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Work (physics): definition, formulas, examples, and relation to energy

Work in physics is the transfer of energy by a force acting through a displacement. This article explains definitions, sign conventions, formulas, examples, units, and connections to energy.

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: June 5, 2026

simple.wikipedia.org · CC BY-SA 3.0

In physics, work describes how a force transfers energy to or from an object by producing a displacement. It is a scalar quantity that quantifies the effect of a force acting over a distance; motion in the direction of a force tends to produce positive work, motion opposite the force produces negative work, and motion perpendicular to the force produces no work. For a constant force the magnitude of the work equals the force magnitude multiplied by the displacement magnitude and by the cosine of the angle between them. See a general introduction to the concept of work and its role in mechanics.

Definition and common formula

For a single, constant force F applied while an object moves a straight-line displacement s, the work W is commonly given by W = F s cos θ, where θ is the angle between the force vector and the displacement vector. This scalar expression captures how only the component of the force parallel to the motion contributes to the transfer of mechanical energy. A simple graphical depiction of the geometric relation between force, displacement and the cosine factor is shown here: $W=Fs\cos \theta$ . Additional explanatory material about force and displacement can be found at force.

Vector form, variable forces and line integrals

When forces vary in space or the path is curved, work must be computed from the line integral of the force vector along the path: W = ∫ F · ds. In words, one sums the infinitesimal contributions of the dot product between force and displacement along the trajectory. This form emphasizes that work depends on both the force field and the actual path taken, especially for nonconservative forces. Sources that discuss vector calculus formulations and line integrals provide background: vector dot product and line integrals. Practical calculations often break the path into segments and apply the constant-force formula piecewise.

Sign conventions, examples and special cases

Positive work: A force with a component in the direction of motion (e.g., pushing a box forward) adds kinetic energy to the object.
Negative work: If the force has a component opposite to the motion (e.g., friction slowing a sliding object or weight acting on a rising object), it removes mechanical energy from the object.
Zero work: Forces perpendicular to displacement, such as a centripetal force on an object in uniform circular motion, do no work because they change direction but not speed.
Holding an object stationary under a constant force produces no work because there is no displacement. See the example of lifting a book and the role of gravitational weight at weight.
Tug-of-war illustrates sign: a team that is being dragged toward the center experiences negative work from its effort while the opposing team does positive work on the rope.

Relationship to energy, units and the work–energy theorem

Work is closely tied to kinetic energy through the work–energy theorem: the net work done on a rigid object equals the change in its kinetic energy. In everyday notation, W_net = ΔK = K_final − K_initial, where kinetic energy K = ½ m v² for an object of mass m and speed v. This relationship explains why doing positive work usually increases speed and why negative work reduces it. Work and energy share the same SI unit, the joule (J); one joule equals one newton-meter. For further reading about kinetic energy and its calculation see kinetic energy and general material about energy.

Conservative vs nonconservative forces and thermodynamic distinctions

For conservative forces (such as ideal gravity or spring forces), the work done depends only on the initial and final positions and can be expressed in terms of a potential energy difference: W_conservative = −ΔU. Nonconservative forces (friction, air resistance) dissipate mechanical energy and make work path-dependent. It is important to distinguish mechanical work from heat transfer: heat involves microscopic, random motion and is characterized in thermodynamics by different conventions and measurements, while mechanical work involves macroscopic forces and displacements. Discussions about heat and work in thermal contexts are linked to thermodynamics and heat transfer.

History, notation and further remarks

The formal modern use of the term "work" in mechanics was introduced in the early 19th century; French mathematician Gaspard-Gustave de Coriolis helped popularize the specific mechanical meaning associated with force and displacement. Work is a scalar, despite involving vector ingredients, because it results from a dot product. When studying work it is helpful to consult basic texts and articles on classical mechanics, the mathematics of the dot product, and applied topics such as engineering mechanics. A concise visual summary of work and energy relations is provided here: $W=\Delta E_{k}=E_{k_{2}}-E_{k_{1}}={\frac {mv_{2}^{2}}{2}}-{\frac {mv_{1}^{2}}{2}}$ .

Notes: In practical calculations, carefully resolve vectors into components, check angle conventions, and remember that only the force component parallel to the displacement contributes to work. When multiple forces act, the total (net) work is the algebraic sum of the works done by each force.

