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What is the thermal stress developed ins...

What is the thermal 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` and bulk modulus for copper `= 14 xx 10^(10) N//m^(2)`.

A

`1.3 xx 10^(7) N//m^(2)`

B

`1.4 xx 10^(7) N//m^(2)`

C

`1.9 xx 10^(7) N//m^(2)`

D

`1.1 xx 10^(7) N//m^(2)`

Text Solution

Verified by Experts

The correct Answer is:
B
Let V be the initial volume of the cavity.
When the temperature increases by `(Delta theta)`, its volume increases by `Delta V`.
`because gamma = (Delta V)/(V(Delta theta)) :. (Delta V)/(V) = gamma (Delta theta)`
where `gamma` = coefficient of volume expansion of copper `= 3 alpha (Delta V)/(V)` is known as the volume strain.
Bulk Modulus `(B) = ("Stress")/("Volume Strain")`
`:.` Thermal stress `= B xx "Volume strain"`
`= B((Delta V)/(V)) = B gamma (Delta theta)`
`gamma = 3 alpha = 3 xx 1.7 xx 10^(-5) = 5.1 xx 10^(-5)`
and `Delta theta = 57 - 37 = 20^(@)C`
`:.` Thermal stress `= 14 xx 10^(10) xx 5.1 xx 10^(-5) xx 20`
`= 1.428 xx 10^(7) N//m^(2) div 1.4 xx 10^(7) N//m^(2)`
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