Physics 4311: Thermal Physics – HW Solution 5

Problem 1: Exact differentials (25 points)

a)

$$\begin{array}{rcll}d{u}_a& =& 2x\mathit{dx}+\mathit{dy}\frac{\partial }{\mathit{\partial y}}(2x)=\frac{\partial }{\mathit{\partial x}}(1)=0\Rightarrow \mathit{ d}{u}_a\mathrm{\text{ is exact}}& \\ d{u}_b& =& 2y\mathit{dx}+\mathit{dy}\frac{\partial }{\mathit{\partial y}}(2y)\ne \frac{\partial }{\mathit{\partial x}}(1)\Rightarrow \mathit{ d}{u}_b\mathrm{\text{ is not exact}}& \end{array}$$

b)

$$u_a=x^2+y+\mathrm{const}$$

c)

Analayze $d{u}_a$:

$$\begin{array}{rcll}{I}_1& =& {\int \; <!--\; nolimits\; -->}_0^2\mathit{dx}2x+{\int \; <!--\; nolimits\; -->}_0^2\mathit{dy}=4+2=6& \\ I_2& =& {\int \; <!--\; nolimits\; -->}_0^2\mathit{dy}+{\int \; <!--\; nolimits\; -->}_0^2\mathit{dx}2x=2+4=6& \\ & & u_a(2,2)-u_a(0,0)=6& \end{array}$$

d)

Analayze $d{u}_b$:

$$\begin{array}{rcll}{I}_1& =& {\int \; <!--\; nolimits\; -->}_0^2\mathit{dx}0+{\int \; <!--\; nolimits\; -->}_0^2\mathit{dy}=0+2=2& \\ I_2& =& {\int \; <!--\; nolimits\; -->}_0^2\mathit{dy}+{\int \; <!--\; nolimits\; -->}_0^2\mathit{dx}2y=2+8=10& \end{array}$$

Problem 2: Rubber elasticity (15 points)

a)

$$\begin{array}{rcll}W& =& {\int \; <!--\; nolimits\; -->}_{L_i}^{L_f}F\mathit{dL}=\mathit{aT}{\int \; <!--\; nolimits\; -->}_{L_i}^{L_f}\left[(L∕L_0)-(L_0^2∕L^2)\right]\mathit{dL}& \\ & =& \mathit{aT}\left[\frac{1}{2}{L}_f^2∕L_0-\frac{1}{2}{L}_i^2∕L_0+L_0^2∕L_f-L_0^2∕L_i\right]& \\ & =& \mathit{aT}{L}_0\left[\frac{1}{2}{L}_f^2∕L_0^2-\frac{1}{2}{L}_i^2∕L_0^2+L_0∕L_f-L_0∕L_i\right]& \\ & =& -0.1758J& \end{array}$$

b)

At fixed length, the tension is proportional to the tenperature, so $F$ increases with increasing $T$?

c)

Product of $T$ and the bracket on the r.h.s. is constant $\Rightarrow$ $L$ decreases with increasing $T$