An aluminium and steel rods having same lengths and cross-sections are joined to make total length o

Question

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

Professor PC Thomas Classes · Answer

Initial length of each rod $L = 60 	ext{ cm}$. Change in temperature $\Delta T = 100^\circ	ext{C} - 30^\circ	ext{C} = 70^\circ	ext{C}$. Total expansion $\Delta L = \Delta L_{Al} + \Delta L_{steel} = L \alpha_{Al} \Delta T + L \alpha_{steel} \Delta T = L \Delta T (\alpha_{Al} + \alpha_{steel})$. Substituting values: $\Delta L = 60 	imes 70 	imes (24 	imes 10^{-6} + 12 	imes 10^{-6}) = 4200 	imes 36 	imes 10^{-6} = 0.1512 	ext{ cm}$. Final length $= 120 + 0.1512 = 120.15 	ext{ cm}$.