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

The problem involves determining the new length of a composite rod made of aluminium and steel when the temperature is increased. We solve this using the concept of linear expansion.

The formula for linear expansion is given by:

L = L_0 (1 + \alpha \Delta T)

where:

• L_0 is the original length,
• \alpha is the coefficient of linear expansion,
• \Delta T is the change in temperature.

The given problem specifies that both aluminium and steel rods have the same initial length, denoted by L_0. Since the total length is 120 cm, the length of each rod initially is:

L_0 = \frac{120}{2} = 60 \text{ cm}

For aluminium, we have:

• \alpha_{Al} = 24 \times 10^{-6}/^\circ\text{C}

For steel, we have:

• \alpha_{Steel} = 1.2 \times 10^{-5}/^\circ\text{C}

The temperature change is:

\Delta T = 100^\circ\text{C} - 30^\circ\text{C} = 70^\circ\text{C}

Calculate the new lengths:

1. Expansion of Aluminium Rod

L_{Al} = 60 \left(1 + 24 \times 10^{-6} \times 70 \right)

L_{Al} = 60 \left(1 + 0.00168\right)

L_{Al} = 60 \times 1.00168 = 60.1008 \text{ cm}

2. Expansion of Steel Rod

L_{Steel} = 60 \left(1 + 1.2 \times 10^{-5} \times 70 \right)

L_{Steel} = 60 \left(1 + 0.00084\right)

L_{Steel} = 60 \times 1.00084 = 60.0504 \text{ cm}

Total Length of the Composite Rod

Sum the expanded lengths:

L_{total} = L_{Al} + L_{Steel} = 60.1008 + 60.0504 = 120.1512 \text{ cm}

Rounding off to two decimal places, the total length of the composite rod at 100°C is 120.15 cm.

Therefore, the correct answer is: 120.15

Choosing this option over others stems directly from detailed calculation rather than estimation, confirming the exact increase in length for specified temperature change.