Question

Two rods of equal length 60 cm each are joined together end to end. Coefficient of linear expansions of the rods are \( 24 \times 10^{-6} \, \text{°C}^{-1} \) and \( 1.2 \times 10^{-5} \, \text{°C}^{-1} \). Their temperatures are the same and equal to \( 30^\circ \text{C} \), which is increased to \( 100^\circ \text{C} \). Find final length of the combination (in cm).

• A. 120.1321
• B. 120.1123
• C. 120.1512
• D. 120.1084

Hint:

When combining rods with different coefficients of linear expansion, calculate the individual length changes and sum them to get the total change in length.

Answer 1

The Correct Option is C

Step 1: Formula for linear expansion.
The linear expansion of each rod is given by \( \Delta L = L \alpha \Delta T \). The total change in length is the sum of the changes in length of each rod.
Step 2: Calculate the change in length.
For rod 1: \[ \Delta L_1 = 60 \times (3.6 \times 10^{-5} \times 70) = 0.1512 \, \text{cm} \] For rod 2: \[ \Delta L_2 = 60 \times (1.2 \times 10^{-5} \times 70) = 0.1512 \, \text{cm} \] Thus, the total change in length is \( 0.1512 \, \text{cm} \). The final length of the combination is: \[ L_f = 120 + 0.1512 = 120.1512 \, \text{cm} \] Step 3: Conclusion.
The final length of the combination is \( 120.1512 \, \text{cm} \), which corresponds to option (3).