A) \[\frac{YA{{x}^{2}}}{2L}\]
B) \[YA{{x}^{2}}L\]
C) \[\frac{YAx}{2L}\]
D) \[\frac{YA{{x}^{2}}}{L}\]
Correct Answer: A
Solution :
If a force F acts along the length L of the wire of corss ? section A and stretches it by\[x,\]then \[Y=\frac{stress}{strain}=\frac{F/A}{x/L}=\frac{FL}{Ax}\Rightarrow F=\frac{YA}{L}x\] So, the work done for an additional small increase dx in length \[dW=F.dx=\frac{YA}{L}x.dx\] Hence the total work done in increasing the length by \[l\] \[W=\int_{0}^{x}{dW}=\int_{0}^{x}{Fdx}=\int_{0}^{x}{\frac{YA}{L}x}.dx=\frac{1}{2}\frac{YA}{L}{{x}^{2}}\]
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