Finding the Error: ε = dL/L Calculation

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The error in the ε = dL/L calculation arises from neglecting the rod's own weight, which contributes to its elongation. When a rod hangs vertically, its weight causes a change in length (dL) that must be factored into the strain calculation. The correct formula for strain is ε = P/(2*E*S), reflecting the influence of both the applied weight and the rod's elongation. This adjustment is crucial for accurate results in determining strain in vertically suspended specimens. Understanding this concept is essential for proper application of Young's modulus in such scenarios.
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I have the following problem:
The specimen (rod) is hanging vertically. Weight=P. Young's modulus = E. Area = S. What is the ε-?
ε = dL/L

As I know Energy U=V*(E*ε ^2)/2
And the work is A=P*dL(dL of center of mass)
In this case A=p*dL/2
So it must be that ε =P/(E*S)
But the correct answer is ε =P/(2*E*S)
Where is the mistake?
 
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The mistake in your calculation is that you are not taking into account the change in length of the rod due to its own weight. When a rod is hanging vertically, the weight of the rod itself causes it to elongate slightly, resulting in a change in length (dL). This change in length needs to be included in the calculation for ε, which is why the correct formula is ε = P/(2*E*S). This takes into account both the weight of the rod and its own elongation.
 

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