Is dL an Exact Differential for a Stressed and Heated Wire?

[meteorologist1]

I have trouble doing a problem involving exact differentials:

Consider a uniform wire of length L and cross-section area A. A force F is applied to the wire. We can write the relationship:

dL = (L/YA)dF + (aL)dT

where Y is the Young's modulus, a the coefficient of thermal expansion, and T the wire temperature. For small deformation and temperature change, we can assume A, Y, and a to be constant. Determine if dL is an exact differential.

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My problem is with L. L is a function of F and T, I'm sure. If dL is an exact differential, I want to check if the partial derivative of (L/YA) w.r.t. T is equal to the partial derivative of of (aL) w.r.t. F. But I'm running into the problem of what the partial derivatives of L w.r.t. to T and F are.

Thanks in advance.

 

[lippyka]

Yes, dL is an exact differential. The partial derivatives of L with respect to T and F can be found by implicit differentiation, since L is a function of both F and T. To do this, you need to solve for L in terms of F and T and then take the partial derivatives of the equation with respect to both F and T. This should give you the values you need to check if the partial derivatives are equal.