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A blacksmith fixes iron ring on the rim ...

A blacksmith fixes iron ring on the rim of the wooden wheel of a bullock cart. The diameter of the rim and the iron ring are 5.243m and 5.231 m respectively at `27" "^(@)C`. To what temperature should the ring be heated so as to fit the rim of the wheel ? `(alpha_(1)=1.20xx10^(-5)K^(-1))`

Text Solution

Verified by Experts

`t_(1)=27^(@)CrArrT_(1)=27+273=300K`
`D_(T1)=5.231m`
`D_(T2)=5.243m`
`L_(T1)=piD_(T_(1))` and `L_(T2)=piD_(T_(2))" "["circumference" L_(T2)=L_(T1)[1+alpha_(l)(T_(2)-T_(1))]" "=2pir=piD]`
`piD_(T2)=piD_(T1)[1+alpha_(l)(T_(2)-T_(1))]`
`:.(D_(T2))/(D_(T1))=1+alpha_(l)(T_(2)-T_(1))`
`:.(5.243)/(5.231)-=alpha_(l)(T_(2)-300)`
`:.1.002294-1=1.20xx10^(-5)(T_(2)-300)`
`:.(0.002294)/(1.2xx10^(-5))=T_(2)-300`
`:.(2294)/(12)+300=T_(2)`
`:.191.16+300=T_(2)`
`:.491.16=T_(2)`
`:.t_(2)=T_(2)-273.15`
`=491.16-273.15`
`:.t_(2)=218.01^(@)C`.
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