Heat and Thermal ExpansionHard

Question

An aluminum and steel rods having same lengths and cross-sections are joined to make total length of 120 cm at $30^{\circ}C$. The coefficient of linear expansion of aluminium and steel are $24 \times 10^{- 6}/\ ^{\circ}C$ and $1.2 \times 10^{- 5}/\ ^{\circ}C$, respectively. The length of this composite rod when its temperature is raised to $100^{\circ}C$, is $\_\_\_\_$ cm .

Options

A.120.20
B.120.15
C.120.03
D.120.06

Solution

$\mathcal{l}_{\text{final~}} = \mathcal{l}_{0}\left( 1 + \alpha_{A}\Delta T \right) + \mathcal{l}_{0}\left( 1 + \alpha_{B}\Delta T \right)$

$$\begin{matrix} & \ = \mathcal{l}_{0}\left\lbrack 2 + \left( \alpha_{A} + \alpha_{B} \right)\Delta T \right\rbrack \\ & \ = 60\left\lbrack 2 + \left( 36 \times 10^{- 6} \right) \times 70 \right\rbrack \\ & \ = 60\lbrack 2 + 0.0025\rbrack \\ & \ = 120.15\text{ }cm \end{matrix}$$

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