# Two uniform bars of length L_1 and L_2 with the coefficient of linear expansion \alpha_1 and...

## Question:

Two uniform bars of length {eq}L_1 {/eq} and {eq}L_2 {/eq} with the coefficient of linear expansion {eq}\alpha_1 {/eq} and {eq}\alpha_2 {/eq} are joined to form a longer rod of length {eq}L_1 + L_2 {/eq}. Determine the effective coefficient of linear expansion of the composite bar.

## Conduction:

The conduction heat transfer occurs when two bodies are in direct physical contact with each other. Generally, this process occurs in the solids particles due to the collision of the molecules. This process does not follow the law of reflection.

## Answer and Explanation:

Given data:

• Length of first bar, {eq}{L_1} {/eq}
• Length of second bar, {eq}{L_2} {/eq}
• Thermal coefficient of expansion of first bar, {eq}{\alpha _1} {/eq}
• Thermal coefficient of expansion for second bar, {eq}{\alpha _2} {/eq}

The expression for the linear expansion for the first bar can be written as,

{eq}{L'_1} = {L_1}\left( {1 + {\alpha _1}\Delta T} \right) {/eq}

The expression for the linear expansion for the second bar can be written as,

{eq}{L'_2} = {L_2}\left( {1 + {\alpha _2}\Delta T} \right) {/eq}

The expression for the linear expansion for the composite bar can be written as,

{eq}{L_f} = {L'_1} + {L'_2} = \left( {{L_1} + {L_2}} \right)\left[ {1 + \alpha \Delta T} \right] {/eq}

Therefore,

{eq}\begin{align*} \left( {{L_1} + {L_2}} \right)\left[ {1 + \alpha \Delta T} \right] &= {L_1}\left( {1 + {\alpha _1}\Delta T} \right) + {L_2}\left( {1 + {\alpha _2}\Delta T} \right)\\ \left( {{L_1} + {L_2}} \right) + \left[ {\left( {{L_1} + {L_2}} \right)\alpha \Delta T} \right] &= {L_1} + {L_1}{\alpha _1}\Delta T + {L_2} + {L_2}{\alpha _2}\Delta T\\ \left( {{L_1} + {L_2}} \right) + \left[ {\left( {{L_1} + {L_2}} \right)\alpha \Delta T} \right] &= \left( {{L_1} + {L_2}} \right) + \left( {{L_1}{\alpha _1} + {L_2}{\alpha _2}} \right)\Delta T\\ \left[ {\left( {{L_1} + {L_2}} \right)\alpha \Delta T} \right] &= \left( {{L_1}{\alpha _1} + {L_2}{\alpha _2}} \right)\Delta T \end{align*} {/eq}

Solve further as,

{eq}\alpha = \dfrac{{\left( {{L_1}{\alpha _1} + {L_2}{\alpha _2}} \right)}}{{\left( {{L_1} + {L_2}} \right)}} {/eq}

Therefore, the effective coefficient of linear expansion of the composite bar is {eq}\dfrac{{\left( {{L_1}{\alpha _1} + {L_2}{\alpha _2}} \right)}}{{\left( {{L_1} + {L_2}} \right)}} {/eq} .