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 ${37}^{\circ }\text{C}$ and ${\alpha }_{Cu}=1.7\times {10}^{-5}{/}^{\circ }\text{C}$, bulk modulus for copper $K_{Cu}=140\times {10}^9{\text{N/m}}^2$

• A. $1.428\times {10}^8{\text{N/m}}^2$
• B. $42\times {10}^6{\text{N/m}}^2$
• C. $36\times {10}^7{\text{N/m}}^2$
• D. $4.28\times {10}^6{\text{N/m}}^2$

Answer 1

The correct option is A $1.428\times {10}^8{\text{N/m}}^2$

Given, increase in temperature,
$\Delta t=57-37={20}^{\circ }\text{C}$
Coefficient of volume expansion,
${\gamma }_{Cu}=3{\alpha }_{Cu}=5.1\times {10}^{-5}{/}^{\circ }\text{C}$
Let initial volume of the capacity be $V$. Also, let its volume increase by $\Delta V$ due to increase in temperature.
$\therefore \Delta V=\gamma V\Delta t$
$\Rightarrow \frac{\Delta V}{V}=\gamma \Delta t$
Thermal stress produced $=K\times$ Volumetric strain
$=K\times \frac{\Delta V}{V}=K\times \gamma \Delta t$
$=140\times {10}^9\times (5.1\times {10}^{-5}\times 20)=1.428\times {10}^8{\text{N/m}}^2$