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Question
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 α = 1.7 × 10–5/°C, bulk modulus for copper = 140 × 109 N/m2.
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Solution
According to the problem, decrease in temperature
(Δt) = 57 – 37 = 20°C
Coefficient of linear expansion
(α) = 1.7 × 10–5/°C
Bulk modulus for copper (B) = 140 × 109 N/m2
Coefficient of cubical expansion
(γ) = 3α = 5.1 × 10–5/°C
Let the initial volume of the cavity be V and its volume increases by ΔV due to increasing in temperature.
∴ `ΔV = γVΔt`
⇒ `(ΔV)/V = γΔt`
We know, B = `"Stress"/"Volume strain"`
∴ Thermal stress = `B xx ((ΔV)/V) = B(γΔT)`
= `B(3αΔT)` .....(∵ γ = 3α)
= 140 × 109 × 3 × 1.7 × 10–5 × 20
= 1.428 × 108 Nm–2
This is about 103 times of atmospheric pressure.
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