Question

show that bita = 2 alpha?​

Answer 1

Answer:

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Explanation:

Denote by

α : the coefficient of linear expansion

β : the coefficient of surface expansion

γ : the coefficient of volumetric expansion

Then a length increases as

L → L ( 1 + α ΔT)

But this means that for isotropic (same in every direction) expansion a surface (length x length) increases as

A → A ( 1 + α ΔT)( 1 + α ΔT) ≈ A (1 +2 α ΔT)

where we have neglected the (usually very small) square term (α ΔT)² .

Comparing with the (definition of β) expression

A → ( 1 + βΔT) , we see the relation

β = 2α .

Likewise

V → V ( 1 + γ ΔT) from the definition of volumetric expansion coefficient.

But also we can approximate (volume = length x length x length)

V → V ( 1 + α ΔT)³ ≈ V ( 1 + 3 α ΔT) , neglecting higher powers of α ΔT.

Hence

γ = 3 α