Question

show that beta=2x whaere beta α are coefficient of aeal expantion and linaer expantion rapected​

Answer 1

Correct Question:-

Show that $\sf{\beta=2\alpha}$ where $\sf{\alpha}$ and $\sf{\beta}$ are coefficients of linear and areal expansions respectively.

Solution:-

Let the area $\sf{A=lb}$ of a rectangular sheet of length l and breadth b be thermally expanded to $\sf{A'=l'b'}$ by changing length to l' and breadth to b'.

The thermal expansion of length is given by,

$\longrightarrow\sf{l'=l(1+\alpha\,\Delta T)\quad\quad\dots(1)}$

The thermal expansion of breadth is given by,

$\longrightarrow\sf{b'=b(1+\alpha\,\Delta T)\quad\quad\dots(2)}$

The thermal expansion of area is given by,

$\longrightarrow\sf{A'=A(1+\beta\,\Delta T)\quad\quad\dots(3)}$

where $\sf{\Delta T}$ is the change in temperature.

Multiplying (1) and (2),

$\longrightarrow\sf{l'b'=lb(1+\alpha\,\Delta T)^2}$

$\longrightarrow\sf{A'=A\left(1+2\alpha\,\Delta T+(\alpha\,\Delta T)^2\right)}$

But $\sf{\alpha}$ is very small in value and $\sf{\Delta T}$ can be small too, so that $\sf{(\alpha\,\Delta T)^2}$ can be neglected.

Hence,

$\longrightarrow\sf{A'=A\left(1+2\alpha\,\Delta T\right)\quad\quad\dots(4)}$

On comparing (3) and (4), we get,

$\longrightarrow\sf{\underline{\underline{\beta=2\alpha}}}$

Hence the Proof!