First slide

Elastic Modulie

Question

A smooth uniform string of natural length $L_{0}$ , cross sectional area A and Young's modulus Y is pulled along its length by a force F on a horizontal smooth surface. The elastic potential energy stored in the string is

Difficult

A

$\frac{2 F^{2} L_{0}}{AY}$
B

$\frac{F^{2} L_{0}}{3 AY}$
C

$\frac{F^{2} L_{0}}{2 AY}$
D

$\frac{F^{2} L_{0}}{6 AY}$

Solution

Tension, $T = \frac{F}{L_{0}} . x$
Stress, $σ = \frac{T}{_{A}} = \frac{F}{{AL}_{0}} x$
$dU = \frac{1}{2} . \frac{σ^{2}}{Y} Adx = \frac{1}{2} \frac{F^{2}}{A^{2} L_{0}^{2}} . x^{2} \frac{A}{Y} dx$
$or dU = \frac{F^{2}}{2 A^{2} L_{0}^{2} Y} . x^{2} dx$
$\Rightarrow U = \frac{F^{2}}{2 {AYL}_{0}^{2}} \int_{0}^{L_{0}} x^{2} dx$
$U = \frac{F^{2}}{2 AY L_{0}^{2}} . \frac{L_{0}^{3}}{3} = \frac{F^{2} L_{0}}{6 AY}$

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