Two steel rods and an aluminium rod of equal length `l_(0)` and equal corss-section are joined rigidly at at their ends as shown in the figure. All the rods are in 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 `Deltal_(1) and Deltal_(2)` be the increase in length of aluminium and steel respectively (if they are free).
then `Delta l_(1)=l_(0) alpha_(1) theta and Deltal_(2)=l_(0) alpha_(2)theta`
Suppose `Delta l_(1) lt Deltal_(2)`
Therefore, the composite rod will increase in between `Deltal_(1) and Deltal_(2)`. Say it is `Deltal`, where `Deltal_(1) lt Deltal lt Deltal_(2)` Due to this, aluminium rod has a length `(Deltal-Deltal_(1))` more than its natural length at temperature `theta` and steel rod (s) will have a length `(Deltal_(2)-Deltal)` less than its natural length at temperature `theta`. 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((Deltal-Deltal_(1))/(l_(0)))=2Y_(2)A((Deltal_(2)-Deltal)/(l_(0)))`
Solving this we get, `Deltal=(Y_(1)Deltal_(1)+2Y_(2)Deltal_(2))/(Y_(1)+2Y_(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_(1)+2Y_(2)alpha_(2)))/(Y_(1)+2Y_(2))`