Question

define an expression for Linear expression for a metal rod due to heating hence define the coefficient of linear expression​

Answer 1

Consider a uniform metal rod of length L at a temperature T. Let the rod expand to a length (L + ∆L) by increasing the temperature from T to (T + ∆T). Then the change in length is ∆L and that in temperature is ∆T.

Here we can say that the change in temperature is directly proportional to the linear strain, or ratio of change in length to the original length, but not directly on change in length, i.e.,

$\dfrac {\Delta L}{L}\propto\Delta T$

For changing the proportionality to equality, we use a constant,

$\dfrac {\Delta L}{L}=\alpha\Delta T$

This proportionality constant $\alpha$ is called coefficient of linear expansion. Then,

$\dfrac {(L+\Delta L)-L}{L}=\alpha\Delta T\\\\\\(L+\Delta L)-L=L\alpha\Delta T\\\\\\L+\Delta L=L+L\alpha\Delta T\\\\\\L+\Delta L=L(1+\alpha\Delta T)$

Here $L+\Delta L$ is the new length of the rod. Let it be $L',$ then,

$\boxed {L'=L(1+\alpha\Delta T)}$

Hence this is the expansion for linear expansion.