Prove that the coefficient of volume expansion is three times of coefficient of linear expansion : or prove that β = 3α :

 

Prove that the coefficient of volume expansion is three times of coefficient of linear expansion :

                                        OR

                                  prove that  β = 3α :

consider a body  of  length  'L'  Breath  'b'  and  Height  'h'  the volume of  a body is  'V' at initial temperature  `'T_1^'` as given as:

V = lbh

When the temperature increases from `'T_1^'` to `'T_2^'` .Its volume become  ' v′ ' :

`v′=l′b′h′`

 v′= l ( 1+αΔT) × b( 1+αΔT) × h( 1+αΔT)

`v′=lbh(1+\alpha\Delta T)³`

                                           `\because(a+b)³=a³+3a²b+3ab²+b³\;`

`v′=v(1+3\alpha\triangle T+3\alpha²\triangle T²+\alpha³\triangle T³)`

Since α is very small ,o we can neglect the higher term

v′ = v(1+3α∆T)

v′ =v+3vα∆T

v′-v = 3vα∆T

∆v = 3vα∆T

                                           ∵∆v=βv∆T

βv∆T= 3Vα∆T

β = 3α

 OR

α = 1/3 β