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 increased temperature, let `Deltal_(1)andDeltal_(2)` be the increase in length of aluminium and steel respectively (if they are free).
Then `Deltal_(1)=l_(0)alpha_(1)thetaandDeltal_(2)=l_(0)alpha_(2)theta`
Suppose `Deltal_(1)ltDeltal_(2)`
Therefore, the composite rod will increase in between `Deltal_(1)andDeltal_(2)`. Say it is `Deltal`, where `Deltal_(1)ltDeltalltDeltal_(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,


`thereforeY_(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))`