# Work (Thermodynamics)

In physics, mechanical work is the amount of energy transferred by a force acting through a distance. Like energy, it is a scalar quantity, with SI units of joules.

If the resultant force F on an object acts while the object is displaced a distance d, and the force and displacement act parallel to each other, the mechanical work done on the object is the product of F multiplied by d:

W = F * d

If the force and the displacement are parallel and in the same direction, the mechanical work is positive. If the force and the displacement are parallel but in opposite directions (i.e. antiparallel), the mechanical work is negative.

However, if the force and the displacement act perpendicular to each other, zero work is done by the force:

W = 0

## Mathematical calculation

### Force and displacement

Force and displacement are both vector quantities and they are combined using the dot product to evaluate the mechanical work, a scalar quantity: $W = \bold{F} \cdot \bold{d} = F d \cos\theta$           (1)

where \textstyle\theta is the angle between the force and the displacement vector.

In order for this formula to be valid, the force and angle must remain constant. The object's path must always remain on a single, straight line, though it may change directions while moving along the line.

In situations where the force changes over time, or the path deviates from a straight line, equation (1) is not generally applicable although it is possible to divide the motion into small steps, such that the force and motion are well approximated as being constant for each step, and then to express the overall work as the sum over these steps.

The general definition of mechanical work is given by the following line integral: $W_C = \int_{C} \bold{F} \cdot \mathrm{d}\bold{s}$             (2)

where: $\textstyle _C$ is the path or curve traversed by the object; $\bold F$ is the force vector; and $\bold s$ is the position vector.

The expression $\delta W = \bold{F} \cdot \mathrm{d}\bold{s}$ is an inexact differential which means that the calculation of $\textstyle{ W_C}$ is path-dependent and cannot be differentiated to give $\bold{F} \cdot \mathrm{d}\bold{s}$.

Equation (2) explains how a non-zero 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. This is what happens during 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 illustrates the difference between work and a related quantity, impulse, which is 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.

### Torque and rotation

Work done by a torque can be calculated in a similar manner. A torque $\tau\;$ applied through a revolution of $\theta\;$, expressed in radians, does work as follows: $W= \tau \theta\$

## Frame of reference

The work done by a force acting on an object depends on the inertial frame of reference, because the distance covered while applying the force does. Due to Newton's law of reciprocal actions there is a reaction force; it does work depending on the inertial frame of reference in an opposite way. The total work done is independent of the inertial frame of reference.

## Units

The SI 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. This definition is based on Sadi Carnot's 1824 definition of work as "weight lifted through a height", which is based on the fact that early steam engines were principally used to lift buckets of water, through a gravitational height, out of flooded ore mines. 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.

Heat conduction is not considered to be a form of work, since the energy is transferred into atomic vibration rather than a macroscopic displacement

Work can be zero even when there is a force. The centripetal force in a uniform circular motion, for example, does zero work since the kinetic energy of the moving object doesn't change. This is because the force is always perpendicular to the motion of the object; only the component of a force parallel to the velocity vector of an object can do work on that object. Likewise when a book sits on a table, the table does no work on the book despite exerting a force equivalent to mg upwards, because no energy is transferred into or out of the book.