Question

A solid rectangular sheet has two different coefficients of linear expansion ${\alpha }_1$ and ${\alpha }_2$ along its length and breadth respectively. The coefficient of surface expansion is (for ${\alpha }_{1}t<<1,{\alpha }_{2}t<<1$)

• A. $\frac{{\alpha }_1+{\alpha }_2}{2}$
• B. $2\left({\alpha }_1+{\alpha }_2\right)$
• C. $\frac{4{\alpha }_1{\alpha }_2}{{\alpha }_1+{\alpha }_2}$
• D. ${\alpha }_1+{\alpha }_2$

Answer 1

Answer: (D)

The coefficient of linear expansion along its length $={\alpha }_1$

The coefficient of linear expansion along its breadth $={\alpha }_2$
Increase in length,
$L_t=l_0\left(1+{\alpha }_1\Delta t\right)$
Increase in breadth,
$B_t=b_0\left(1+{\alpha }_2\Delta t_2\right)$
Let coefficient of surface expansion is $\beta$
Area = length $\times$ breadth
$=l_0\left(1+{\alpha }_1\Delta t\right)\times b_0\left(1+{\alpha }_2\Delta t\right)$
$=l_0{b}_0\left(1+{\alpha }_1\Delta t\right)\left(1+{\alpha }_2\Delta t\right)$
$=S_0\left(1+{\alpha }_1\Delta t+{\alpha }_2\Delta t+\dots \right)$
where, $S_0=l_0\cdot b_0$
$=$ Initial area of surface
In state of expansion,
$S_t=L_t\times B_t$
$=l_0{b}_0\left(1+{\alpha }_1\Delta t\right)\left(1+{\alpha }_2\Delta t\right)$
$=S_0\left(1+{\alpha }_1\Delta t+{\alpha }_2\Delta t+\dots \right)$
$S_t=S_0(1+\beta \Delta t)$
$\therefore S_0(1+\beta \Delta t)=S_0\left(1+{\alpha }_1\Delta t+{\alpha }_2\Delta t+\dots \right)$
$\beta \cdot \Delta t={\alpha }_1\Delta t+{\alpha }_2\Delta t$
$\beta ={\alpha }_1+{\alpha }_2$