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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:
When a force is applied on a spring, and the length of the spring changes by a differential amount dx, the work done is = For linear elastic springs, the displacement x is proportional to the force applied =, where K is the spring constant and has the unit of N/m.
Since work is defined as force multiplied by displacement, the area of the graph shows the mechanical work output of the muscle. In a typical work-generating instance, the muscle shows a rapid curvilinear rise in force as it shortens, followed by a slower decline during or shortly before the muscle begins the lengthening phase of the cycle.
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 ...
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 kinetic energy of an object is equal to the work, or force in the direction of motion times its displacement , needed to accelerate the object from rest to its given speed. The same amount of work is done by the object when decelerating from its current speed to a state of rest. [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.
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.