History

The mechanical concept of work developed from the study of power transmission with levers, ropes and pulleys. It was already observed in antiquity that a heavy load can be lifted by means of force transducers with different amounts of force, whereby the product of force and distance is always the same if the same "work" is done, i.e. the same load is lifted by the same amount. The extended concept of work arose after the invention of the steam engine from the question of how much mechanical work can be obtained from the supply of a certain amount of heat, given by burning a certain amount of coal. A deeper microscopic interpretation of the concepts of work and heat arises in statistical physics when describing a system of very many particles.

Mechanical work

In the early days of mechanics in the 17th century, work was not yet defined by all scholars as force times displacement. Instead, there was confusion about the appropriate definition. Thus, in the 17th century, Leibniz's view in particular was opposed to that of Descartes: Leibniz favoured a prefiguration of today's definition, in which work is proportional to distance; Descartes advocated proportionality to time. Descartes' view thus corresponded to the everyday perception of work as an effort acting over a certain time.

Both views coexisted undisturbed as long as the machines of antiquity (levers, pulleys or the inclined plane) were only considered in a state of equilibrium. Here it is irrelevant whether the Golden Rule of Mechanics is referred to the saved distance or the saved time. However, when Leibniz compared both definitions in 1686 in the Acta Eruditorum using the example of free fall, i.e. an example from dynamics, he received different statements. Because of the quadratic increase in the distance of the fall with time, which has been known since Galileo, the impact energies of the two definitions do not agree. A weight that falls from four times the height only needs twice the fall time, but gains four times the kinetic energy in today's terms.

But this did not settle the dispute. Even Leibniz said that the variants "force times time" or "mass times velocity", i.e. impulse in today's parlance, could also be used with caution to determine kinetic energy. The concept of mechanical work with its current definition was only given in 1829 by Gaspard Gustave de Coriolis.

Relationship to warmth

The fact that heat can in any case be partially converted into mechanical work and is itself a form of energy that can also be generated by mechanical work was known through the steam engine (and its precursors) as well as through the inexhaustible generation of heat by mechanical work (see Benjamin Thompson) in the first half of the 19th century. Sadi Carnot recognised in 1824 that the highly variable efficiency of producing work from heat had not only to do with friction and heat losses, but had to be explained by a fundamental difference between heat and work. James Prescott Joule proved from 1843 onwards in a series of experiments that the reverse conversion of mechanical or electrical work into heat always has the same efficiency, the mechanical heat equivalent. After Hermann von Helmholtz had formulated the general law of conservation of energy in 1847, Rudolf Clausius found the equation for the 1st law of thermodynamics in 1850, in today's notation Δ $\Delta U=W+Q$ . In 1873, Josiah Willard Gibbs succeeded in inserting the energy conversions of chemical reactions into the 1st law.

Work and power

Introduction

A descriptive meaning of the physical quantity is the "effort" one has when lifting a heavy object. It is true that one can make this task seemingly easier by using an inclined plane, a pulley block, a hydraulic jack or a similar aid. This actually reduces the force required. Such aids are therefore also called force converters. However, this comes at the price of having to travel a longer distance. For example, the inclined path on the inclined plane is longer than the vertical height difference by which the object is lifted. It can be seen that the distance increases to the same extent as the force is reduced by the aid. (see Golden Rule of Mechanics). The product of the two quantities "force times distance" is therefore the same in all cases, if one disregards friction and similar interfering influences.

Therefore, it seems reasonable to define a physical quantity that quantifies this amount of work independently of the method used. This quantity is given the name work with the calculation equation:

W = Fs

Here the work, force and distance travelled. (First, it is assumed that the force is constant and points in the direction of motion. A more general definition follows below).

The unit of work results from the definition equation:

$[W]=[F][s]=1\,\mathrm {N\cdot m} =1\,\mathrm {J(Joule)}$

(Note: Formally, the unit of work is similar to that of torque: "Newton metres". However, since the physical backgrounds are completely different, the units should not be equated).

Work thus has the dimension of energy.

