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

Short Answer

Expert verified

Answer

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 the new length and we further calculate the change in volume and find the new volume.

02

formula for linear expansion and volume expansion

The formula for length expansion of solidsΔL=αLΔT

The formula for volume expansion of solids ΔV=βVΔT

Here,are the coefficient of linear and volume expansion respectively, its original length and volume, and is the temperature change.

New length=L+ΔL

New volume=V+ΔV

But for the cube,

Volume = (Length)3

03

Equate new volume

(L+ΔL)3=V+ΔV(L+αLΔT)3=V+βVΔTL3(1+αΔT)3=V(1+βΔT)L3(1+α3ΔT3+3αΔT+3α2ΔT2)=V(1+βΔT)

Since α is very small, second and fourth terms in the bracket on LHS are negligible. Also, we can write volume as,.So the equation becomes

(1+3αΔT)=(1+βΔT)3αΔT=βΔT3α=β

Therefore, it is proved thatβ=3α

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