line through the d).

From a mathematical point of view, that is to say, $d W$ is not an exact one-form. This line through is merely a flag to warn us there is actually no function ( 0-form) $W$ which is the potential of $d W$ . If there were, indeed, this function $W$ , we should be able to just use Stokes Theorem, and evaluate this putative function, the potential of $d W$ , at the boundary of the path, that is, the initial and final points, and therefore the work would be a state function. This impossibility is consistent with the fact that it does not make sense to refer to the work on a point; work pressuposes a path.

PV work is often measured in the (non-SI) units of litre-atmospheres, where 1 L·atm = 101.3 J.

Mechanical energy

The mechanical energy of a body is that part of its total energy which is subject to change by mechanical work. It includes kinetic energy and potential energy. Some notable forms of energy that it does not include are thermal energy (which can be increased by frictional work, but not easily decreased) and rest energy (which is constant so long as the rest mass remains the same).

The relation between work and kinetic energy

If an external work W is acted upon a body, causing its kinetic energy to change from E_k1 to E_k2, then:

$W = \Delta E_k = E_{k2} - E_{k1}\,$

Conservation of mechanical energy

The principle of conservation of mechanical energy states that, if a system is subject only to conservative forces (e.g. only to a gravitational force), its mechanical energy remains constant.

For instance, if an object with constant mass is in free fall, the total energy of position 1 will be equal position 2.

$(E_k + E_p)_1 = (E_k + E_p)_2 \,\!$

where

$E k$ is the kinetic energy, and
$E p$ is the potential energy.