Question: Derive an expression for the energy stored in a stretched string. Define the term energy density of a body under strain.
Ans: Energy stored in a stretched string: when a given wire is stretch due to application deforming force then work is done which is stored in the form of potential energy known as energy stored on a stretched wire.
Let us consider a wire of length L and area of cross-section A. Let F be the stretching force applied on the wire as shown in the figure. If l be the length through which the wire is stretched, thenNormal stress = F/A
Longitudinal strain = l/L
Then young’s modulus of elasticity (γ)
= Normal stress/longitudinal strain
= F/A * L/l
= FL/Al
therefore F = γAl/L
Let ‘dw’ be the small workdone for increasing length of the wire ‘dl’ due to the application of force ‘f’ which can be written as
DW = γAl/L dl
Now, total workdone can be obtained by integrating small workdone from 0 to l as
Since, total workdone is in the form of potential energy so we can write as,
Total workdone (W) = Potential energy (E) = 1/2 F.l
Therefore Energy stored (E) = 1/2 F.l Thus it is the required equation for energy store on a stretched wire.
Energy density: The energy stored on a stretched wire per unit volume is known as Energy density.
So, mathematically we can write as,
Energy density (U) = E/V = 1/2 * F/A * l/L = 1/2 stress strain
Therefore Energy stored (E) = 1/2 stress strain volume
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