Question

Calculate the stress developed inside a tooth cavity filled with copper when hot tea at temperature of $57{}^{\circ }\text{C}$ is drunk. You can take body (tooth) temperature to be $37{}^{\circ }\text{C}$. Coefficient of linear expansion $\alpha =1.7\times \left({10}^{-5}\right){}^{\circ }{\text{C}}^{-1}$ and bulk modulus for copper $=140\times {10}^9{\text{N/m}}^2$.

• A. $5\times {10}^{10}{\text{N/m}}^2$
• B. $6\times {10}^5{\text{N/m}}^2$
• C. $1.5\times {10}^8{\text{N/m}}^2$
• D. $15\times {10}^{10}{\text{N/m}}^2$

Answer 1

The correct option is C $1.5\times {10}^8{\text{N/m}}^2$

Coefficient of linear expansion $\alpha =1.7\times {10}^{-5}{}^{\circ }{\text{C}}^{-1}$

So, coefficient of cubical expansion $\nu =3\alpha =5.1\times {10}^{-5}{}^{\circ }{\text{C}}^{-1}$

Change in temperature $=57-37=20{}^{\circ }\text{C}$

Let the volume is $V$ then change in volume

$\Delta V=V\nu \Delta T$

and $\frac{\Delta V}{V}=\nu \Delta T$

$\text{Thermal stress produced = Bulk modulus}\times \text{Volumetric strain}$

$\therefore \text{Thermal stress}=\beta \times \frac{\Delta V}{V}$

$=\beta \times \nu \times \Delta T$
$=140\times {10}^9\times (5.1\times {10}^{-5}\times 20)$

$=1.428\times {10}^8$

$\Rightarrow \text{Thermal stress}=1.5\times {10}^5{\text{N/m}}^2$

Hence, option $\left(\text{C}\right)$ is correct.