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Work by a spring

Consider a spring that exerts a horizontal force F = (− kx, 0, 0) that is proportional to its deflection in the x direction independent of how a body moves. The work of this spring on a body moving along the space with the curve X (t) = (x (t), y (t), z (t)), is calculated using its velocity, v = (v, v, v), to obtain W = ∫ 0 t F ⋅ v d t = − ∫ 0 t k x v x d t = − 1 2 k x 2. {\displaystyle W=\int _{0}^{t}\mathbf {F} \cdot \mathbf {v} dt=-\int _{0}^{t}kxv_{x}dt=-{\frac {1}{2}}kx^{2}.} For convenience, consider contact with the spring occurs at t = 0, then the integral of the product of the distance x and the x-velocity, xv dt, over time t is 1 / 2 x. The work is the product of the distance times the spring force, which is also dependent on distance; hence the x result.