First slide

Thermal expansion

Question

The coefficient of linear expansion of brass and steel are $α_{1}$ and $α_{2}$ . If we take a brass rod of length $l_{1}$ and steel rod of length $l_{2}$ at 0^oC, their difference in length $(l_{2} - l_{1})$ will remain the same at all temperatures if

Moderate

A

$α_{1} l_{2} = α_{2} l_{1}$
B

$α_{1} l_{2}^{2} = α_{2} l_{1}^{2}$
C

$α_{1}^{2} l_{1} = α_{2}^{2} l_{2}$
D

$α_{1} l_{1} = α_{2} l_{2}$

Solution

$\begin{array}{l} L_{2} = l_{2} (1 + α_{2} Δθ) and L_{1} = l_{1} (1 + α_{1} Δθ) \\ \Rightarrow (L_{2} - L_{1}) = (l_{2} - l_{1}) + Δθ (l_{2} α_{2} - l_{1} α_{1}) \\ Now (L_{2} - L_{1}) = (l_{2} - l_{1}) so, l_{2} α_{2} - l_{1} α_{1} = 0 \\ α_{1} l_{1} = l_{2} α_{2} \end{array}$

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