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Calculate the stress developed inside a tooth cavity filled with copper when hot tea at temperature of $57^{\circ}C$ is drunk. You can take body temperature to be $37^{\circ}C$ and $\alpha=1.7\times 10^{-5}$ , bulk modulus for copper = $140 \times 10^9 N/m^{2}.$

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Let $\Delta T$ be the change in temperature, $\Delta T = 57 - 37 = 20^{\circ}C$

Let $\alpha$ be linear expansion of body, $\alpha = \frac{1.7 \times 10^{-5}}{K}$

& ϒ be the cubical expansion $=3\alpha =3\times 1.7 \times 10^{-5}$

$= \frac{5.1 \times 10^{-5}}{K}$

Let V be the volume of the cavity and $\Delta V$ be the increase in its volume which is a result of increase in temperature by $\Delta T$ .

$\Delta V = \gamma V. \Delta T$

$\frac{\Delta V}{V} = \gamma \Delta T$

Now, we know that,

Thermal stress production = B x Volumetric strain

$=B.\frac{\Delta V}{V}$

$=B.\gamma \Delta T$

$= 140 \times 10^9 \times 5.1 \times 10^{-5} \times 20$

$= 1.428 \times 10^{8} Nm^{-2}$

Thus, the stress is $1.01 \times 10^{5} Nm^{-2}$ , viz., $10^{3}$ times of atmospheric pressure.

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