Question

Prove that:

$\\ \tt{\rightarrow{\beta = 2\alpha}}$
$\tt{\rightarrow{\gamma = 3\alpha}}$


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Answer 1

Prove that:—

→ β = 2α

→ γ = 3α

Sølütìôñ:—

Denotes 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 = 3α

Answer 2

Answer:

Hey Mate!,

Explanation:

Refer the Attachment then You'll get the answer..

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