To tackle the problem of finding the effective coefficient of linear expansion for a composite bar made of two different materials, we can break it down into two parts as you've outlined. Let's dive into each part step by step.

Understanding the Effective Coefficient of Linear Expansion

When two materials are joined together, each material expands differently when subjected to temperature changes. The effective coefficient of linear expansion (α) for the composite bar can be derived from the individual coefficients of linear expansion of the materials involved.

Part (a): Deriving the Effective Coefficient of Linear Expansion

Let’s denote:

• α₁: Coefficient of linear expansion for material 1
• α₂: Coefficient of linear expansion for material 2
• L₁: Length of material 1
• L₂: Length of material 2
• L: Total length of the composite bar (L = L₁ + L₂)

When the temperature changes by ΔT, the change in length for each material can be expressed as:

• ΔL₁ = α₁ * L₁ * ΔT
• ΔL₂ = α₂ * L₂ * ΔT

The total change in length (ΔL) of the composite bar is the sum of the changes in length of both materials:

ΔL = ΔL₁ + ΔL₂ = α₁ * L₁ * ΔT + α₂ * L₂ * ΔT

Factoring out ΔT, we get:

ΔL = (α₁ * L₁ + α₂ * L₂) * ΔT

The effective coefficient of linear expansion (α) for the composite bar is defined as:

α = ΔL / (L * ΔT)

Substituting our expression for ΔL, we have:

α = (α₁ * L₁ + α₂ * L₂) / L

This shows that the effective coefficient of linear expansion for the composite bar is indeed given by:

α = (α₁ L₁ + α₂ L₂) / (L₁ + L₂)

Part (b): Designing a Composite Bar with Specific Properties

Now, let’s design a composite bar made of steel and brass that has a total length of 52.4 cm and an effective coefficient of linear expansion of 13 × 10^-6 /°C.

We know the coefficients of linear expansion for the materials:

• α_steel ≈ 11 × 10^-6 /°C
• α_brass ≈ 19 × 10^-6 /°C

Let L₁ be the length of the steel part and L₂ be the length of the brass part. We can express the total length as:

L₁ + L₂ = 52.4 cm

Substituting into the effective coefficient formula:

13 × 10^-6 = (11 × 10^-6 * L₁ + 19 × 10^-6 * L₂) / 52.4

Multiplying both sides by 52.4 gives:

13 × 10^-6 * 52.4 = 11 × 10^-6 * L₁ + 19 × 10^-6 * L₂

Calculating the left side:

13 × 10^-6 * 52.4 = 0.000682

Now we have:

0.000682 = 11 × 10^-6 * L₁ + 19 × 10^-6 * L₂

We can express L₂ as:

L₂ = 52.4 - L₁

Substituting this into the equation gives:

0.000682 = 11 × 10^-6 * L₁ + 19 × 10^-6 * (52.4 - L₁)

Expanding and rearranging leads to:

0.000682 = 11 × 10^-6 * L₁ + 0.0009996 - 19 × 10^-6 * L₁

0.000682 - 0.0009996 = (11 - 19) × 10^-6 * L₁

-0.0003176 = -8 × 10^-6 * L₁

Solving for L₁ gives:

L₁ = 39.7 cm

Substituting back to find L₂:

L₂