# Work

Glossary Definition for 16-19

## Description

A force is said to do work when there is a displacement of the point of application of the force in the direction of the force. Some common examples of work include the mechanical work done in compressing a spring and the electrical work done in moving a charged particle.
Work is usually represented by the symbol *W* or Δ*W*.
When a force, ** F**, of magnitude

*F*, produces a displacement

**Δ**of magnitude Δ

*x**x*, the work done is

Δ

*W*=

*F*Δ

*x*

provided that

**is in the direction of the resulting displacement. In the more general case, as shown in figure 1, the work done is defined as Δ**

*F**W*=

*F*Δ

*x*cosθ

where θ is the angle between the force and the resulting displacement.

*Figure 1: A force F is applied to an object, which is then displaced by Δx. The angle between the force and the displacement is θ.*

Work is a scalar quantity. It has magnitude and sign but no direction.

## Discussion

*Heat, work and energy conservation*

A system’s internal energy can be changed as a result of heating or working – or a combination of both. Thermodynamics, the branch of physics concerned with heat, temperature, energy and work, has its origins in attempts to quantify heat and work in the context of machinery.

*Work, force, displacement and direction*

While work is a scalar quantity, the work done in a given situation depends on two vector quantities: force and displacement. Work was originally defined, in Sadi Carnot’s 1824 book Reflections on the Motive Power of Fire in terms of lifting a load through a height. In such situations, both the force, ** F**, and displacement

**Δ**are in the same direction (vertical) and the work Δ

*x**W*is

Δ

*W*=

*F*Δ

*x*

If there is an angle θ between the displacement and the line of action of the force, then Δ

*W*=

*F*Δ

*x*cosθ

Mathematically, the equation for work can be written as the dot product (scalar product) of the force and displacement vectors:

Δ*W* = *F*•Δ*x*

If the force is at right-angles to the displacement, so that the angle θ is 90 °, then cosθ is zero and no work is done by the force. For example, an object moving in a circle at constant speed requires the action of a force (centripetal force) directed to the centre of the circle to change its direction of motion. But the object’s displacement during any (infinitesimally) small time interval is always at right-angles to the direction of the force at that time. The force does no work on the object. No energy is transferred to or from the object; its kinetic energy and hence its speed remain constant. Similarly, when a body is moved in a horizontal direction no work is done by the gravitational force upon it, as this acts in a vertical direction.

*Electrical work*

The definition of work does not specify the type of force – it applies equally to any force that produces a displacement. The term ‘electrical work’ is sometimes used in the context of electric circuits and fields, and refers to the work done on charged particles by electrostatic forces and fields, conceived as an emf. In an electric field ** E**, the electrostatic force

**on a particle with charge**

*F**q*is

**=**

*F**q*

*E*so for motion through a distance Δ

*x*parallel to the field of magnitude

*E*:

Δ

*W*=

*q*

*E*Δ

*x*

This equation can be written in terms of an electrostatic potential difference, Δ

*V*, where Δ

*V*=

*E*Δ

*x*

hence the equation for electrical work becomes

Δ

*W*=

*q*Δ

*V*

Note that this is just a way of writing the more general equation for work, which applies specifically to charged particles acted on by electrostatic forces; it is not a different definition of work.

Also note that if a charged particle is moving in a magnetic field, it experiences a force at right-angles to its direction of motion. Such a force therefore does no work.

## SI unit

joule, J

## Expressed in SI base units

kg m^{2} s^{-2}

## Mathematical expressions

- Δ
*W*=*m**g*Δ*x* - Δ
*W*=*q**E*Δ*x*

is the work done by an electrostatic force, due to an electric field of magnitude*E*, upon a particle of charge*q*as it moves through a distance Δ*x*in the direction of.*E* - Δ
*W*=*q*Δ*V*

is the work done upon a particle with charge*q*as it moves through an electrostatic potential difference Δ*V*.

is the work done by the force due to the Earth’s gravitational field,

*g*, on an object of mass

*m*as it falls through a vertical distance Δ

*x*near the surface of the Earth.

## Related entries

- Energy of a system
- Heat
- Internal energy
- Power

## In context

Tasks performed by people and machines in everyday situations may involve anything from a few joules of work to several thousands. For example, lifting an object of mass 100 g (e.g. a small apple) through 1 m involves doing about 1 J of work. Picking up a 10 kg bag of books and raising it through 1 m involves about 100 J. If a crane lifts a 1 tonne concrete block through 20 m (about the height of a five-storey building), the work done is about 200 kJ.