Energy stored in stretched string

During my course of teaching of elasticity ,i teach about the energy stored in the stretch string.One of the example is a sling in which when rubber string is stretched elastic
potential energy get stored and when it is released it converts into kinetic energy.
WORK DONE AND ELASTIC ENERGY


Let us take the example of a wire. Suppose the original length of the wire is L, When not deformed, there are no internal forces. As the length of the wire increase from L to L + DL, internal forces increase from 0 to F. Suppose at any instant when the length is (L +x), the internal force is Fx. The work done to further increase the length through dx is Fxdx, hence total work done



W = Fx dx = EA/L x dx

So the total work done is

W = ½ EA/L (DL)2

W = ½ F (DL)

Thus, the elastic energy per unit volume is given by-

U = W/AL = ½ stress ´ strain

(1) The above results is true for all types of stress and strain.

(2) The work done per unit volume or the elastic energy per unit volume is given by half the product of stress and strain.

(3) If we draw a graph between stress and strain then as the area between the graph and strain axis is ½ stress ´ strain, it gives the work done per unit volume.

There is a beautiful video which the maximum energy which can be given particle which is being thrown with the help of sling

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