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

Explanation

`Deltal=l_Salpha_SDeltaT+l_calpha_cDeltaT`
`0.002=[1.5alpha_S+0.5xx1.6xx10^-5]xx100`
`alpha_s=(1.2xx10^-5)/(1.5)=8xx10^(-6^@) C`
there is no change in component length
For steel,
`x=l_salpha_sDeltaT-(Fl_S)/(AY_S)=0`
`(F)/(AY_S)=alpha_SDeltaT`
For copper,
`x=(Fl_S)/(Ay_c)-l_calpha_cDeltaT=0`
`(F)/(Ay_c)=alphaDeltaT` .(ii)
On dividing Eq. (i) by Eq. (ii), we get
`(y_S)/(y_c)=(alpha_c)/(alpha_S)`
`y_S=y_cxx(alpha_c)/(alpha_S)=(1.5xx10^(13)xx1.6xx10^-5)/(8xx10^-6)`
`y_S=3xx10^(13) N//m^(2)`