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Calculate the stress developed inside a ...

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_(Cu) = 1.7 xx 10^(-5)//^(@)C` bulk modulus for copper `B_(Cu) = 140 xx 10^(9) N//m^(2)`.

Text Solution

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Given, decrease in temperature `(Delta t) = 57 - 37 = 20^(@)C`
Coefficient of linear expansion `(alpha) = 1.7 xx 10^(-5)//^(@)C`
Bulk modulus for copper (B) `= 140 xx 10^(9) N//m^(2)`
Coefficient of cubical expansion `(gamma) = 3 alpha = 5.1 xx 10^(-5)//^(@)C`
Let initial volume of the cavity be V and its volume increases by `Delta V` due to increase in temperature.
`:. Delta V = gamma V Delta t`
`rArr (Delta V)/(V) = gamma Delta t`
Thermal stress produced `= B xx` Volumetric strain
`= B xx (Delta V)/(V) = B xx gamma Delta t`
`= 140 xx 10^(9) xx (5.1 xx 10^(-5) xx 20)`
`= 1, 428 xx 10^(8)N//m^(2)`
This is about `10^(3)` times of atmospheric pressure.
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