First slide

Thermal expansion

Question

Two rods of different materials and identical cross sectional area, are joined face to face at one end and their free ends are fixed to the rigid walls. If the temperature of the surroundings is increased by 30°C, the magnitude of the displacement of the joint of the rod is (length of rods l₁=l₂=1unit, ratio of their young's moduli, Y₁/Y₂=2, coefficients of linear expansion are α₁ and α₂)

Moderate

A

$5 (α_{2} - α_{1})$
B

$10 (α_{1} - α_{2})$
C

$10 (α_{2} - 2 α_{1})$
D

$5 (2 α_{1} - α_{2})$

Solution

$\begin{array}{l} Y_{1} (Strain)_{1} = Y_{2} (Strain)_{2} \\ Y_{1} [\frac{a_{1} {lΔT}_{1} - x}{l}] = Y_{2} [\frac{a_{2} {lΔT}_{2} + x}{l}] \end{array}$
$Displacement of the rod x = \frac{(Y_{2} a_{2} - Y_{1} a_{1})}{Y_{1} + Y_{2}} \times l \times ΔT$

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