Problem 28 - Arihant AIEEE Physics

Question

An equilateral triangle \(A B C\) is formed by joining three rods of equal length and \(D\) is the mid-point of \(A B\). The coefficient of linear expansion for \(A B\) is \(\alpha_{1}\) and for \(A C\) and \(B C\) is \(\alpha_{2}\). Find the relation between \(\alpha_{1}\) and \(\alpha_{2}\), if distance \(D C\) remains constant for small changes in temperature (a) \(\alpha_{1}=\alpha_{2}\) (b) \(\alpha_{1}=4 \alpha_{2}\) (c) \(\alpha_{2}=4 \alpha_{1}\) (d) \(\alpha_{1}=\frac{1}{2} \alpha_{2}\)

Step-by-Step Solution

Answer

The relationship is \(\alpha_1 = 4\alpha_2\).

Step 1: Identify Initial Setup

An equilateral triangle \(ABC\) has sides \(AB = AC = BC = L\), and \(D\) is the midpoint of \(AB\), so \(AD = DB = \frac{L}{2}\). The coefficient of linear expansion for \(AB\) is \(\alpha_1\), and for \(AC\) and \(BC\) is \(\alpha_2\). We need to find the relation between \(\alpha_1\) and \(\alpha_2\) such that the distance \(DC\) remains constant with temperature change.

Step 2: Calculate Initial Distance DC

The point \(D\) being the midpoint of \(AB\), we have \(AD = \frac{L}{2}\). By symmetry in an equilateral triangle, initial distance \(DC\) (using the Pythagorean theorem in triangle \(ADC\)) is \(DC = \sqrt{AC^2 - AD^2} = \sqrt{L^2 - \left(\frac{L}{2}\right)^2} = \frac{\sqrt{3}}{2}L\).

Step 3: Consider Thermal Expansion

When temperature changes, the length of \(AB\) becomes \(L \times (1+\alpha_1\Delta T)\) and the lengths of \(AC\) and \(BC\) both become \(L \times (1+\alpha_2\Delta T)\). The new length of \(DC\) should remain the same for small \(\Delta T\).

Step 4: Calculate Changed Distance DC

If \(DC\) remains constant, the new \(DC\) using the expanded lengths should also be \(\frac{\sqrt{3}}{2}L\). With thermal expansion, \(AD\) becomes \(\frac{L}{2}(1+\alpha_1\Delta T)\) and \(AC\) becomes \(L(1+\alpha_2\Delta T)\). The new \(DC\) is given by:\\(DC_{new} = \sqrt{(AC_{new})^2 - (AD_{new})^2} = \sqrt{[L(1+\alpha_2\Delta T)]^2 - [\frac{L}{2}(1+\alpha_1\Delta T)]^2}\).

Step 5: Equate the Distances and Simplify

Since \(DC\) remains constant: \ \(\sqrt{[L(1+\alpha_2\Delta T)]^2 - [\frac{L}{2}(1+\alpha_1\Delta T)]^2} = \frac{\sqrt{3}}{2}L\).\Squaring both sides and simplifying gives: \ \(L^2(1+2\alpha_2\Delta T) - \frac{L^2}{4}(1+2\alpha_1\Delta T) = \frac{3}{4}L^2\). \ This simplifies to \(2\alpha_2\Delta T = \frac{1}{2} \alpha_1\Delta T\).

Step 6: Solve for Relationship Between Coefficients

Simplifying \(2\alpha_2\Delta T = \frac{1}{2} \alpha_1\Delta T\) gives\(\alpha_1 = 4\alpha_2\), after cancelling \(\Delta T\) from both sides and rearranging terms.

Key Concepts

Understanding an Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are of equal length, and consequently, all internal angles are equal, each measuring 60 degrees. This symmetry makes these triangles particularly interesting in mathematics and engineering.

• Each internal angle being 60 degrees provides a base for easy calculations in many geometric problems.
• The properties also imply that if you split an equilateral triangle down the middle, you end up with two congruent right triangles.

In the context of triangles formed by rods, such as in our exercise, the equilateral triangle allows for straightforward initial calculations when setting up problems that involve changes due to thermal expansion. The basic equality of sides helps engineers and physicists easily predict changes because all sides react similarly to external conditions, assuming the material is uniform.

Coefficient of Linear Expansion Explained

The coefficient of linear expansion, often denoted by the Greek letter alpha (\(\alpha\)), is a material property that quantifies the change in length per unit length of a material for a unit change in temperature. In simpler terms, it's a measure of how much a material expands or contracts when it's heated or cooled. Here are some key points to understand:

• This coefficient is crucial in scenarios where precise measurements are important, such as in the construction of structures, bridges, and even electronic circuit boards.
• The coefficient varies from material to material, meaning different materials expand at different rates under the same temperature changes.
• In our exercise, the rods forming sides of the triangle have different coefficients of linear expansion: \(\alpha_1\) for side AB and \(\alpha_2\) for sides AC and BC.

Understanding the coefficient is necessary to predict how different parts of a structure will behave when subjected to temperature variations. It helps assure the structure's integrity by allowing for compensations or adjustments in design.

Basics of Distance Calculation in Triangles

Calculating distances in triangles, specifically with thermal expansion considered, involves understanding the original geometry and then applying mathematical transformations that account for change in dimension.

• In our original setup, we computed the initial distance from point D to C using the Pythagorean theorem. This theorem is especially useful in right triangles, but it also extends to other triangle types when broken down into components.
• With thermal expansion, new lengths are calculated based on the original lengths modified by their respective coefficients of linear expansion and temperature change, \(\Delta T\).
• When calculating the modified distance DC, the new lengths are plugged into the formula derived from the Pythagorean theorem: \(DC_{new} = \sqrt{(AC_{new})^2 - (AD_{new})^2}\).

Understanding these calculations ensures accurate determination of longer-term structural changes in the material when subjected to varying environmental conditions. By substituting the expanded lengths into this distance calculation, problems like the one in the exercise can find the constraints or relationships required for certain conditions, such as unchanged distance.