Work (physics)

In physics, work is the energy that is transferred to a body when it is moved along a path by a force. When the force is conservative (non-dissipative) the work is independent of the path. Work done on a body is accomplished not only by a displacement of the body as a whole from one place to another but also, for example, by compressing a gas, by rotating a shaft, and even by directing small magnetic particles within a body along an external magnetic field.

The use of the terms "work", "energy", "force" have a well-defined, quantitative, meaning in physics, which differs somewhat from their more qualitative meaning in daily life. For instance, in physics work can be negative.

Definition

As stated, work is force times path length: When a constant force F acts on a body along a straight path and the body is moved over a length |s| along this path, then

$W=\mathbf {F} \cdot \mathbf {s} =|\mathbf {F} |\,|\mathbf {s} |\,\cos \alpha \,.$

Here s is vector of magnitude |s| in direction of the path. The inner product between the two vectors F and s is the product of their magnitudes |F| and |s| times the cosine of the angle, α, between the vectors.

When the force is directed along the path, α = 0 and the work is simply W = |F| |s|.

When a force is perpendicular to the path, α = 90°, cos(α) = 0, and the force performs no work. Examples of such a situation are the centrifugal force on a mass in uniform circular motion and the gravitational force acting on a satellite in a circular orbit. If a body is in uniform (i.e., with constant speed) straight motion and the only force acting on it is perpendicular to its path, then the body will persist in its uniform straight motion (Newton's first law).

When a force F is antiparallel to the path, it performs negative work (and usually another force than F will move the body and perform positive work).

Work being a form of energy, it is conserved; the work done by the force is converted into kinetic energy, if the speed of the body increases, or into potential energy, or a mixture of both.

When the force is not constant along the path, or the path is not straight, it is possible to compute the work by infinitesimal calculus. One divides the path in N pieces Δs_i, which are small enough to assume that the force is constant and the piece is straight. The (approximate) total work is obtained by summing the work done along the individual small pieces,

$W\approx \sum _{i=1}^{N}W_{i}=\sum _{i=1}^{N}\mathbf {F} (\mathbf {s} _{i})\Delta \mathbf {s} _{i}\,.$

To improve the approximation one makes the pieces Δs smaller and smaller, so that their lengths go to zero. The limit is the path integral

$W=\int _{\mathbf {s} _{1}}^{\mathbf {s} _{2}}\mathbf {F} (\mathbf {s} )\,\mathrm {d} \mathbf {s} \,,$

where s₁ is the start and s₂ the end point of the path.

When the the force is conservative, the work is independent of path, and when furthermore the path is a closed curve, the total work is zero (one way the work is positive and the other way the work is equally large in absolute value, but negative).

Example of mechanical work

• Work to lift a mass m in the gravitational field of the earth. Close to the surface of the earth, the attractive force is constant and equal to the gravitational acceleration g. The work to lift the mass to a height h is,

$W(h)=F_{g}\,h=m\,g\,h\,.$

When h is positive, and the mass is at rest before and after the lifting (no kinetic energy), the work is completely converted into potential energy.

Numeric examples:

$m=1\,\mathrm {kg} ,g=9.81\,\mathrm {m} /\mathrm {s^{2}} ,F_{g}=m\,g=9.81\,\mathrm {N} ,h=40\,\mathrm {cm} ,W(h)=9.81\,\mathrm {N} \cdot 0.4\,\mathrm {m} \approx 4\,\mathrm {J} \,.$

Here N is the unit newton and J is the unit joule.