Question

define coefficient of linear expansion and coefficient of volume expansion. how are they related ​

Answer 1

$\small\bold{Relation\: between\: Alpha\: and\: Gamma}$

We have , for the cube

New volume , V' = V(1 + gamma△T)

(L')³ = (L)³(1 + gamma△T)

we can also write (L')³ as (L)³(1 + alpha△T)³,

(L)³(1 + alpha△T)³ = (L³)(1 + gamma△T)

(L)³ will be cancelled out,

then the eqn we will get,

(1 + gamma△T) = (1 + alpha△T)³

.•. (1 + alpha△T)³ => 1 + 3alpha△T + 3(alpha)²(△T)² + alpha³(△T)³

So by following this,

Since alpha is very small quantity., the terms alpha³(△T)³ and 3(alpha)²(△T)² will be negligibly small , as compared to alpha△T, so that

1 + gamma△T = 1 + 3alpha△T

then,

gamma = 3alpha

alpha = gamma/3

Hope it helps you out♡♡