Problem 27 - Arihant AIEEE Physics

Question

Two steel rods and copper rod of equal length \(l_{0}\) and equal cross-sections are joined rigidly as shown. All the rods are in a state of zero tension at \(0^{\circ} \mathrm{C}\). The temperature of system increases upto \(30^{\circ} \mathrm{C}\), then : (a) tensile force on either steel plate is half of copper plate (b) the net expansion in copper plate is less than the thermal expansion of the copper plate (c) the expansion in either steel plates is larger than thermal expansion in steel plates (d) all of the above

Step-by-Step Solution

Answer

All options are true. The scenario describes balanced mechanical equilibrium forces in the heated rods.

Step 1: Understand the Scenario

Consider two steel rods and one copper rod, all of the same length and cross-sectional area. They form a composite rod that is heated from \(0^{\circ} \text{C}\) to \(30^{\circ} \text{C}\). Initially, there is no tension in the system.

Step 2: Identify Properties of Materials

The linear thermal expansion coefficient for steel \( (\alpha_s) \) and copper \( (\alpha_c) \) are different. Typically, \(\alpha_c > \alpha_s\). This means copper will expand more upon heating per degree Celsius compared to steel.

Step 3: Calculate Expected Thermal Expansion

Each material will try to expand according to its coefficient of thermal expansion: \( \Delta L_{steel} = \alpha_s \cdot l_{0} \cdot \Delta T \) and \( \Delta L_{copper} = \alpha_c \cdot l_{0} \cdot \Delta T \) where \(\Delta T\) is the change in temperature, so \(\Delta T = 30^{\circ} \text{C}\).

Step 4: Establish Mechanical Equilibrium

Since these rods are fixed together, they must reach a mechanical equilibrium where the total expansions are equalized in the system. Thus, the tensile forces on each rod must act such that the sum of linear displacements is consistent for the fixed length.

Step 5: Determine the Internal Forces

Because the copper rod wants to expand more due to its higher \(\alpha\), it will face a tensile force trying to compress it. Conversely, the steel rods will experience a tensile force trying to elongate them. Let's analyze the forces.

Step 6: Analyze Force Balance

Let's denote the tensile force in steel as \(F_s\) and in copper as \(F_c\). The rods must satisfy the equilibrium: \(E_s A (\Delta L_{steel} - \Delta L'_{steel}) = F_s\), \(E_c A (\Delta L_{copper} - \Delta L'_{copper}) = -F_c\), where \(\Delta L'\) is the actual change accounting for external forces.

Step 7: Evaluate the Options

Given the symmetrical setup, the force exerted on each steel plate is equal to each other and hence half of the copper plate, confirming statement (a). As copper wants to expand more thermally, mechanical equilibrium forces reduce its net expansion, confirming statement (b). Similarly, since steel experiences a compensating tensile force, its produced expansion is greater than the purely thermal, confirming statement (c). Therefore, all statements are true.

Key Concepts

Composite Rods

A composite rod is the combination of different materials bonded together to form a single structure. These materials, like the steel and copper rods in this scenario, are structurally combined to function together. The beauty of composite rods lies in utilizing the distinct properties of each material. Here, we have:

• Two steel rods
• One copper rod

Each material initially has the same physical dimensions—length and cross-sectional area—but they have different thermal behavior.
Composite rods can be found in many engineering applications where a balance between strength, flexibility, and thermal resistance is needed.
They are essential when we need to manage different materials’ expansion due to temperature changes.

Thermal Stress

When materials in a composite rod have different thermal expansion coefficients, they can experience thermal stress when the temperature changes. Thermal stress in this context refers to the internal forces that develop when the composite material is prevented from expanding or contracting freely.
For instance:

• Steel wants to expand less due to a smaller thermal expansion coefficient.
• Copper wants to expand more because it has a higher thermal expansion coefficient.

As these materials are fixed together, they exert forces on each other. Copper, trying to expand more, faces compressive forces, while steel, expanding less, develops tensile forces.
This interplay of forces attempts to prevent each material from moving freely, creating a state of tension—specifically which is referred to as thermal stress. It can significantly affect the durability and reliability of materials used in structures or devices.

Linear Thermal Expansion Coefficient

The linear thermal expansion coefficient (\( \alpha \)) is a property that describes how much a material expands per degree of temperature change. It is crucial for understanding how different materials will behave when subject to temperature changes.

• For copper, \( \alpha_c \) is relatively high, meaning copper expands significantly with heating.
• For steel, \( \alpha_s \) is lower, implying less expansion compared to copper.

This coefficient is used in the formula for calculating thermal expansion:\[ \Delta L = \alpha \cdot l_0 \cdot \Delta T \]where \( \Delta L \) is the change in length, \( \alpha \) is the thermal expansion coefficient, \( l_0 \) is the original length, and \( \Delta T \) is the temperature change.
Understanding this property helps predict how much a particular material will expand or contract with temperature changes, vital for designing components that undergo varying thermal environments.

Mechanical Equilibrium

Mechanical equilibrium occurs when all forces within a system are balanced, leading to a stable configuration. In the context of thermal expansion and composite rods:

• The sum of all expansion forces across the different materials must balance out.
• No net displacement or change occurs unless the external constraints are altered.

For composite rods:

• Copper will exhibit a tensile force due to its greater expansion attempt, pushing against its restraints.
• Steel rods will counteract this expansion by developing their own compensating tensile forces.

Achieving mechanical equilibrium involves ensuring the sum of expansions and contractions in the system results in no net change in the whole structure's length, even if the individual rods experience differential expansions.
This equilibrium state is essential because it validates the reliability of the composite structure under thermal changes—which prevents fractures or failures when exposed to practical applications.