Further clarification results:

Without the point of application of the force travelling any distance, i.e. no mechanical work is done (for example, not by the stationary support if it simply supports a weight at rest).
If the path is covered in several sections, the sum of the corresponding partial work is always the same work regardless of the division made.
If the force is applied in a direction other than the (current) direction of movement of the point of application, only the force component parallel to the path counts for the work. If force and path are perpendicular to each other, the work is (e.g. no work is done against gravity when a trolley rolls horizontally). If this force component is opposite to the movement, the work is to be taken as negative. In this case, energy is not supplied to the system on which the force acts, but is withdrawn.

There are some differences to the everyday experience of physical work:

Even just holding a heavy object tires the muscles, although no work in the physical sense is being done here.
Dividing a path into several pieces can significantly reduce the perceived effort.

The differences are explained by the fact that the mere production of muscle power costs the body (chemical) energy.

Examples

Acceleration work: A underground with a mass of 60 t is accelerated over a distance of 100 m by the constant force of 60 kN. The work done by the drive motors is $W=Fs=60\,\mathrm {kN} \cdot \,100\mathrm {m} =6000\,\mathrm {kJ}$ . The kinetic energy of the orbit increases by the energy amount $6000\,\mathrm {kJ}$ .
Lifting work: A crane lifts a pallet with stones of 500 kg onto the roof at a height of 10 m on a building site. The weight force is $G=mg=500\,\mathrm {kg} \cdot 9,81\,\mathrm {ms^{-2}} \approx 5000\,\mathrm {N}$ In doing so, the crane performs a work of $W=Gh=5000\,\mathrm {N} \cdot 10\,\mathrm {m} =50\,\mathrm {kJ}$ . The potential energy of the stones increases by $50\,\mathrm {kJ}$ .
Acceleration work: If the rope of the crane breaks at a height of 10 m, the pallet falls down accelerated by $h=10$ m. The weight force then performs the acceleration work $W=Gh=50\,\mathrm {kJ}$ .

General definition of mechanical work

A mechanical work is always given when a body travels a distance and a force acts on it. If the force and the path do not have the same direction, but include an angle α $\alpha$ (with $0^{\circ }\leq \alpha \leq 180^{\circ }$ ), then only the component $|{\vec {F}}|\cos \alpha$ of the force ${\vec F}$ parallel to the path is to be considered, or - with the same result - the component of the path parallel to the force. The definition for the work done by the force ${\vec {F}}$ is:

$W=|{\vec {F}}|\,|\Delta {\vec {s}}|\,\cos \alpha ={\vec {F}}\cdot \Delta {\vec {s}}$

The work done is positive if the force is in the direction of the movement, negative if the force is opposite to the movement, and zero if it is at right angles to the direction of movement. If the work is positive ( $W>0$ energy is supplied to the body. If it is negative ( $W<0$ means that the body under consideration is $|W|$ energy | W | {\displaystyle |W|} to the system that exerts the force F it.

If several forces ${\vec {F}}_{i}$ a body, the equation for calculating the work can also be applied to one of them. In this way, one determines the work done by this force on the body. The total work done by the resulting force work done ${\vec {F}}_{\mathrm {res.} }=\Sigma {\vec {F}}_{i}$ $W_{\mathrm {ges.} }$ is then the sum of all the individual works:

$\sum _{i}W_{i}=\sum _{i}({\vec {F}}_{i}\cdot \Delta {\vec {s}})=\left[\sum _{i}{\vec {F}}_{i}\right]\cdot \Delta {\vec {s}}={\vec {F}}_{\mathrm {res.} }\cdot \Delta {\vec {s}}=W_{\mathrm {ges.} }$

The total work should not be confused with the "work done". Thus, when lifting a weight slowly, the total work is zero because the downward weight force ${\vec {F}}_{G}=m{\vec {g}}$ and the upward lifting force ${\vec {F}}_{h}=-{\vec {F}}_{G}$ equilibrium at all times. However, the work supplied by the lifter is determined solely from the lifting force and is $W={\vec {F}}_{h}\cdot \Delta {\vec {s}}=-m{\vec {g}}\cdot \Delta {\vec {s}}=+mgh$ .

