Question

Two rods of equal length \(60\,\text{cm}\) each are joined together end to end. The coefficients of linear expansion of the rods are \(24\times10^{-6}\^{\circ}\text{C}^{-1}\) and \(1.2\times10^{-5}\^{\circ}\text{C}^{-1}\). Their initial temperature is \(30^{\circ}\text{C}\), which is increased to \(100^{\circ}\text{C}\). Find the final length of the combination (in cm).

• A. \(120.1321\)
• B. \(120.1123\)
• C. \(120.1512\)
• D. \(120.1084\)

Hint:

For thermal expansion of composite rods:
Calculate expansion of each rod separately
Use \( \Delta L = L\alpha\Delta T \)
Add individual expansions to get total increase

Answer 1

The Correct Option is C

Step 1: Given: \[ L_1 = L_2 = 60\,\text{cm}, \alpha_1 = 24\times10^{-6}, \alpha_2 = 12\times10^{-6} \] \[ \Delta T = 100 - 30 = 70^{\circ}\text{C} \]
Step 2: Linear expansion of first rod: \[ \Delta L_1 = L_1 \alpha_1 \Delta T = 60 \times 24\times10^{-6} \times 70 = 0.1008\,\text{cm} \]
Step 3: Linear expansion of second rod: \[ \Delta L_2 = L_2 \alpha_2 \Delta T = 60 \times 12\times10^{-6} \times 70 = 0.0504\,\text{cm} \]
Step 4: Total increase in length: \[ \Delta L = 0.1008 + 0.0504 = 0.1512\,\text{cm} \]
Step 5: Final length of the combination: \[ L_{\text{final}} = 120 + 0.1512 = 120.1512\,\text{cm} \]