Question

Two metal rods of lengths `L_(1)` and `L_(2)` and coefficients of linear expansion `alpha_(1)` and `alpha_(2)` respectively are welded together to make a composite rod of length `(L_(1)+L_(2))` at `0^(@)C.` Find the effective coefficient of linear expansion of the composite rod.
A. `(L_(1)alpha_(1)^(2)-L_(2)alpha_(2)^(2))/(L_(1)^(2)+L_(2)^(2))`
B. `(L_(1)^(2)alpha_(1)-L_(2)^(2)alpha_(2))/(L_(1)^(2)+L_(2)^(2))`
C. `(L_(1)alpha_(1)+L_(2)alpha_(2))/(L_(1)-L_(2))`
D. `(L_(1)alpha_(1)+L_(2)alpha_(2))/(L_(1)+L_(2))`

Answer 1

Correct Answer - D
Given, the length of rods are `L_(1) and L_(2)` Coefficient of linear expansions are `alpha_(1) and alpha_(2)` As we know that
`DeltaL_(1)=alpha_(1)L_(1)DeltatrArrDeltaL_(2)=alpha_(2)L_(2)Deltat`
and for combined rod
`DeltaL=alpha(L_(1)+L_(2))Deltat`
Now, `DeltaL=DeltaL_(1)+DeltaL_(2)`
`rArr" "alpha(L_(1)+L_(2))Deltat=alpha_(1)L_(1)Deltat+alpha_(2)L_(2)Deltat`
`rArr" "alpha(L_(1)+L_(2))Deltat=alpha_(1)L_(1)Deltat+alpha_(1)L_(2)`
`rArralpha=(alpha_(1)L_(1)+alpha_(2)L_(2))/(L_(1)+L_(2))`