If the path consists of different sections, the corresponding partial works along the individual sections shall be added. If the force component is not constant $|{\vec {F}}|\cos \alpha$ along the path, think of the path $|{\vec {F}}|\cos \alpha$ divided into sufficiently small pieces, each with a constant value of , and sum all contributions to the work. This leads to the general formula for the mechanical work in the form of a path or curve integral:

$W=\int _{C}{\vec {F}}({\vec {s}})\cdot \mathrm {d} {\vec {s}}$

Where is the path given in space that the point of application ${\vec {s}}$ the force ${\vec F}({\vec s})$ travels from start to finish.

Connection to energy

If one starts from the basic equation of mechanics

$\vec F = m \vec a$

and integrates along a path, you get

$\int _{C}{\vec {F}}\cdot \mathrm {d} {\vec {s}}=m\int _{C}{\vec {a}}\cdot d{\vec {s}}={\frac {1}{2}}m(v_{2}^{2}-v_{1}^{2})$

The left-hand side therefore gives exactly the work while the right side is the change in kinetic energy the body. This relationship can be summarised in the so-called work theorem:

$W=\Delta T$ .

If the force can be derived from a potential field ( ${\vec {F}}=-\nabla V({\vec {s}})$ ) - one then speaks of a conservative force - the work done just corresponds to the decrease in potential energy:

$W=-\Delta V({\vec {s}})$

Combining both equations, Δ $\Delta T=-\Delta V$ the relation

$T+V=\mathrm {konst.}$

This is nothing other than the law of conservation of energy in its simplest form for a mass point in the potential field. The work that a potential field does on a mass point changes its kinetic energy, but not its total energy.

Work as energy transfer through system boundaries

If we consider a system consisting of several bodies, we can distinguish between internal and external forces. Internal forces are those that act in pairs between two bodies of the system, whereby Newton's third law applies. In the case of external forces, one body is outside the system boundaries. If we assume that all internal forces are conservative forces, i.e. can be derived from potential fields as described above, then the work of all internal forces causes a change in the total potential energy of the system: $W_{\mathrm {int.} }=-\Delta V$ . However, according to the work theorem mentioned above, the work of all forces causes a change in kinetic energy: $W_{\mathrm {alle} }=\Delta T$ . From this follows for the work of the external forces:

$W_{\mathrm {ext.} }=W_{\mathrm {alle} }-W_{\mathrm {int.} }=\Delta T+\Delta V=\Delta E$

In words, the work done on the system by external forces causes a change in the total energy of the system. This leads to the idea that work can be understood as the supply of energy by means of external forces.

Example: Work in the gravitational field

According to the purely mechanical definition, when a body of mass sinks by the difference in height gravity performs the work $W=mgh$ . Whether it falls freely (acceleration work), slides down an inclined plane (friction work) or lifts another load via a lever is irrelevant for the calculation of the work

However, one only obtains these statements if one considers gravity as an external force. It does not belong to the system, but acts on the system formed by the body. This system for itself is then characterised by the mass and the height coordinate of the body, but not by the potential energy in the gravitational field or gravity. If the height coordinate changes by the external force does the work $W=mgh$ on this system. If, on the other hand, gravity and potential energy are included in the system under consideration, falling is an internal process in which potential energy is converted into kinetic energy, but no work is done. In the examples with sliding and the use of levers, the determination of the work depends on whether the inclined support or the other lever arm is considered part of the system or not.

If the weight is ${\vec {F}}=-m{\vec {g}}$ lifted by the external force work $W=mgh$ done on it. If the system boundaries include the gravitational field and thus also the potential energy, this work adds the corresponding energy to the system. However, if gravity is also understood as an external force, the energy of the system does not change because both forces compensate each other.

Special cases

Lifting work: work that must be performed on a body at rest of mass in order to lift it in a homogeneous gravitational field with acceleration due to gravity by the lifting height

The force needed to lift (against gravity) is: $F=m\,g$ ,

The distance travelled corresponds to the height .

Thus the lifting work done is: $W=F\,s=m\,g\,h.$

Work in rotary motion: In a rotary motion under the action of a torque, the mechanical work $W=M\,\Delta \varphi$ where the torque on the body and Δ $\Delta \varphi$ the angle (in radians) through which it is rotated. The formula results from $W=F\,s$ if the force at a distance from the axis of rotation $M=F\,r$ generates the torque and its point of application on the circle $s=r\,\Delta \varphi$ covers the arc

Tension work, also spring work, to stretch an initially untensioned spring by the distance

The (tensioning) force of a spring of spring constant is at spring extension : $F(x)=D\,x$ .

