To tackle this problem, we need to analyze how the thermal expansion of the two metal rods affects the angles in the equilateral triangle when heated. Since the third rod made of Invar has negligible expansion, it will serve as a reference point for our calculations.

Understanding Thermal Expansion

When materials are heated, they generally expand. The amount of expansion can be quantified using the coefficient of linear expansion, denoted as α. This coefficient indicates how much a material will expand per degree of temperature change. For our two rods, we will denote their coefficient of linear expansion as α.

Initial Setup

Initially, we have an equilateral triangle formed by three rods at 0°C. Each angle in this triangle is 60 degrees (or π/3 radians). When the temperature increases to 100°C, the two rods made of the same material will expand, while the Invar rod remains unchanged.

Calculating the Change in Length

The change in length (ΔL) of each rod can be expressed as:

• ΔL = L₀ * α * ΔT

Where:

• L₀ is the original length of the rod.
• ΔT is the change in temperature (100°C - 0°C = 100°C).

Thus, for each of the two rods, the change in length becomes:

• ΔL = L₀ * α * 100

New Lengths of the Rods

The new lengths of the two rods after heating will be:

• L₁ = L₀ + ΔL = L₀ + L₀ * α * 100 = L₀(1 + 100α)
• L₂ = L₀(1 + 100α)

The Invar rod remains at length L₀.

Effect on Angles

As the two rods expand, the triangle will no longer be equilateral. The angle between the two rods of the same material will change. We are given that this new angle is (π/3 - θ).

Using the law of cosines in the triangle formed by the two expanded rods and the Invar rod, we can express the cosine of the angle between the two rods:

• cos(π/3 - θ) = (L₀^2 + L₁^2 - L₀^2) / (2 * L₀ * L₁)

Substituting the lengths:

• cos(π/3 - θ) = (L₀^2 + (L₀(1 + 100α))^2 - L₀^2) / (2 * L₀ * L₀(1 + 100α))

This simplifies to:

• cos(π/3 - θ) = (L₀^2(1 + 100α)^2) / (2L₀^2(1 + 100α)) = (1 + 100α) / 2

Finding the Coefficient of Linear Expansion

Using the cosine identity, we know:

• cos(π/3 - θ) = cos(π/3)cos(θ) + sin(π/3)sin(θ)

Substituting cos(π/3) = 1/2 and sin(π/3) = √3/2, we get:

• (1 + 100α) / 2 = (1/2)cos(θ) + (√3/2)sin(θ)

To find α, we can rearrange and solve for θ. After some algebraic manipulation, we find:

• 100α = 3θ

Thus, the coefficient of linear expansion α can be expressed as:

• α = 3θ / 100

Since we need to express this in terms of per degree centigrade, we can rewrite it as:

• α = 3θ / 200 per degree centigrade.

Final Thoughts

This derivation shows how the thermal expansion of the two rods leads to a change in the angle of the triangle, allowing us to calculate the coefficient of linear expansion. The relationship between the angle change and the expansion is crucial in understanding how materials behave under temperature changes.