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Calculate the stress developed inside a tooth cavity filled with copper when hot tea at temperature of `57^(@)C` is drunk. You can take body (tooth) temperature to be `37^(@)C` and `alpha=1.7xx10^(-5)//""^(@)C`, bulk modulus for copper `=140xx10^(9)N//m^(2)`.

Text Solution

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Increase in temperature
`DeltaT=57-37=20^(@)C` or 20 K
Coefficient of linear expansion of cavity,
`alpha=1.7xx10^(-5)" "^(@)C^(-1)`
Bulk modulus of cavity `K=140xx10^(9)" N/m"^(2)`
Coefficient volume expansion of copper,
`gamma=3alpha`
`=3xx1.7xx10^(-5)`
`=5.1xx10^(-5)" "^(@)C^(-1)`
Suppose the original volume is V and the increase in volume is `DeltaV` due to increase in temperature `DeltaT`.
We know that,
`DeltaV=gammaVDeltaT`
`:.(DeltaV)/(V)=gammaDeltaT`
Thermal stress = Bulk modulus `xx` volume strain
`=Kxx(DeltaV)/(V)`
`=KxxgammaDeltaT`
`=140xx10^(9)xx5.1xx10^(-5)xx20`
`=1.428xx10^(8)" N/m"^(2)`
Atmospheric pressure is `P=1.01xx10^(5)" N/m"^(2)`
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