Since the force along the path is not constant, the integral takes the place of the product W $W=F\,s$ $W=\int _{0}^{s}F(x)\,\mathrm {d} x$ .

Thus the work done is: $W=\int _{0}^{s}Dx\,\mathrm {d} x={\tfrac {1}{2}}\,D\,s^{2}$ .

Acceleration work: A body of mass with velocity $v_{0}$ is accelerated to a velocity and covers a distance Its kinetic energy changes by Δ $\Delta E_{\text{kin}}$ :

$W=\Delta E_{\text{kin}}={\tfrac {1}{2}}\,m\,v^{2}-{\tfrac {1}{2}}\,m\,v_{0}^{2}={\tfrac {1}{2}}\,m\,(v^{2}-v_{0}^{2}).$

The formula results from $W=F\,s$ because the force on the body produces the acceleration ${\tfrac {F}{m}}$ and to reach the final velocity a time Δ $\Delta t={\tfrac {m}{F}}\,(v-v_{0})$ must act. Meanwhile, the body travels the distance $s=v_{0}\,\Delta t+{\tfrac {1}{2}}{\tfrac {F}{m}}\,\Delta t^{2}$

Volume work or compression work: work that must be done on a gas to compress it from volume to volume

$W=-\int _{V_{1}}^{V_{2}}p\,\mathrm {d} V.$

The negative sign comes from the fact that the force on the area the piston $F=-p\,A$ must be opposite to the internal pressure of the gas. The pressure can be variable or constant (depending on the type of change of state).

At constant pressure, this becomes the pressure-volume work, e.g. when pumping a volume of liquid against a constant pressure.

$W=p\,V\,.$

Deformation work: Work done by an external force when it deforms a body.

Electrical work: To move the quantity of charge from one point to another, between which the electrical voltage prevails, the work must be

$W=-\,Q\,U$

are performed. The formula results from $W=F\,s$ because $F=-\,Q\,{\tfrac {U}{s}}$ (when the electric field points directly from the starting point to the end point).

Magnetic work: If there is a magnetic dipole ${\vec {B}}$ in a magnetic field ${\vec m}$ , the work to be done on the dipole when the magnetic field is increased must be

$W=-{\vec {m}}\cdot {\vec {\Delta B}}$

be performed.

Friction work: product of friction force and displacement, i.e. $W={\vec F}_{{\mathrm {Reib}}}\cdot {\vec s}$ . In general, this mechanical energy is distributed by dissipation and abrasion to both sides of the friction surface and is converted into internal energy (e.g. heating) and surface work.

Surface work: To increase a surface in which the surface tension σ $\sigma$ prevails, to increase Δ $\Delta A$ , the work to be done is

$W_{\text{A}}=\sigma \Delta A$ . For the derivation of the formula see Surface tension#Mechanical definition.

An example from physiology: The work of the heart is composed of the pressure-volume work and the acceleration work by adding the work of the two ventricles.

Forces of constraint (unless they explicitly depend on time) do no work because they are always directed orthogonally to the path curve.

Work and heat

Definition of work as energy transfer

The extended definition of work is based on the basic concepts of system, energy and heat. Every physical system has a certain energy content at every point in time. A change Δ $\Delta E$ of the energy content can only happen through interactions with a second system (see conservation of energy). An example of such an interaction is the transfer of heat when the two systems have different temperatures. All other changes in energy together represent the total work done on the system

$W=\Delta E-Q$ .

This definition agrees with the mechanical concept given above in all cases of purely mechanical work. It represents a generalisation of this concept in that it can also be applied to processes such as chemical reactions in which a mechanical force and a spatial movement cannot be identified.

To apply the law of conservation of energy, the system must have clearly defined boundaries. However, the system boundary is not always to be understood only spatially. A component of a mixture of substances can also be considered a system, e.g. in a chemical reaction with another component.

The forms of work derived above from the mechanical concept are always accompanied by a change in at least one external parameter: e.g. position and orientation of the system in an external field, size and shape of the spatial extension, strength and direction of an electric or magnetic field prevailing in the system. In contrast, the general concept of work also includes when the energy of a system is changed by the transfer of matter from a second system. This can also occur through a chemical reaction between different substances present in the system, with each substance being treated as a separate system. The relevant contribution to the change Δ $\Delta E$ the total energy is called the heat of reaction or, sometimes more physically precise, chemical work.

