Question

prove alpha=1/3 beta​

Answer 1

• β = 2α . V → V ( 1 + γ ΔT) from the definition of volumetric expansion coefficient. V → V ( 1 + α ΔT)³ ≈ V ( 1 + 3 α ΔT) , neglecting higher powers of α ΔT.
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Answer 2

According to the Question we have to Prove that, α= ⅓ β

We know that,

β is the coefficient of volumetric expansion and α is the coefficient of linear expansion.

Let us now consider a cube with dimensions l,b,h as the length breadth and height respectively.

Now, let assume that V is the Volume.

and t is the initial temperature.

Next let's change the temperature t to T.

Let the dimensions before expansion be, l,b,h,v

and the dimensions after expansion will be: L,B,H,V.

[Where - L/l= length; b/B= breadth; h/H = Height and v/V= Volume]

From linear expression theory we know that,

L= l( 1+α∆T)

B= b(1+α∆T) and

H= h(1+α∆T).

V= L*B*H _______(1)

Using the values of L,B,H we find,

V= l( 1+α∆T) * b(1+α∆T) * h(1+α∆T)

Now the volume of the cube before expansion is given by v= l*b*h ________(2)

By using this, V= v(1+α∆T)³

so, from equation 2 we can get

V= v( 1+3α△T)

or, V= v + v* 3α△T

or, V-v= v*3α△T

so we consider (V-v) as ∆V

so, ∆V = v*3α△T

or, ∆V/v△T= 3α

or, β= 3α

or, α= ⅓ β

Hence, α= ⅓ β [Proved]

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