Work (Physics)

In physics, work is a mode of transferring energy, rather than a substance or a property stored in an object. In classical mechanics, a force does work when its point of application moves with a component of displacement along the force. Work connects forces and motion with changes in energy; in thermodynamics, the concept also describes energy transfers through processes such as compression and expansion. Its value can be positive, negative, or zero, depending on the interaction and the adopted sign convention. (openstax.org)

Definition and mathematical formulation

For a constant force acting through a displacement, the work is

where $F$ is the force magnitude, $d$ is the displacement magnitude, and $\theta$ is the angle between them. The dot product is the ordinary Euclidean inner product of the two vectors. Consequently, work is a scalar, although force and displacement have directions. Only the component of force parallel to the displacement contributes. (openstax.org)

For a varying force or a curved trajectory, work is calculated using a line integral:

Here $C$ is the actual path followed by the point of application. In one-dimensional motion, this becomes the integral $\int_{x_i}^{x_f}F_x\,dx$, corresponding to the signed area under a force–position graph. The result generally depends on the path, not merely on the endpoints. (ocw.mit.edu)

The unit of work in the International System of Units is the joule (J):

One joule is the work done by a one-newton force through one metre in the force’s direction. (nist.gov)

Positive, negative, and zero work

A force does positive work when its component along the motion points in the direction of displacement. It does negative work when that component opposes displacement. A force perpendicular to motion does no work. Thus, the ideal centripetal force in uniform circular motion changes the direction of velocity without changing kinetic energy. Exerting a force on an immovable wall likewise produces no mechanical work on the wall if its point of application remains stationary. Physical effort is therefore not synonymous with mechanical work. (openstax.org)

These distinctions also apply to electromagnetic interactions. The magnetic part of the Lorentz force on a moving point charge is perpendicular to its velocity, so it does no work on that charge. An electric force, by contrast, can have a component along the motion and change the particle’s energy. (openstax.org)

The work–energy theorem

The work–energy theorem states that the net work done by all forces on a particle equals its change in kinetic energy:

For a particle of constant mass, this follows from Newton’s laws of motion by integrating $\mathbf F_{\mathrm{net}}\cdot d\mathbf r$. It applies to varying forces as well as constant ones. Positive net work increases kinetic energy, while negative net work decreases it. (openstax.org)

The word “net” is essential: the work of one force need not equal the total kinetic-energy change. An object moving at constant speed can receive positive work from an applied force while friction does equal negative work. Its kinetic energy remains unchanged, although energy is transferred and may become thermal energy. For extended bodies, rotation, deformation, and internal motion must also be included in the energy accounting. (openstax.org)

Conservative forces and potential energy

A conservative force performs work that depends only on the initial and final positions. Its work around a closed path is zero, and an associated potential energy $U$ can be defined so that

Examples include gravity and the restoring force of an ideal spring. Near Earth’s surface, raising a mass through height $h$ increases gravitational potential energy by $mgh$, while gravity does work $-mgh$. Stretching an ideal spring from equilibrium stores energy $\tfrac12kx^2$. (openstax.org)

When only conservative forces do work, kinetic and potential energy exchange while their sum remains constant. Sliding friction is generally nonconservative: its work depends on the distance travelled and converts mechanical energy into internal energy. This does not violate energy conservation; it changes the form in which energy is accounted for. (openstax.org)

Power and rotational work

Power measures the rate of doing work. Average power is $W/\Delta t$, while instantaneous power is

The velocity is that of the force’s point of application. Power is measured in watts, with one watt equal to one joule per second. Performing the same work in less time requires greater average power. (openstax.org)

For rotation about a fixed axis, torque supplies the corresponding expression:

A constant torque gives $W=\tau\Delta\theta$, and rotational power is $P=\tau\omega$. Angles are expressed in radians. Net rotational work equals the change in rotational kinetic energy of a rigid body about the fixed axis. (openstax.org)

Work in thermodynamics

In thermodynamics, work and heat are distinct modes of energy transfer across the boundary of a thermodynamic system. Heat is transfer associated with a temperature difference; work includes macroscopic interactions such as moving a piston or turning a shaft. Neither is a state function: their amounts depend on the process connecting two states. (openstax.org)

For quasistatic expansion of a gas, the work done by the gas is

where $p$ is the gas pressure. Expansion gives positive work under this convention; compression gives negative work. For a closed system with negligible changes in bulk kinetic and potential energy, the first law relates work to internal energy:

An alternative convention counts work done on the system as positive, giving $\Delta U=Q+W_{\mathrm{on}}$. The physical energy balance is identical when the convention is used consistently. (openstax.org)