First slide

Thermal expansion

Question

Two straight metallic strips each of thickness t and length l are rivertted together. Their coefficients of linear expansions are $α_{1} and α_{2}$ and ar. If they are heated through temperature $∆ T$ , the bimetallic strip will bend to form an arc of radius

Moderate

A

$t / \{(α_{1} + α_{2}) ΔT\}$
B

$t / \{(α_{2} - α_{1}) ΔT\}$
C

$t (α_{1} + α_{2}) ΔT$
D

$t (α_{2} - α_{1}) ΔT$

Solution

Let the angle subtended by the arc formed be $θ$ . Then
$θ = \frac{l}{r} or θ = \frac{ΔL}{Δr} = \frac{l_{2} - l_{1}}{r_{1} - r_{2}}$
$∴ θ = \frac{l (α_{2} - α_{1}) ΔT}{t} or \frac{l}{r} = \frac{l (α_{2} - α_{1}) ΔT}{t}$
So, $r = \frac{t}{(α_{2} - α_{1}) ΔT}$

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