To determine the distance \( x \) that the center of the bar rises due to thermal expansion, we can use the formula for linear expansion. The formula is given by:

Understanding Linear Expansion

The linear expansion of a material can be expressed as:

\( \Delta L = L_0 \cdot \alpha \cdot \Delta T \)

Where:

• \( \Delta L \) = change in length
• \( L_0 \) = original length of the bar
• \( \alpha \) = coefficient of linear expansion
• \( \Delta T \) = change in temperature

Given Values

From your question, we have:

• \( L_0 = 3.77 \, \text{m} \)
• \( \alpha = 25 \times 10^{-6} \, \text{°C}^{-1} \)
• \( \Delta T = 32 \, \text{°C} \)

Calculating the Change in Length

Now, we can substitute these values into the linear expansion formula:

\( \Delta L = 3.77 \, \text{m} \cdot (25 \times 10^{-6} \, \text{°C}^{-1}) \cdot (32 \, \text{°C}) \)

Calculating this step-by-step:

• First, calculate \( 25 \times 10^{-6} \cdot 32 = 0.0008 \)
• Now, multiply this by \( 3.77 \, \text{m} \): \( 3.77 \cdot 0.0008 = 0.003016 \, \text{m} \)

Finding the Rise at the Center

The total change in length \( \Delta L \) represents the total expansion of the bar. However, since the bar has a crack at its center and is fixed at both ends, the upward buckling at the center will be half of this total expansion. Therefore, the distance \( x \) to which the center rises is:

\( x = \frac{\Delta L}{2} = \frac{0.003016 \, \text{m}}{2} = 0.001508 \, \text{m} \)

Final Result

Thus, the distance \( x \) that the center of the bar rises is approximately:

\( x \approx 0.00151 \, \text{m} \) or \( 1.51 \, \text{mm} \)

This calculation illustrates how thermal expansion can lead to structural changes in materials, especially when they are constrained. Understanding these principles is crucial in fields like engineering and materials science, where temperature variations can significantly impact the integrity of structures.

To solve the problem of how much the center of the bar rises due to thermal expansion, we can use the formula for linear expansion, which relates the change in length of an object to its original length, the coefficient of linear expansion, and the change in temperature. The formula is given by:

Understanding Linear Expansion

The formula for linear expansion is:

ΔL = L₀ × α × ΔT

• ΔL is the change in length.
• L₀ is the original length of the bar.
• α is the coefficient of linear expansion.
• ΔT is the change in temperature.

Given Values

From the problem, we have:

• L₀ = 3.77 m
• α = 25 × 10^-6 /°C
• ΔT = 32 °C

Calculating the Change in Length

Now, we can plug these values into the linear expansion formula:

ΔL = 3.77 m × (25 × 10^-6 /°C) × 32 °C

Calculating this step-by-step:

• First, calculate the product of the coefficient of linear expansion and the change in temperature:

25 × 10^-6 × 32 = 800 × 10^-6 = 0.0008

• Next, multiply this result by the original length:

ΔL = 3.77 m × 0.0008 = 0.003016 m

Finding the Rise at the Center

The total change in length, ΔL, represents how much the bar expands. However, since the bar has a crack at its center, it will buckle upward. The upward displacement at the center, x, can be approximated as half of the total change in length, assuming the bar bends symmetrically.

x = ΔL / 2 = 0.003016 m / 2 = 0.001508 m

Final Result

Thus, the distance to which the center of the bar rises is approximately:

x ≈ 0.001508 m or 1.508 mm.

This calculation illustrates how thermal expansion can lead to structural changes in materials, especially when they are constrained in certain ways, such as having a crack. Understanding these principles is crucial in fields like engineering and materials science, where temperature changes can significantly affect the integrity of structures.