# Mechanical work

 related topics {math, energy, light} {work, book, publish} {math, number, function}

W = τ θ

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. The term work was first coined in 1826 by the French mathematician Gaspard-Gustave Coriolis.[1][2]

According to the work-energy theorem if an external force acts upon a rigid object, causing its kinetic energy to change from Ek1 to Ek2, then the mechanical work (W) is given by:[3]

where m is the mass of the object and v is the object's velocity.

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 dot product of the vectors F and d:[4]

If the force and the displacement are parallel and in the same direction (θ = 0), the mechanical work is positive. If the force and the displacement are parallel but in opposite directions (i.e. antiparallel, θ = 180⁰), the mechanical work is negative. If a force F is applied at an angle θ, only the component of the force in the same direction as the displacement (Fcosθ) does work. Thus, if the force acts perpendicular to the displacement (θ = 90⁰ or 270⁰), zero work is done by the force.[4]

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### 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.