To tackle the problem of the Golden Gate Bridge cable's behavior under temperature changes, we need to consider how temperature affects the length and sag of the cable. The cable behaves like a parabolic curve, and we can use the coefficient of linear expansion to find the changes in length and sag due to temperature variations. Let's break this down step by step.

Understanding the Parameters

First, we have the following information:

• Distance between the towers (span) = 4200 ft
• Sag of the cable at 50º F = 470 ft
• Coefficient of linear expansion (α) = 6.5 × 10^-6 /ºF
• Initial temperature = 10º F
• Final temperature = 90º F

Calculating Change in Length of the Cable

The change in length of the cable due to temperature can be calculated using the formula:

ΔL = L₀ × α × ΔT

Where:

• ΔL = change in length
• L₀ = original length of the cable
• α = coefficient of linear expansion
• ΔT = change in temperature

First, we need to determine the original length of the cable (L₀). For a parabolic cable, the length can be approximated using the formula:

L₀ ≈ span + (8/3)(sag)

Substituting the values:

L₀ ≈ 4200 ft + (8/3)(470 ft)

L₀ ≈ 4200 ft + 1253.33 ft ≈ 5453.33 ft

Now, we can find the change in temperature:

ΔT = 90º F - 10º F = 80º F

Now substituting into the length change formula:

ΔL = 5453.33 ft × (6.5 × 10^-6 /ºF) × 80º F

ΔL ≈ 5453.33 ft × 0.0000065 × 80 ≈ 2.84 ft

Determining Change in Sag

Next, we need to calculate the change in sag due to the temperature increase. The sag of a parabolic cable is influenced by its length and the tension in the cable. As the cable expands, the tension changes, which can affect the sag. However, for simplicity, we can use a linear approximation for small changes.

The change in sag can be approximated using the formula:

ΔSag = (ΔL / L₀) × sag

Substituting the values we have:

ΔSag = (2.84 ft / 5453.33 ft) × 470 ft

ΔSag ≈ 0.000520 × 470 ft ≈ 0.244 ft

Final Results

In summary, after a temperature change from 10º F to 90º F:

• The change in length of the cable is approximately 2.84 ft.
• The change in sag is approximately 0.244 ft.

This analysis shows how temperature variations can significantly affect structural components like the cables of the Golden Gate Bridge, emphasizing the importance of considering thermal expansion in engineering design.