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.
Similar Questions
In figure which strip brass or steel have higher coefficient of linear expansion
In figure which strip brass or steel have higher coefficient of linear expansion
A beaker is filled with water at $4\,^oC$. At one time the temperature is increased by few degrees above $4\,^oC$ and at another time it is decreased by a few degrees below $4\,^oC$. One shall observe that:
A beaker is filled with water at $4\,^oC$. At one time the temperature is increased by few degrees above $4\,^oC$ and at another time it is decreased by a few degrees below $4\,^oC$. One shall observe that:
On heating a uniform metallic cylinder length increases by $3 \%$. The area of cross-section of its base will increase by ........... $\%$
On heating a uniform metallic cylinder length increases by $3 \%$. The area of cross-section of its base will increase by ........... $\%$
A glass flask contains some mercury at room temperature. It is found that at different temperatures the volume of air inside the flask remains the same. If the volume of mercury in the flask is $300 \,\,cm^3$, then volume of the flask is ........ $cm^3$. (given that coefficient of volume expansion of mercury and coefficient of linear expansion of glass are $1.8 × 10^{-4} (^o C)^{-1}$ and $9 × 10^{-6} (^o C)^{-1}$ respectively)
A glass flask contains some mercury at room temperature. It is found that at different temperatures the volume of air inside the flask remains the same. If the volume of mercury in the flask is $300 \,\,cm^3$, then volume of the flask is ........ $cm^3$. (given that coefficient of volume expansion of mercury and coefficient of linear expansion of glass are $1.8 × 10^{-4} (^o C)^{-1}$ and $9 × 10^{-6} (^o C)^{-1}$ respectively)
The diagram below shows the change in the length $X$ of a thin uniform wire caused by the application of stress $F$ at two different temperatures $T_1$ and $T_2$. The variation shown suggests that
The diagram below shows the change in the length $X$ of a thin uniform wire caused by the application of stress $F$ at two different temperatures $T_1$ and $T_2$. The variation shown suggests that