Find Work Done with Increasing Tension of a Wire

[James Wine]

Homework Statement

Let a wire's tension increase quasi-statically from [tex]F_{1}[/tex][tex]\rightarrow[/tex][tex]F_{2}[/tex], with Young's Modulus [tex]Y[/tex], Cross-sectional Area [tex]A[/tex], and Length [tex]L[/tex]. Find the work done

Homework Equations

[tex]F=\frac{YA}{L}[/tex] *Delta L, tex having issues
W= Integral from L to L_2

The Attempt at a Solution

W = YA*Integral[dL/L]
W = YA ln(L_2/L)

Question is if the length changes then the cross-sectional area should also change and would require this to be a double integral, 2nd question would be if I even thought about the first integral in the right way.

 

[vela]

If you really need to take into account the effect of A changing, you could use the approximation that the total volume remains constant, or you could use ratio[/url].

Note your calculation assumes that F₁=0 because you used L, the initial length of the wire, for the lower limit of the integral. Your work looks fine otherwise.