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 Homework Statement:
 A wire is replaced by a different wire with the same material, length but double the diameter. The same load is attached to the wire. What is the change to the elastic potential energy
 Relevant Equations:

F=kx
E= 1/2kx^2
Young's Modulus = Stress/strain
I think the answer is that the elastic potential energy will be a 1/16th of the original value. This is my reasoning:
1) If the diameter doubles, the cross sectional area is 4 times the original value. (from A= πr^{2}).
2) F= stress/area. Force (load is the same). If cross sectional area quadruples, then the stress must quarter.
3) E= stress/strain. E is the same. If stress quarters, than strain must quarter. If strain quarters, extension must quarter (since original length is still the same).
4) From E= 1/2 kx^{2}, with k being the same and extension being a quarter of the original value, the energy must be a 1/16th of the original value.
However, the answers say the elastic potential energy is a quarter of the original value. Which part of my reasoning is incorrect ?
1) If the diameter doubles, the cross sectional area is 4 times the original value. (from A= πr^{2}).
2) F= stress/area. Force (load is the same). If cross sectional area quadruples, then the stress must quarter.
3) E= stress/strain. E is the same. If stress quarters, than strain must quarter. If strain quarters, extension must quarter (since original length is still the same).
4) From E= 1/2 kx^{2}, with k being the same and extension being a quarter of the original value, the energy must be a 1/16th of the original value.
However, the answers say the elastic potential energy is a quarter of the original value. Which part of my reasoning is incorrect ?