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. 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_(1) and alpha_(2)` respectively. Young's modulus of aluminium is `Y_(1) `and of steel is `Y_(2)` .

Explanation

At increased temperature, let `Delta l_(1) and Delta l_(2)` be the increase in length of aluminium and steel respectively (if they are free).
Then `Delta l_(1) = l_(0) alpha theta and " " Delta l_(2) = l_(0) alpha_(2) theta`
suppose `Delta l_(1) lt Delta l_(2)`
Therefore, the composite rod will increase in between `Delta l_(1) and Delta l_(2)` . Say it is `Delta l` , where `Delta l_(1) lt Delta l lt Delta l_(2)`
Due to this, aluminium rod has a length `(Delta l - Delta l_(1))` more than its natural length al temperature e and steel rod (s) will have a length `(Delta l_(2) - Delta l)` less than its natural length at temperature e . Due to this, steel rods will exert force `F_(2)` on aluminium rod from two sides, which in equilibrium be balanced by internal restoring force `F_(1)`. Thus,

`therefore Y_(1) A ((Delta l - Delta l_(1))/(l_(0))) = 2 Y_(2) A ((Delta l_(2) - Delta l)/(l_(0)))`
Solving this we get ,`Delta l = (Y_(1) Delta l_(1) + 2 Y_(2) Delta l_(2))/( Y_(1) + 2 Y_(2)) `
= `(Y_(1) l_(0) alpha_(1) theta + 2Y_(2) l_(0) alpha_(2) theta )/(Y_(1) + 2Y_(2)) = (l_(0) theta (Y_(1) alpha + 2Y_(2) alpha_(2)))/( Y_(1) + 2Y_(2))`