= − ∫ kx x x m ( . ) This expression may also be obtained by considering the area of the triangle as in Fig. . (d).
Note that the work done by the external pulling force F is positive since it overcomes the spring force. ( . ) Fig. .
Illustration of the spring force with a block attached to the free end of the spring. (a) The spring force F s is zero when the displacement x from the equilibrium position is zero. (b) For the stretched spring x > and F s < (c) For the compressed spring x < and F s > .(d) The plot of F s versus x. The area of the shaded triangle represents the work done by the spring force.
Due to the opposing signs of F s and x, this work done is negative, W kx / s . The same is true when the spring is compressed with a displacement x c (< ). The spring force does work / c s kx while the Fig. .
Parabolic plots of the potential energy V and kinetic energy K of a block attached to a spring obeying Hooke’s law. The two plots are complementary, one decreasing as the other increases. The total mechanical energy E = K + V remains constant. external force F does work + kx c / .
If the block is moved from an initial displacement x i to a final displacement x f , the work done by the spring force W s is k x x k x k x s ( . ) Thus the work done by the spring force depends only on the end points. Specifically, if the block is pulled from x i and allowed to return to x i ; k x x k x k x s = ( . ) The work done by the spring force in a cyclic process is zero.
We have explicitly demonstrated that the spring force (i) is position dependent only as first stated by Hooke, ( F s = − kx