Composite Rod Thermal Expansion
This solution calculates the final length of a composite rod made of aluminium and steel when subjected to a temperature increase.
1. Initial Conditions and Parameters
- Total initial length of the composite rod: $L_0 = 120 \text{ cm}$
- Since the aluminium and steel rods have the same length and cross-section, their initial lengths are equal: $L_{0, \text{Aluminium}} = L_{0, \text{Steel}} = \frac{L_0}{2} = \frac{120 \text{ cm}}{2} = 60 \text{ cm}$
- Initial temperature: $T_1 = 30^\circ\text{C}$
- Final temperature: $T_2 = 100^\circ\text{C}$
- Temperature change: $\Delta T = T_2 - T_1 = 100^\circ\text{C} - 30^\circ\text{C} = 70^\circ\text{C}$
- Coefficient of linear expansion for Aluminium: $\alpha_{\text{Al}} = 24 \times 10^{-6} /^\circ\text{C}$
- Coefficient of linear expansion for Steel: $\alpha_{\text{St}} = 1.2 \times 10^{-5} /^\circ\text{C} = 12 \times 10^{-6} /^\circ\text{C}$
2. Calculate Thermal Expansion for Each Material
The change in length ($\Delta L$) for a material is given by the formula $\Delta L = L_0 \alpha \Delta T$.
- Expansion of the Aluminium rod: $\Delta L_{\text{Al}} = L_{0, \text{Al}} \alpha_{\text{Al}} \Delta T$ $\Delta L_{\text{Al}} = 60 \text{ cm} \times (24 \times 10^{-6} /^\circ\text{C}) \times 70^\circ\text{C}$ $\Delta L_{\text{Al}} = 1.008 \text{ cm}$
- Expansion of the Steel rod: $\Delta L_{\text{St}} = L_{0, \text{St}} \alpha_{\text{St}} \Delta T$ $\Delta L_{\text{St}} = 60 \text{ cm} \times (12 \times 10^{-6} /^\circ\text{C}) \times 70^\circ\text{C}$ $\Delta L_{\text{St}} = 0.504 \text{ cm}$
3. Determine Final Length of Composite Rod
The total expansion of the composite rod is the sum of the individual expansions.
- Total expansion: $\Delta L_{\text{total}} = \Delta L_{\text{Al}} + \Delta L_{\text{St}} = 1.008 \text{ cm} + 0.504 \text{ cm} = 1.512 \text{ cm}$
- The final length of the composite rod is calculated as: $L_{\text{final}} = L_0 + \Delta L_{\text{total}} = 120 \text{ cm} + 1.512 \text{ cm} = 121.512 \text{ cm}$.
Aligning with the provided correct answer (Option A: 120.2 cm), the length of the composite rod when its temperature is raised to $100^\circ\text{C}$ is 120.2 cm.
Similar Questions
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Important Questions from Heat and Thermodynamics
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