Two rods of equal cross-sections, one of copper and the other of steel, are joined to form a composite rod of length `2.0m`. At `20^(@)C` the length of the copper rod is `0.5m`. When the temperature is raised to `120^(@)C` the length of the composite rod increases to `2.002m`. If the composite rod is fixed between two rigid walls and is, thus, not allowed to expand, it is found that the lengths of the two component rods also do not change with the increase in temperature. Calculate the Young's modulus and the coefficient of linear expansion of steel. Given `Y_(Cu)=1.3xx10^(13)N//m^(2)` and `alpha_(Cu)=1.6xx10^(-5)//^(@)C`

Explanation

(a) As with increase in temperature due to thermal expansion the length of a rod changes,
i.e., `L.=L(1+alphaDeltatheta)`
So, for composite rod
`L_(S).+L_(C ).=L_(C )(1+alpha_(C )Deltatheta)+L_(S)(1+alpha_(S)Deltatheta)`
or `L_(S).+L_(C ).=(L_(S)+L_(C ))+(L_(S)alpha_(S)+L_(C )alpha_(C ))Deltatheta`
According to given problem,
`2.002=2+[0.5xx1.6xx10^(-5)+1.5alpha_(S)]xx100`
or `1.5alpha_(S)=2xx10^(-5)-0.8xx10^(-5)`, i.e.,`alpha_(S)=0.8xx10^(-5)//^(@)C`
(b) When the rods are fixed between the walls and its length remains unchanged,
`(DeltaL_(S)+DeltaL_(C ))_(H)=(DeltaL_(S)+DeltaL_(C ))_(C )`
However, as here length of individual rods also remains unchanged,
`(DeltaL_(S))=(DeltaL_(S))_(C )` and `(DeltaL_(C ))_(H)=(DeltaL_(C ))_(C )`
Now as `(DeltaL)_(H)=LalphaDeltatheta` and `(DeltaL)_(C )=(FL//AY)`
So, `L_(S)alpha_(S)Deltatheta=(FL_(S)//AY_(S))` amd `L_(C )alpha_(C )Deltatheta=(FL_(C )//AY_(C ))`
Dividing one by the other `((alpha_(C ))/(alpha_(S)))=((Y_(S))/(Y_(C )))`
So, `Y_(S)=(alpha_(C ))/(alpha_(S))xxY_(C )=(1.6xx10^(-5))/(0.8xx10^(-5))xx1.3xx10^(13)=2.6xx10^(13)N//m^(2)`