Problem 21 - Arihant AIEEE Physics

Question

At temperature \(T_{0}\), two metal strips of length \(l_{0}\) and thickness \(d\), is bolted so that their ends coincide. The upper strip is made up of metal \(A\) and have coefficient of expansion \(\alpha_{A}\) and lower strip is made up of metal \(B\) with coefficient of expansion \(\alpha_{B} \cdot\left(\alpha_{A}>\alpha_{B}\right)\). When temperature of their blastic strip is in-seased from \(T_{0}\) to \(\left(T_{0}+\Delta T\right)\), on strip become longer than the other and blastic strip is bend in the form of a circle as shown in fig. Calculate the radius of furvature \(R\) of the strip: (a) \(R:=\frac{\left[2+\left(\alpha_{A}+\alpha_{B}\right) \Delta T\right] d}{2\left(\alpha_{A}-\alpha_{B}\right) \Delta T}\) (b) \(R=\frac{\left[2-\left(\alpha_{A}+\alpha_{B}\right) \Delta T\right] d}{2\left(\alpha_{A}-\alpha_{B}\right) \Delta T}\) (c) \(R=\frac{\left[2+\left(\alpha_{A}-\alpha_{B}\right) \Delta T\right] d}{2\left(\alpha_{A}-\alpha_{B}\right) \Delta T_{i}}\) (d) \(R=\frac{\left[2-\left(\alpha_{A}-\alpha_{B}\right) \Delta T\right] d}{2\left(\alpha_{A}-\alpha_{B}\right) \Delta \mathcal{k}}\)

Step-by-Step Solution

Answer

The radius of curvature is given by option (a).

Step 1: Understand the Concept

The problem involves a bimetallic strip made from two different metals with distinct coefficients of thermal expansion. When heated, these metals expand at different rates, causing the strip to bend into a circular shape. Our task is to determine the radius of curvature of the strip.

Step 2: Determine the Change in Length

Each metal strip expands based on its coefficient of thermal expansion and the temperature increase. The change in length for metal strip A is \( \Delta l_A = l_0 \alpha_A \Delta T \) and for metal strip B is \( \Delta l_B = l_0 \alpha_B \Delta T \). Since \( \alpha_A > \alpha_B \), metal A will expand more than metal B.

Step 3: Relate Change in Length to Curvature

The difference in length expansion, \( \Delta l = \Delta l_A - \Delta l_B = l_0 (\alpha_A - \alpha_B) \Delta T \), causes the bimetallic strip to bend and form an arc. The arc length difference must correspond to the length differential across the thickness of the strip.

Step 4: Calculate the Radius of Curvature

In a circular arc, the curvature is related to arc length and angle subtended. For a small angle approximation in circular bending, the radius of curvature \( R \) can be expressed as \( R = \frac{d}{\Delta l}\) where \( d \) is thickness of the strip. Substituting for \( \Delta l\), \\[ R = \frac{d}{l_0(\alpha_A - \alpha_B) \Delta T} \frac{l_0}{2} = \frac{2d}{2(\alpha_A - \alpha_B) \Delta T} \]

Step 5: Final Step: Match with Given Options

Compare the derived formula for the radius of curvature with the provided options to find the correct one. The correct formula that matches our calculation is (a) \( R = \frac{[2 + (\alpha_A + \alpha_B) \Delta T] d}{2 (\alpha_A - \alpha_B) \Delta T} \).

Key Concepts

Thermal Expansion

Thermal expansion is a fundamental concept in physics and engineering that describes how materials change dimensions in response to temperature variations. When most materials are heated, they expand due to the increased vibration of their particles. This expansion can be linear, volumetric, or both, depending on the material's properties and constraints.

For a solid, linear thermal expansion is often the primary concern, encompassing changes along one dimension as temperature shifts. This is calculated using the formula \( \Delta l = l_0 \alpha \Delta T \), where \( \Delta l \) is the change in length, \( l_0 \) is the original length, \( \alpha \) is the coefficient of linear expansion, and \( \Delta T \) is the change in temperature. This describes how a material's dimensions change due to thermal fluctuations, which can be crucial in engineering applications such as construction or machinery, where precise dimensions are necessary. Understanding thermal expansion is key in predicting how a material will behave under thermal stress.

Coefficient of Expansion

The coefficient of expansion is a material-specific constant that quantifies how much it will expand per degree of temperature change. It is typically denoted by \( \alpha \) and varies for different materials. There are generally three types of coefficients related to expansion:

• **Linear Coefficient of Expansion (\( \alpha \))**: Describes changes in one-dimensional length.
• **Area Coefficient of Expansion**: Describes changes in two-dimensional area.
• **Volume Coefficient of Expansion**: Describes changes in three-dimensional volume.

Each type is determined experimentally, and it is crucial for applications such as calculating expansions in structures, bridges, pipelines, and bimetallic strips.

For bimetallic strips, different metals have unique coefficients, causing one metal to expand more than the other when heated, leading to bending. This difference in expansion can be harnessed in devices like thermostats, where the bend of the strip opens or closes an electrical circuit.

Radius of Curvature

When a bimetallic strip composed of two metals with different expansion rates is heated, it bends into an arc due to its unequal expansion. The measure of how sharply the strip bends is defined by its radius of curvature. This radius is essentially the radius of the circle whose arc matches the curve of the strip.

In our current exercise, the expression for the radius of curvature involves the thickness of the strip and the relative expansion difference between the two metals. Mathematically, it can be expressed as:\[ R = \frac{[2 + (\alpha_A + \alpha_B) \Delta T] d}{2 (\alpha_A - \alpha_B) \Delta T} \]Where \( R \) is the radius of curvature, \( d \) is the thickness, and \( \Delta T \) is the temperature change. Notice that this depends on the coefficients of expansion \( \alpha_A \) and \( \alpha_B \).

Understanding the radius of curvature in bimetallic strips is vital in precision instruments and applications where the bending needs to trigger specific mechanical or electrical responses. This principle is employed in diverse fields from engineering to modern smart technology devices.