Calculate the stress developed inside a tooth cavity filled with copper when hot tea at temperature of $57\,^oC$ is drunk. You can take body (tooth) temperature to be $37\,^oC$ and $\alpha = 1.7 \times 10^{-5}/^oC$, bulk modulus for copper $ = 140 \times 10^9\, N/m^2 $.
Calculate the stress developed inside a tooth cavity filled with copper when hot tea at temperature of $57\,^oC$ is drunk. You can take body (tooth) temperature to be $37\,^oC$ and $\alpha = 1.7 \times 10^{-5}/^oC$, bulk modulus for copper $ = 140 \times 10^9\, N/m^2 $.
Increase in temperature
$\Delta \mathrm{T}=57-37=20^{\circ} \mathrm{C} \text { or } 20 \mathrm{~K}$
Coefficient of linear expansion of cavity, $\alpha=1.7 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}$
Bulk modulus of cavity $\mathrm{K}=140 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}$
Coefficient volume expansion of copper,
$\gamma=3 \alpha$
$=3 \times 1.7 \times 10^{-5}$
$=5.1 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}$
Suppose the original volume is $\mathrm{V}$ and the increase in volume is $\Delta \mathrm{V}$ due to increase in temperature $\Delta \mathrm{T}$.
We know that,
$\Delta \mathrm{V}=\gamma \mathrm{V} \Delta \mathrm{T}$
$\therefore \frac{\Delta \mathrm{V}}{\mathrm{V}}=\gamma \Delta \mathrm{T}$
Thermal stress $=$ Bulk modulus $\times$ volume strain
$=\mathrm{K} \times \frac{\Delta \mathrm{V}}{\mathrm{V}}$
$=\mathrm{K} \times \gamma \Delta \mathrm{T}$
$=140 \times 10^{9} \times 5.1 \times 10^{-5} \times 20$
$=1.428 \times 10^{8} \mathrm{~N} / \mathrm{m}^{2}$
Atmospheric pressure is $\mathrm{P}=1.01 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}$. Hence, the pressure in cavity of teeth is approximately $1000$ times than that of atmosphere.
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