Two steel rods and an aluminium rod of equal length `l_0` and equal cross- section are joined rigidly at their ends as shown in the figure below. All the rods are in a state of zero tension at `0^@ C`. Find the length of the system when the temperature is raised to `theta`. Coefficient of linear expansion of aluminium and steel are `alpha_(a)` and `alpha_(s)` respectively. Young's modulus of aluminium is `Y_(a)` and of steel is `Y_(s)`.

Explanation

At `theta_1`
At `theta_2`
If rods are free to expand,
`l_S=l_0(1+alpha_Stheta)`
`l_(alpha)=l_0(1+alpha_atheta)`
`Y=(F//A)/(x//l)`
`x=(Fxxl)/(AY)` if `alpha_s gt alpha_a`
`(Fxxl_0)/(AY_S)=l_0(1+alpha_S Delta theta)-1` ..(i)
`(ZFxxl_0)/(AY_a)=l-l_0(alpha+alpha_aDeltatheta)`..(ii)
By soving (i) and (ii) we get
`l=l_0[1+(alpha_aY_a+Zalpha_SY_S)/(Y_a+ZY_S)]`