Question

derive the relation between the coefficient of linear and cubical expansion of a solid​

Answer 1

Derivation of the relation between the coefficient of linear and cubic expansion of a solid:

The following is the relationship between the coefficients of linear and real expansion:

At 0°C, consider a thin rectangular parallelopiped solid with the dimensions lo bo ho Vo. Let's imagine the solid is heated to a temperature of C.

Let l,b,h, V represent the length, width, height, and volume at C.

α=coefficient of linear expansion

First equation is as follows:

Then,

The original volume Vo=loboho 

Consider linear expansion

length l=lo(1+αt)

breadth b=bo(1+αt)

height h=ho(1+αt)

Final volume V=lbh=lo(1+αt)×bo(1+αt)×ho(1+αt)

V=loboho(1+3αt+3α2t2+α3t3)

\alpha

Now, 

because α is very small hence α2 is still small, hence quantity α^2t^2 α^3t^3 can be ignored.

The second equation is as follows:

V=Vo(1+3αt) 

Consider the cubical expansion of the solid

V=Vo(1+γt)

From (1) and (2),

γ=3α

Hence, the relation between the coefficient of linear and cubical expansion is derived.