Question

An aluminium and steel rods having same lengths and cross-sections are joined to make total length of 120 cm at 30\(^\circ\)C. The coefficient of linear expansion of aluminium and steel are \(24 \times 10^{-6}\)/\(^\circ\)C and \(1.2 \times 10^{-5}\)/\(^\circ\)C, respectively. The length of this composite rod when its temperature is raised to 100\(^\circ\)C, is________ cm.

• A. 120.20
• B. 120.03
• C. 120.15
• D. 120.06

Hint:

For composite systems undergoing thermal expansion, the total change in length is simply the sum of the individual changes in length of each component.
It's often helpful to unify the units and powers of 10 for constants (like the two \(\alpha\) values here) before calculating to minimize errors.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We have a composite rod made of an aluminium part and a steel part of equal initial length.
The rod is heated, and we need to find its new total length.
The total final length will be the sum of the final lengths of the two individual parts.
Step 2: Key Formula or Approach:
The formula for linear thermal expansion gives the final length \(L_f\) of a material after a temperature change \(\Delta T\):
\[ L_f = L_0(1 + \alpha \Delta T)
\] where \(L_0\) is the initial length and \(\alpha\) is the coefficient of linear expansion.
For a composite rod, the total final length is \(L_{total, f} = L_{Al, f} + L_{St, f}\).
Step 3: Detailed Explanation:
Initial total length at \(T_0 = 30^\circ\)C is \(L_{total,0} = 120\) cm.
Since the aluminium (Al) and steel (St) rods have the same initial length:
\(L_{Al,0} = L_{St,0} = \frac{120 \text{ cm}}{2} = 60\) cm.
The final temperature is \(T_f = 100^\circ\)C.
The change in temperature is \(\Delta T = T_f - T_0 = 100^\circ\text{C} - 30^\circ\text{C} = 70^\circ\)C.
The coefficients of linear expansion are given:
\(\alpha_{Al} = 24 \times 10^{-6} /^\circ\)C.
\(\alpha_{St} = 1.2 \times 10^{-5} /^\circ\)C = \(12 \times 10^{-6} /^\circ\)C.
Now, we calculate the final length of each rod:
Final length of Aluminium rod:
\(L_{Al,f} = L_{Al,0}(1 + \alpha_{Al} \Delta T) = 60(1 + 24 \times 10^{-6} \times 70) = 60(1 + 0.00168) = 60.1008\) cm.
Final length of Steel rod:
\(L_{St,f} = L_{St,0}(1 + \alpha_{St} \Delta T) = 60(1 + 12 \times 10^{-6} \times 70) = 60(1 + 0.00084) = 60.0504\) cm.
The total final length of the composite rod is the sum of the final lengths of the two parts:
\[ L_{total,f} = L_{Al,f} + L_{St,f} = 60.1008 \text{ cm} + 60.0504 \text{ cm} = 120.1512 \text{ cm}.
\] Alternatively, we can calculate the total expansion \(\Delta L_{total} = L_{Al,0}\alpha_{Al}\Delta T + L_{St,0}\alpha_{St}\Delta T\).
\[ \Delta L_{total} = (60 \times 24 \times 10^{-6} \times 70) + (60 \times 12 \times 10^{-6} \times 70) = 0.1008 + 0.0504 = 0.1512 \text{ cm}.
\] \[ L_{total,f} = L_{total,0} + \Delta L_{total} = 120 + 0.1512 = 120.1512 \text{ cm}.
\] The closest option is 120.15 cm.
Step 4: Final Answer:
The length of the composite rod at 100\(^\circ\)C is approximately 120.15 cm.