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)`.

Answer

A, B

Explanation

Let `Delta l` be the change in length. `("let" Delta l_s gt Delta l gt Delta l_a)`
Strain in steel = `(Delta l_s - Delta l_a)/(l _0)`
and strain in aluminium= `(Delta l - Delta l_a)/(l_0)`
In equilibrium, `2 F_s = F_a`
`2[(Delta l_s - Delta l)/(l_0)] Y_sA = [(Delta l - Delta l_a)/(l_0)]Y_a A`
or `2(l_0 alpha_s theta - Delta l) Y_s = (Delta l - l_0 prop_a theta) Ya`
Solving the equation, we get
`Delta l = ((2 alpha_s Y_s + alpha _a Y_a)/(2 Ys + Y_a))l_0 theta`
`:.` Total length
= `l_0 + Delta l = l_0[1 + ((alpha_aY_a +2 alpha_s Y_s)/(2Y_s + Y_a))theta]`.