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Show that β ≈ 3α, by calculating the change in volume ΔV of a cube with sides of length L.

Short Answer

Expert verified

The formulaβ=3α is proved below.

Step by step solution

01

Introduction

We calculate the change in length by the formula for linear expansion in solids and find new length and we further calculate change in volume and find new volume.

02

formula for linear expansion and volume expansion

Formula for length expansion of solids L=αLT

Formula for volume expansion of solids V=βVT

Here,α,βare the coefficient of linear and volume expansion respectively, L, V is original length and volume respectively, and Tis the change in temperature.

New length = L+L

New volume = V+V

But for cube,

Volume = (Length)3

03

Equate new volume

(L+L)3=V+V(L+αLT)3=V+βVTL3(1+αT)3=V(1+βT)L3(1+α3T3+3αT+3α2T2)=V(1+βT)

Since α is very small, second and fourth terms in bracket on LHS are negligible. Also, wkt.V=L3. So the equation becomes

(1+3αT)=(1+βT)3αT=βT3α=β

Therefore, it is proved thatβ=3α

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