9The daily temperature variation of the Golden Gate bridge in San Francisco can be in excess of 20 C° .The bridge is approximately 2 km long and is made of steel (with an asphalt covering on the roadway).

(a) What is the approximate change in length of the bridge with this temperature variation?

(A) 4.4 cm (B) 44cm (C) 4.4 m (D) 44 m

(b) if the bridge builders neglected to include expansion joints, then approximately how large of a “bump” would form in the middle of the bridge when it expanded?

(A) 2.1 cm (B) 21 cm (C) 2.1 m (D) 21 m

To tackle the questions about the Golden Gate Bridge and its temperature-induced expansion, we need to apply some principles of thermal expansion. Let's break this down step by step.

Understanding Thermal Expansion

When materials like steel are subjected to temperature changes, they expand or contract. The formula for linear expansion is:

ΔL = L₀ × α × ΔT

• ΔL = change in length
• L₀ = original length of the material
• α = coefficient of linear expansion for the material (for steel, it's approximately 11 × 10^-6 °C^-1)
• ΔT = change in temperature

Calculating the Change in Length

Given that the length of the Golden Gate Bridge is about 2 km (or 2000 meters) and the temperature variation is 20 °C, we can substitute these values into the formula.

First, let's identify the values:

• L₀ = 2000 m
• α = 11 × 10^-6 °C^-1
• ΔT = 20 °C

Now, substituting these into the formula:

ΔL = 2000 m × (11 × 10^-6 °C^-1) × (20 °C)

ΔL = 2000 × 11 × 20 × 10^-6

ΔL = 440000 × 10^-6 m = 0.44 m = 44 cm

Thus, the approximate change in length of the bridge with this temperature variation is (B) 44 cm.

Estimating the Bump Formation

If the builders neglected to include expansion joints, the bridge would not be able to accommodate this expansion, leading to a "bump" in the middle. To estimate the height of this bump, we can consider how the bridge would deform under the expansion.

Assuming the bridge expands uniformly, the bump can be approximated as a triangle where the base is the length of the bridge and the height is the amount of expansion. The maximum height of the bump can be estimated by considering the change in length divided by the length of the bridge.

Using the formula for the height of a triangle:

Height = ΔL / 2

Substituting the values we found:

Height = 44 cm / 2 = 22 cm

Since we are looking for the closest option, we can round this to approximately (B) 21 cm.

Summary of Answers

• Change in length of the bridge: (B) 44 cm
• Height of the bump formed: (B) 21 cm

Understanding these concepts of thermal expansion helps us appreciate the engineering challenges faced in constructing large structures like the Golden Gate Bridge. Proper planning for temperature variations is crucial to ensure safety and longevity.