A composite bar of length L=L1+L2 is made up from a rod of material 1 and of length L1 attached to a rod of material 2 and of length L2 as shown. If α1 and α2 are their respective coefficients of linear expansion, then equivalent coefficient of linear expansion for the composite rod is:

Question

A composite bar of length L=L1+L2 is made up from a rod of material 1 and of length L1 attached to a rod of material 2 and of length L2 as shown. If α1 and α2 are their respective coefficients of linear expansion, then equivalent coefficient of linear expansion for the composite rod is: (A) (α1 L2+α2 L1/L) (B) (α1 L2+α2 L2/L) (C) (α1 L1+α2 L2/L) (D) (α1 α2(L12+L2)/(α1 L1+α2 L2))

Tardigrade Admin · Accepted Answer

When it has been given a temperature change

Δ T

, the new length of the composite bar is the sum of the individual final lengths (Let '

α

' be required coefficient of linear expansion)

L^{'} = L_{1}^{'} + L_{2}^{'}

L (1 + α Δ T) = L_{1} (1 + α_{1} Δ T) + L_{2} (1 + α_{2} Δ T)

\Rightarrow L + Lα Δ T = L_{1} + L_{1} α_{1} Δ T + L_{2} + L_{2} α_{2} Δ T

\Rightarrow L + Lα Δ T = L + L_{1} α_{1} Δ T + L_{2} α_{2} Δ T

(∵ L_{1} + L_{2} = L)

\Rightarrow Lα Δ T = L_{1} α_{1} Δ T + L_{2} α_{2} Δ T

\Rightarrow α = \frac{L _{1} α _{1} + L _{2} α _{2}}{L}

$\frac{α _{1} L _{2} + α _{2} L _{1}}{L}$

$\frac{α _{1} L _{2} + α _{2} L _{2}}{L}$

$\frac{α _{1} L _{1} + α _{2} L _{2}}{L}$

$\frac{α _{1} α _{2} ( L _{1}^{2} + L _{2} )}{( α _{1} L _{1} + α _{2} L _{2} )}$

Lα1​L2​+α2​L1​​

Lα1​L2​+α2​L2​​

Lα1​L1​+α2​L2​​

(α1​L1​+α2​L2​)α1​α2​(L12​+L2​)​

$\frac{α _{1} L _{2} + α _{2} L _{1}}{L}$

$\frac{α _{1} L _{2} + α _{2} L _{2}}{L}$

$\frac{α _{1} L _{1} + α _{2} L _{2}}{L}$

$\frac{α _{1} α _{2} ( L _{1}^{2} + L _{2} )}{( α _{1} L _{1} + α _{2} L _{2} )}$