First slide

Thermal expansion

Question

A bimetallic strip is formed out of two identical strips one of copper and the other of brass. The coefficients of linear expansion of the two metals are $α_{C} and α_{B}$ . On heating, the temperature of the strip goes up by $∆ T$ and the strip bends to form an arc of radius of curvature R. Then R is

Moderate

A

proportional to $∆ T$
B

inversely proportional to $∆ T$
C

proportional to $|α_{B} - α_{C}|$
D

inversely proportional to $|α_{B} - α_{C}|$

Solution

From figure, $(R + \frac{t}{2}) θ = l_{0} (1 + α_{C} ΔT) . . . . . (1)$
and $(R - \frac{t}{2}) θ = l_{0} (1 + α_{B} ΔT) . . . . (2)$
Dividing eq. (2) by eq. (1), we 8et
$\frac{(R + \frac{t}{2})}{(R - \frac{t}{2})} = \frac{1 + α_{C} ΔT}{1 + α_{B} ΔT}$
Solving this equation and neglecting $\frac{t}{2} α_{C} ΔT$ and $\frac{t}{2} α_{B} ΔT,$ We get
$R = \frac{t}{(α_{C} - α_{B}) ΔT}$
$∴ R \propto \frac{1}{ΔT} and F \propto \frac{1}{|α_{B} - α_{C}|}$

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