Question

Prove that the relationship between the coefficients of linear,areal and volume expansion is 1:2:3

Answer 1

Let alpha,beta and gamma be coefficient of linear,areal and volume expansion respectively

Consider a cube of length l, dt is the change in temp, dl is the change in length, ds is the change in area, dv is the change in volume.

Linear coefficient of expansion Alpha- α
coefficient of area expansion Beta- β
coefficient of volume expansion Gama- γ

Relation between α and β :-

From thermal expansion of solids,

dl= αldt.(1)

ds=βsdt..(2)

ds=new area - original area

=(l+dl)2-l2

=l2+2ldl+(dl)2-l2

=2ldl( dl is very small and the square of it will be very very small so it can be neglected)

=2l αdtl( from (1))

=2l2 αdt

=2s αdt..(3)

From (2) and (3)

βsdt=2s αdt

β=2 α

hence, α = β2

Relation between a and γ :-

dl= αldt.(1)

dv=γvdt.(2)

dv=new volume-original volume

=(l+dl)3-l3

=l3+3(l)2dl+3l(dl)2+(dl)3-l3

=3l2dl (dl2 and dl3 can be neglected)

Substituting dl = αldt

=3l22 αldt

=3l3 αdt

Since l3 = v
=3v αdt..(3)

From (2) and (3)

γvdt=3v αdt

γ=3 α

α=γ/3

SO α=β/2= γ/3

Hence proved