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

To calculate the final length of the combination of two rods when their temperature is increased, we can apply the formula for linear expansion. Linear expansion is given by the formula: 

\(L_f = L_i + \Delta L\)

where:

• \(L_f\) is the final length
• \(L_i\) is the initial length
• \(\Delta L = \alpha L_i \Delta T\) is the change in length due to thermal expansion
• \(\alpha\) is the coefficient of linear expansion
• \(\Delta T\) is the change in temperature

Here is the step-by-step solution:

1. Initial length of each rod, \(L_i = 60\,\text{cm}\).
2. Temperature change, \(\Delta T = 100^{\circ}\text{C} - 30^{\circ}\text{C} = 70^{\circ}\text{C}\).
3. Calculate the expansion of the first rod:
   • Coefficient of linear expansion, \(\alpha_1 = 24\times10^{-6}\^{\circ}\text{C}^{-1}\)
   • Change in length, \(\Delta L_1 = \alpha_1 \times L_i \times \Delta T = 24\times10^{-6} \times 60 \times 70\)
   • \(\Delta L_1 = 0.1008\,\text{cm}\)
4. Calculate the expansion of the second rod:
   • Coefficient of linear expansion, \(\alpha_2 = 1.2\times10^{-5}\^{\circ}\text{C}^{-1}\)
   • Change in length, \(\Delta L_2 = \alpha_2 \times L_i \times \Delta T = 1.2\times10^{-5} \times 60 \times 70\)
   • \(\Delta L_2 = 0.0504\,\text{cm}\)
5. Calculate the total length of the combination after expansion:
   • Initial combined length, \(L_{i\text{ total}} = 60 + 60 = 120\,\text{cm}\)
   • Total change in length, \(\Delta L_{\text{total}} = \Delta L_1 + \Delta L_2 = 0.1008 + 0.0504 = 0.1512\,\text{cm}\)
   • Final length, \(L_f = 120 + 0.1512 = 120.1512\,\text{cm}\)

Therefore, the final length of the combination of the two rods is \(120.1512\,\text{cm}\).

Thus, the correct answer is 120.1512.