First slide

Thermal expansion

Question

A composite bar of length $L = L_{1} + L_{2}$ , is made up from a rod of material I and of length $L_{1}$ attached to a rod of material 2 and of length $L_{2}$ as shown. If $α_{1} and α_{2}$ are their respeclive coefficients of linear expansion, then equivalent coefficient of linear expansion for the composite rod is:

Moderate

A

$\frac{α_{1} L_{2} + α_{2} L_{1}}{L}$
B

$\frac{α_{1} L_{2} + α_{2} L_{2}}{L}$
C

$\frac{α_{1} L_{1} + α_{2} L_{2}}{L}$
D

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

Solution

When it has been given a temperature change $∆ T$ , the new length of the composite bar is the sum of the individual final lengths (Let ' $α$ ' be required coefficient of linear expansion)
$L^{'} = {L^{'}}_{1} + {L^{'}}_{2}$
$L (1 + α ∆ T) = L_{1} (1 + α_{1} ∆ T) + L_{2} (1 + α_{2} ∆ T)$
$\Rightarrow L + Lα ∆ T = L_{1} + L_{1} α_{1} ∆ T + L_{2} + L_{2} α_{2} ∆ T$
$\Rightarrow L + Lα ∆ T = L + L_{1} α_{1} ∆ T + L_{2} α_{2} ∆ T$
( $∵ L_{1} + L_{2} = L$ )
$\Rightarrow Lα ∆ T = L_{1} α_{1} ∆ T + L_{2} α_{2} ∆ T$
$\Rightarrow α = \frac{L_{1} α_{1} + L_{2} α_{2}}{L}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App