A rubber cord of length L, which is a known state function L (T, F) of the temperature T of the cord and of the forces of magnitude F applied at each end to stretch it. Two physical properties of the cord are :a) the Young modulus, defined as E =L/A(∂L/∂F)^-1 , where A is the cord cross section

Question

A rubber cord of length L, which is a known state function L (T, F) of the temperature T of the cord and of the forces of magnitude F applied at each end to stretch it. Two physical properties of the cord are :

a) the Young modulus, defined as [latex]E = \frac{L}{A} \Bigl(\frac{\partial L}{\partial F} \Bigr)^{-1}[/latex],where A is the cord cross section area.

b) the thermal expansion coefficient [latex]α =\frac{l}{L}\frac{\partial L}{\partial T}[/latex].

Determine how much the length of the cord varies if its temperature changes by ΔT and at the same time the force F changes by ΔF. Assume that [latex]ΔT \ell T[/latex] and [latex]ΔF \ell F[/latex]. Express ΔL in terms of E and α.

Accepted Answer

Applying definition (1.7) to the differential of the length L (T, F) of the rubber cord, the change of length of the rubber cord is given by,

[latex]dƒ(x,y) = \frac{\partial ƒ(x,y) }{\partial x } dx +\frac{\partial ƒ(x,y) }{\partial y }dy [/latex]. (1.7)

[latex]ΔL = \frac{\partial L }{\partial T } Δ T +\frac{\partial L }{\partial F } Δ F [/latex].

which can be recast as,

[latex]\Delta L = L\Bigl(\frac{l}{L} \frac{\partial L}{\partial T} \Bigr)\Delta T +\frac{L}{A} \biggl(\frac{L}{A}\Bigl(\frac{\partial L }{\partial F }\Bigr)^{-1}\biggr)^{-1} \Delta F[/latex].

Using the two physical properties of the cord, we obtain an expression for the change of length of the rubber cord,

[latex]Δ L = L α Δ T +\frac{L}{A E} ΔF[/latex].

Question 1.3: A rubber cord of length L, which is a known state function L...

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