Question

show that coefficient of superficial expansion is twice of coefficient of linear expansion. Can any 1 solve this problem problem

Answer 1

here u go I hope this will help u :)

Answer 2

Relationship between coefficient of superficial expansion and coefficient of linear expansion:

• Let

              $l$ = length of the surface

              $b$ = breadth of the surface

             Δ$l$ = change in length

             Δ$b$ = change in breadth

             ΔT = change in temperature

• Coefficient of linear expansion is given by

            $\alpha =$ Δ$l$/$l$ΔT

            $\alpha$$l$ΔT = Δ$l$

        ⇒ $l$ + Δ$l$ = $l+\alpha l$ΔT

       ⇒  $l$ + Δ$l$ = $l$(1 + $\alpha$ΔT) -----------------(1)

• Similarly

        ⇒ $b$ + Δ$b$ = $b(1+\alpha$ΔT)  --------------(2)

• Now, coefficient of areal expansion is given by

            $\beta$ = ΔA/AΔT          ------------------(3)

           A + ΔA =  $l$(1 + $\alpha$ΔT)$b(1+\alpha$ΔT)

           A + ΔA = $lb$$(1+\alpha$ΔT)^2

           A + ΔA = A(1 + 2$\alpha$ΔT)                  [∵ $\alpha$ is very small]

           A + ΔA = A + 2A$\alpha$ΔT

            ΔA = 2A$\alpha$ΔT      ---------------------(4)

• Substituting (4) in (3), we get

             $\beta$ = 2A$\alpha$ΔT/AΔT

             ∴ $\beta =2\alpha$

• Hence, the coefficient of superficial expansion is twice the coefficient of linear expansion.