A 1.00 -km steel railroad rail is fastened securely at both...

A 1.00 -km steel railroad rail is fastened securely at both ends when the temperature is $20.0^{\circ} \mathrm{C}$ . As the temperature increases, the rail buckles, taking the shape of an arc of a vertical circle. Find the height $h$ of the center of the rail when the temperature is $25.0^{\circ} \mathrm{C}$ . (You will need to solve a transcendental equation.)

Key Concepts

Thermal Expansion

Thermal expansion describes how the dimensions of a material change in response to temperature variations, characterized by the material's coefficient of thermal expansion. In problems like this, the original length of an object increases with temperature, and the linear thermal expansion formula (?L = ?L??T) is used to determine the change in length due to a temperature difference.

Buckling and Constrained Structures

Buckling occurs when a structure, such as a rail fixed at both ends, experiences compressive stresses that lead to lateral deflection. When thermal expansion is prevented by the fixed supports, the induced stress can cause the structure to deform by buckling, adopting a curved shape to relieve the strain.

Circular Arc Geometry

The buckled rail forms a section of a circle, and understanding its geometry involves relationships among the arc length, chord length, the circle radius, and the height (sagitta) of the arc. This geometric interpretation is essential for calculating the deflection or the vertical displacement (height at the center) of the buckled beam.

Transcendental Equations

Transcendental equations are mathematical equations that involve non-algebraic functions, such as trigonometric or exponential functions, which do not yield to standard algebraic solution methods. In this context, finding the height of the buckled rail requires solving a transcendental equation that arises from equating the arc length with the expanded length, necessitating numerical approaches for an accurate solution.

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