Chemical work

Extending the first law to chemical processes, it is Δ $\Delta U=W_{mech}+W_{chem}+Q$ . Therein $W_{chem}=\Sigma _{i}\mu _{i}\Delta N_{i}$ . The index numbers the types of substances present in the system, $N_{i}$ is the quantity of the substance and μ $\mu _{i}$ its chemical potential (all in today's notation). The classification of the chemical energy conversion as work is due to the fact that the expressions $W_{mech}=-p\Delta V$ and $W_{chem}=\mu \Delta N$ formally similar, according to which the quantities $N_{i}$ play the role of the external parameters (here the volume ). Moreover, this energy contribution does not fulfil the criterion for heat used in physics since about 1920. It is not introduced into the system from the outside, but arises inside the system, where it is fed by the difference in the binding energies of the molecules before and after the reaction. Nevertheless, this chemical energy contribution is still often called heat and designated with the symbol , e.g. in everyday life ("combustion generates heat"), but also in the field of chemistry.

Interpretation through statistical physics

The simple model system of non-interacting particles allows a microscopic interpretation of heat and work. If of such particles with occupation numbers $n_{i}$ are distributed on the levels (or on the phase space cells) with energies $E_{i}$ , then the total energy is

$E_{\mathrm {ges} }=\sum _{i=1}^{N}n_{i}\,E_{i}.$

An infinitesimal change of $E_{\mathrm {ges} }$ is then

$\mathrm {d} E_{\mathrm {ges} }=\sum _{i=1}^{N}E_{i}\,\mathrm {d} n_{i}+\sum _{i=1}^{N}n_{i}\,\mathrm {d} E_{i}\ .$

If the particle system is in a thermodynamic equilibrium state, then the total energy is just the internal energy ( $E_{\mathrm {ges} }=U$ ) and it can be shown that the two summands in this equation $\mathrm {d} U=\delta Q+\delta W$ correspond to the two summands in the 1st law in the form The first summand represents the energy supplied by a reversible change of state by heat δ $\delta Q=T\,\mathrm {d} S$ the second summand the work done on the system, in the simplest case e.g. the volume work δ $\delta W=-p\,\mathrm {d} V$ . Here denotes $\mathrm {d}$ the full differential of the state variable named behind it, while δ $\delta$ denotes the inexact differential of the process variable in question. The same result follows for quantum mechanical treatment. Heat without work thus means that the total energy increases or decreases by changing the occupation numbers of the energy levels, while work without heat shifts the position of the levels with unchanged occupation numbers. The latter thus represents the microscopic criterion for an adiabatic process.

Questions and Answers

Q: What is work in physics?

A: Work is the force that an object experiences when a force is applied to it for a certain amount of time.

Q: How is work represented mathematically?

A: Work is represented by the formula W=Fs cos θ, where W represents work, F represents the magnitude of the force, s represents displacement and cos θ represents the angle between the direction of the force and actual direction of displacement.

Q: What happens if there is an angle between the direction of force and displacement?

A: If there is an angle between the direction of force and displacement, then less work will be done as it becomes less efficient than pushing in a parallel direction. The more perpendicular (90°) to the direction of force, then more work approaches zero. If greater than 90°, then overall movement will be in opposite direction from what was intended by force; resulting in negative work.

Q: Is heat conduction considered to be a form of work?

A: No, heat conduction is not considered to be a form of work since no macroscopically measurable forces are present; only microscopic forces occurring in atomic collisions.

Q: Who created 'work' as a term?

A: The term 'work' was created by French mathematician Gaspard-Gustave Coriolis in 1830s.

Q: What does Work-Energy theorem state?

A: According to Work-Energy theorem, if an external force acts upon a rigid object causing its kinetic energy to change from Ek1 to Ek2 then mechanical work (W) can be calculated using mv2/2 - mv1/2 , where m stands for mass and v stands for velocity.

Author

AlegsaOnline.com - Work (physics) - Leandro Alegsa

Creation date: November 13, 2022
Last updated: June 5, 2026

URL: https://en.alegsaonline.com/art/109051