To understand how to derive the effective coefficient of linear expansion for a composite bar made of two different materials, we need to consider how each material expands when subjected to a change in temperature. The effective coefficient of linear expansion is a way to express how the entire composite bar will behave as a single entity when heated or cooled.

Defining Linear Expansion

Linear expansion refers to the change in length of a material when it is subjected to a temperature change. The formula for linear expansion is given by:

ΔL = α × L × ΔT

Where:

• ΔL is the change in length.
• α is the coefficient of linear expansion of the material.
• L is the original length of the material.
• ΔT is the change in temperature.

Understanding the Composite Bar

In our case, we have two bars: metal A with length L_A and coefficient of linear expansion α_A, and metal B with length L_B and coefficient of linear expansion α_B. The total length of the composite bar is:

L = L_A + L_B

Calculating the Change in Length

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

• For bar A: ΔL_A = α_A × L_A × ΔT
• For bar B: ΔL_B = α_B × L_B × ΔT

Finding the Total Change in Length

The total change in length of the composite bar is the sum of the changes in length of both bars:

ΔL = ΔL_A + ΔL_B

Substituting the expressions we derived:

ΔL = (α_A × L_A × ΔT) + (α_B × L_B × ΔT)

This can be factored to:

ΔL = (α_A × L_A + α_B × L_B) × ΔT

Effective Coefficient of Linear Expansion

Now, to find the effective coefficient of linear expansion α_eff for the entire composite bar, we can use the relationship:

ΔL = α_eff × L × ΔT

Substituting the total length L = L_A + L_B into this equation gives:

ΔL = α_eff × (L_A + L_B) × ΔT

Equating the Two Expressions

Now we can set the two expressions for ΔL equal to each other:

(α_A × L_A + α_B × L_B) × ΔT = α_eff × (L_A + L_B) × ΔT

Since ΔT is common on both sides, we can cancel it out (assuming ΔT is not zero):

α_A × L_A + α_B × L_B = α_eff × (L_A + L_B)

Solving for the Effective Coefficient

Finally, we can solve for α_eff:

α_eff = (α_A × L_A + α_B × L_B) / (L_A + L_B)

This expression shows how the effective coefficient of linear expansion for the composite bar is a weighted average of the coefficients of the individual materials, taking into account their respective lengths. Thus, we have derived the formula:

α_eff = [α_A × L_A + α_B × L_B] / L

This derivation illustrates the fundamental principles of thermal expansion and how different materials can combine to exhibit unique properties when subjected to temperature changes.