Finding the Error in Calculating ε

[Yegor]

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, where V-volume
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?

 

[Doc Al]

So it must be that ε =P/(E*S)
But the correct answer is ε =P/(2*E*S)
Where is the mistake?

I can't quite figure out what you're doing with energy and volume. But I suspect that your mistake is in treating the rod as if the tension were the same throughout, which is not the case.

 

[Yegor]

Ok. In final moment Energy is equal U=V*(E*ε ^2)/2=V*(σ*ε/2),
where σ-stress (tension), (σ=F/S=E*ε (Hook's law)
But the "tension is not the same throughout".
It changes from 0 to P/S.
What I should do?

 

[Doc Al]

But the "tension is not the same throughout".
It changes from 0 to P/S.
What I should do?

The stress ranges from 0 (at the bottom) to P/S at the top. So the average stress is 1/2 P/S. Thus the overall strain is P/(2 E S), where E is Young's modulus.