A composite bar of length `L = L_(1) + L_(2)` is made up from a rod of material `1` and of length `L_(1)` attached to a rod of material `2` and of length `L_(2)` as shown. If `alpha_(1)` and `alpha_(2)` are their respective coefficient of linear expansion, then equivalent coefficient of linear expansion for the composite rod is

Options

A. `(alpha_(1) L_(2) + alpha_(2) L_(1))/L`
B. `(alpha_(1) L_(2) + alpha_(2) L_(2))/L`
C. `(alpha_(1) L_(1) + alpha_(2) L_(2))/L`
D. `(alpha_(1) alpha_(2) (L_(1)^(2) + L_(2)))/((alpha_(1)L_(1) + alpha_(2) L_(2)))`

Answer

C

Explanation

When it has been given a temperature change `Delta T`, the new length of teh composite bar is teh sum of the individual lengths (Let `'alpha'` be requried coeffcient of linear expansion)
`L'=L_(1)^(')+L_(2)^(')`
`L(1+alpha Delta T) =L_(1)(1+alpha_(1)Delta T)+L_(2)(1+alpha_(2)Delta T)`
`rArr L + Lalpha Delta T=L_(1)+L_(1)alpha_(1) Delta T+L_(2)+L_(2)alpha_(2) Delta T`
`rArr L + Lalpha Delta T=L+L_(1)alpha_(1) Delta T+L_(2)alpha_(2) Delta T`
` ( :' L_(1)+L_(2)=L)`
`implies Lalpha Delta T = L_(1)alpha_(1) Delta T+L_(2)alpha_(2) Delta T`
`implies alpha=(L_(1)alpha_(1)+L_(2)alpha_(2))/(L)`.