First slide

Thermal expansion

Question

Two rods are joined between fixed supports as shown in the figure. Condition for no change in the lengths of individual rods with the increase of temperature will be
( $α_{1}, α_{2}$ = linear expansion coefficient
$A_{1}, A_{2}$ = Area of rods
$Y_{1}, Y_{2}$ = Young modulus)

Moderate

A

$\frac{A_{1}}{A_{2}} = \frac{α_{1} Y_{1}}{α_{2} Y_{2}}$
B

$\frac{A_{1}}{A_{2}} = \frac{L_{1} α_{1} Y_{1}}{L_{2} α_{2} Y_{2}}$
C

$\frac{A_{1}}{A_{2}} = \frac{L_{2} α_{2} Y_{2}}{L_{1} α_{1} Y_{1}}$
D

$\frac{A_{1}}{A_{2}} = \frac{α_{2} Y_{2}}{α_{1} Y_{1}}$

Solution

$Y = \frac{Stress}{Strain} = \frac{\frac{T}{A}}{\frac{∆ l}{l}}$
$T = \frac{Y . ∆ l}{l} A = Y . Aα ∆ T$
In both the rods tension will be same so
$T_{1} = T_{2}, Hence Y_{1} A_{1} α_{1} ∆ T = Y_{2} A_{2} α_{2} ∆ T$
$\frac{A_{1}}{A_{2}} = \frac{Y_{2} α_{2}}{Y_{1} α_{1}}$

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