Problem 27 - Master Resource Book in JEE Main Physics

Question

A bimetallic is made of two strips \(A\) and \(B\) having coefficients of linear expansion \(\alpha_{A}\) and \(\alpha_{B}\). If \(\alpha_{A}<\alpha_{B}\), then on heating, the strip will (a) bend with \(A\) on outer side (b) bend with \(B\) on outer side (c) not bend at all (d) None of the above

Step-by-Step Solution

Answer

The strip will bend with \(B\) on outer side (option b).

Step 1: Understanding the Problem

A bimetallic strip is composed of two different metals, each with a different coefficient of linear expansion, which causes the strip to bend upon heating. We are given that the coefficient of expansion for material A, \(\alpha_{A}\), is less than for material B, \(\alpha_{B}\). The task is to determine in which configuration the strip will bend.

Step 2: Analyzing Expansion Behavior

When heated, each metal strip expands according to its coefficient of linear expansion. Because \(\alpha_B > \alpha_A\), strip B will expand more than strip A when both are subjected to the same temperature change.

Step 3: Predicting the Bend Direction

As strip B expands more than strip A, the strip will bend such that the metal with the greater expansion (strip B) will form the outer side of the curve. Strip A, having a smaller expansion, will form the inner side.

Step 4: Selecting the Correct Option

Since strip B will be on the outer side due to its greater expansion, the correct option is (b) bend with \(B\) on outer side.

Key Concepts

Linear Expansion

Linear expansion describes how the length of a material changes when its temperature changes.
Essentially, when you heat a material, it tends to expand, and when you cool it, it contracts. This phenomenon is predictable and can be quantified using its coefficient of linear expansion.The coefficient of linear expansion is denoted by the symbol \(\alpha\). It allows us to predict how much a given material will expand or contract with a change in temperature. Mathematically, this can be put into an equation:\[ \Delta L = L_0 \alpha \Delta T \]where:

• \( \Delta L \) is the change in length.
• \( L_0 \) is the original length of the material.
• \( \alpha \) is the coefficient of linear expansion.
• \( \Delta T \) is the change in temperature.

It's crucial to understand that different materials have different coefficients, which means they expand by different amounts for the same temperature change. This variability in expansion is what causes a bimetallic strip to bend.

Thermal Expansion

Thermal expansion is a broader concept that encompasses not just the change in length (linear expansion), but also changes in area and volume of materials as a result of temperature variations. This principle is pivotal in understanding how objects react to thermal conditions in everyday applications. Thermal expansion is a natural response of most materials when exposed to heat, causing them to expand. For solids, such as the strips in a bimetallic strip, the most straightforward form of thermal expansion is linear expansion.
However, if you consider two-dimensional and three-dimensional objects, the area and volume expansions come into play. A practical application of this concept is seen in bridges and railway lines, which have expansion joints to accommodate the changes in length and prevent structural damage during temperature variations. In a bimetallic strip, thermal expansion is what drives the bending action, with each metal's unique coefficient of linear expansion dictating how much they expand and which side will end up on the outer curve.

Coefficients of Expansion

Coefficients of expansion are critical parameters that determine how materials behave when temperatures change. They signify the rate of expansion per degree of temperature change, and they are specific to each material, encompassing linear, area, and volumetric expansion coefficients.For the exercise involving a bimetallic strip, the focus is on the coefficient of linear expansion \(\alpha\), which conveys how much a length of material will extend per degree increase in temperature. To understand this in practical terms:

• If \(\alpha_{A} < \alpha_{B}\), as in our exercise, it means metal B expands more with heat compared to metal A.
• This difference in expansion causes the formation of a curve, with the metal having the larger \(\alpha\) forming the outer side.

Hence, by knowing the coefficients of expansion for different materials, we can predict and control how structures and devices behave in varying thermal conditions, ensuring safety and functionality.