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Thus, the work done for a variable force can be expressed as a definite integral of force over displacement. [24] If the displacement as a variable of time is given by ∆x(t), then work done by the variable force from t 1 to t 2 is:
The non-mechanical work of force fields can have either positive or negative sign, work being done by the system on the surroundings, or vice versa. Work done by force fields can be done indefinitely slowly, so as to approach the fictive reversible quasi-static ideal, in which entropy is not created in the system by the process.
The work done by a conservative force is equal to the negative of change in potential energy during that process. For a proof, imagine two paths 1 and 2, both going from point A to point B. The variation of energy for the particle, taking path 1 from A to B and then path 2 backwards from B to A, is 0; thus, the work is the same in path 1 and 2 ...
Every conservative force has a potential energy. By following two principles one can consistently assign a non-relative value to U: Wherever the force is zero, its potential energy is defined to be zero as well. Whenever the force does work, potential energy is lost.
The work of a force on a particle along a virtual displacement is known as the virtual work. Historically, virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies, [1] but they have also been developed for the study of the mechanics of deformable bodies. [2]
The work done by a variable force acting over a finite linear displacement is given by integrating the force with respect to an elemental linear displacement W = ∫ s 1 s 2 F ⋅ d s {\displaystyle W=\int _{s_{1}}^{s_{2}}\mathbf {F} \cdot \mathrm {d} \mathbf {s} }
Work is dependent on the displacement as well as the force acting on an object. As a particle moves through a force field along a path C, the work done by the force is a line integral: = This value is independent of the velocity /momentum that the particle travels along the path.
Energy is transferred to an object by work when an external force displaces or deforms the object. The quantity of energy transferred is the vector dot product of the force and the displacement of the object. As forces are applied to the system they are distributed internally to its component parts.