Work (physics)

In physics, work describes the transfer of energy that occurs when a force moves an object over a distance. In the simplest case, where a constant force pushes or pulls in the same direction as the motion, the amount of work is simply the strength of that force multiplied by the distance traveled. Work is considered positive when the force has a component pointing in the same direction as the object's displacement, and negative when the force has a component pointing opposite to that displacement. For instance, when a ball is dropped, gravity does positive work on it, equal to the ball's weight times the distance it falls. If the ball is thrown upward, gravity does negative work, equal to the weight times the upward displacement.

Because both force and displacement are vectors, work is calculated using the dot product of these two vectors, which yields a scalar result. For a constant force and a constant angle between the force and displacement, the formula is \( W = F s \cos \theta \). When the force or displacement varies, work is found using a line integral: \( W = \int \mathbf{F} \cdot d\mathbf{s} \), which can also be expressed as \( \int \mathbf{F} \cdot \mathbf{v} \, dt \), where \( d\mathbf{s} \) is an infinitesimal change in displacement, \( dt \) is an infinitesimal time increment, and \( \mathbf{v} \) is velocity. Work itself is a scalar—it has magnitude but no direction—and it transfers energy from one place or form to another. The SI unit for work is the joule (J), the same as for energy.

Historically, ancient Greek physics focused only on the statics of simple machines, like balancing forces, and did not include dynamics or the concept of work. During the Renaissance, the study of simple machines shifted to consider how far they could lift a load, not just the force they applied, which eventually led to the idea of mechanical work. In 1600, Galileo Galilei developed the complete dynamic theory of simple machines in *Le Meccaniche*, showing they share a mathematical basis as force amplifiers and, for the first time, explaining that they transform energy rather than create it.

Before the term "work" was formally adopted in 1826, similar ideas went by names like moment of activity, quantity of action, latent live force, dynamic effect, efficiency, and force. In 1637, René Descartes noted that lifting 100 pounds one foot twice is equivalent to lifting 200 pounds one foot, or 100 pounds two feet. In 1686, Gottfried Leibniz wrote that the same "force" (meaning work) is needed to raise a 1-pound body 4 yards as to raise a 4-pound body 1 yard. In 1759, John Smeaton defined "power" as "the exertion of strength, gravitation, impulse, or pressure, as to produce motion," and said it could be calculated by multiplying the weight raised by the height it is raised in a given time—a definition very close to the modern one.

The term "work" (or "mechanical work") and the work-energy principle were introduced in the late 1820s, independently by French mathematician Gaspard-Gustave Coriolis and French professor Jean-Victor Poncelet. Both were developing a view of mechanics suited to studying the dynamics and power of machines, such as steam engines lifting water from flooded mines.