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

To find the final length of the combined rods, we will calculate the expansion of each rod separately and then sum their lengths.

Step-by-Step Solution:

1. Identify the given values:
   • Initial length of each rod (\(L_1 = L_2\)): \(60 \, \text{cm}\)
   • Coefficient of linear expansion for the first rod (\(\alpha_1\)): \(24 \times 10^{-6} \, \text{°C}^{-1}\)
   • Coefficient of linear expansion for the second rod (\(\alpha_2\)): \(1.2 \times 10^{-5} \, \text{°C}^{-1}\)
   • Initial temperature (\(T_1\)): \(30^\circ \text{C}\)
   • Final temperature (\(T_2\)): \(100^\circ \text{C}\)
2. Calculate the change in temperature (\(\Delta T\)):

\(\Delta T = T_2 - T_1 = 100 - 30 = 70^\circ \text{C}\)

1. Calculate the change in length for the first rod (\(\Delta L_1\)):

\(\Delta L_1 = L_1 \times \alpha_1 \times \Delta T\)

\(\Delta L_1 = 60 \times 24 \times 10^{-6} \times 70\)

\(\Delta L_1 = 0.1008 \, \text{cm}\)

1. Calculate the change in length for the second rod (\(\Delta L_2\)):

\(\Delta L_2 = L_2 \times \alpha_2 \times \Delta T\)

\(\Delta L_2 = 60 \times 1.2 \times 10^{-5} \times 70\)

\(\Delta L_2 = 0.0504 \, \text{cm}\)

1. Calculate the total change in length of the combination:

\(\Delta L = \Delta L_1 + \Delta L_2 = 0.1008 + 0.0504 = 0.1512 \, \text{cm}\)

1. Calculate the final length of the rods combined:

\(\text{Final Length} = (L_1 + L_2) + \Delta L = 120 + 0.1512 = 120.1512 \, \text{cm}\)

Thus, the final length of the combination of rods at 100°C is 120.1512 cm.

This corresponds to the correct answer: 120.1512.