In science, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. A force is said to do positive work if it has a component in the direction of the displacement of the point of application. A force does negative work if it has a component opposite to the direction of the displacement at the point of application of the force. For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). If the ball is thrown upwards, the work done by the gravitational force is negative, and is equal to the weight multiplied by the displacement in the upwards direction. Both force and displacement are vectors. The work done is given by the dot product of the two vectors, where the result is a scalar. When the force F is constant and the angle θ between the force and the displacement s is also constant, then the work done is given by:
W = F ⋅ s = F s cos θ {\displaystyle W=\mathbf {F} \cdot \mathbf {s} =Fs\cos {\theta }}
If the force and/or displacement is variable, then work is given by the line integral:
W = ∫ F ⋅ d s = ∫ F ⋅ d s d t d t = ∫ F ⋅ v d t {\displaystyle {\begin{aligned}W&=\int \mathbf {F} \cdot d\mathbf {s} \\&=\int \mathbf {F} \cdot {\frac {d\mathbf {s} }{dt}}dt\\&=\int \mathbf {F} \cdot \mathbf {v} dt\end{aligned}}}
where d s {\displaystyle d\mathbf {s} } is the infinitesimal change in displacement vector, d t {\displaystyle dt} is the infinitesimal increment of time, and v {\displaystyle \mathbf {v} } represents the velocity vector. The first equation represents force as a function of the position and the second and third equations represent force as a function of time. Work is a scalar quantity, so it has only magnitude and no direction. Work transfers energy from one place to another, or one form to another. The SI unit of work is the joule (J), the same unit as for energy.
History The ancient Greek understanding of physics was limited to the statics of simple machines (the balance of forces), and did not include dynamics or the concept of work. During the Renaissance the dynamics of the Mechanical Powers, as the simple machines were called, began to be studied from the standpoint of how far they could lift a load, in addition to the force they could apply, leading eventually to the new concept of mechanical work. The complete dynamic theory of simple machines was worked out by Italian scientist Galileo Galilei in 1600 in Le Meccaniche (On Mechanics), in which he showed the underlying mathematical similarity of the machines as force amplifiers. He was the first to explain that simple machines do not create energy, only transform it.
Early concepts of work Although work was not formally used until 1826, similar concepts existed before then. Early names for the same concept included moment of activity, quantity of action, latent live force, dynamic effect, efficiency, and even force. In 1637, the French philosopher René Descartes wrote:
Lifting 100 lb one foot twice over is the same as lifting 200 lb one foot, or 100 lb two feet. In 1686, the German philosopher Gottfried Leibniz wrote:
The same force ["work" in modern terms] is necessary to raise body A of 1 pound (libra) to a height of 4 yards (ulnae), as is necessary to raise body B of 4 pounds to a height of 1 yard. In 1759, John Smeaton described a quantity that he called "power" "to signify the exertion of strength, gravitation, impulse, or pressure, as to produce motion". Smeaton continues that this quantity can be calculated if "the weight raised is multiplied by the height to which it can be raised in a given time", making this definition remarkably similar to Coriolis's.
Etymology and modern usage The term work (or mechanical work), and the use of the work-energy principle in mechanics, was introduced in the late 1820s independently by French mathematician Gaspard-Gustave Coriolis and French Professor of Applied Mechanics Jean-Victor Poncelet. Both scientists were pursuing a view of mechanics suitable for studying the dynamics and power of machines, for example steam engines lifting buckets of water out of flooded ore mines. According to Rene Dugas, French engineer and historian, it is to Solomon of Caux "that we owe the term work in the sense that it is used in mechanics now". The concept of virtual work, and the use of variational methods in mechanics, preceded the introduction of "mechanical work" but was originally called "virtual moment". It was re-named once the terminology of Poncelet and Coriolis was adopted.
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