A hole is drilled in a copper sheet. The diameter of the hole is `4.24 cm` at `27.0^(@)C`. What is the change in the diameter of the hole when the sheet is heated to `227^(0)C`? `alpha` for copper `= 1.70 xx 10^(-5)K^(-1)`

Explanation

Initial temperature, `T_(1) = 27.0^(@)C`
Diameter of the hole at `T_(1), d_(1) = 4.24 cm`
Final temperature, `T_(2) = 227^(@)C`
Diameter of the hole at `T_(2) = d_(2)`
Co-efficient of linear expansion of copper, `α(Cu)= 1.70 × 10^(–5) K^(–1)`
For co-efficient of superficial expansion β,and change in temperature ΔT, we have the relation:
`("Change in area "(DeltaA))/("Original area"(A))=betaDeltaT`
`((pi(d_(2)^(2))/(4)-pi(d_(1)^(2))/(4)))/((pi(d_(1)^(2))/(4)))=(DeltaA)/A`
`:.(DeltaA)/A=(d_(2)^(2)-d_(1)^(2))/d_(1)^(2)`
But `beta=2a`
`:.(d_(2)^(2)-d_(1)^(2))/(d_(1)^(2))=2aDeltaT`
`(d_(2)^(2))/(d_(1)^(2))-1=2a(T_(2)-T_(1))`
`(d_(2)^(2))/((4.24)^(2))=2xx1.7xx10^(-5)(227-27)+1`
`d_(2)^(2)=17.98xx1.0068=18.1 `
`:. d_(2)=4.2544 cm`
Change in diameter = `d_(2) – d_(1)` = 4.2544 – 4.24 = 0.0144 cm
Hence, the diameter increases by `1.44 × 10^(–2) cm`.