A machinist bores a hole of diameter 1.35 cm in a steel...

A machinist bores a hole of diameter 1.35 cm in a steel plate at a temperature of $25.0^{\circ} \mathrm{C}$ . What is the cross-sectional area of the hole (a) at $25.0^{\circ} \mathrm{C}$ and $(\mathrm{b})$ when the temperature of the plate is increased to $175^{\circ} \mathrm{C}$ ? Assume that the coefficient of linear expansion remains constant over this temperature range. (Hint: See Exercise $17.20.$)

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Key Concepts

Thermal Expansion

Thermal expansion is the phenomenon where materials change their dimensions in response to variations in temperature. This change can affect length, area, or volume, depending on the dimensionality of the geometry in consideration. It is an important concept in materials science and engineering because it influences the performance and reliability of structures and components under temperature fluctuations.

Coefficient of Linear Expansion

The coefficient of linear expansion is a material-specific constant that quantifies the change in length per unit length per unit temperature change. It is a critical parameter used to predict and calculate how much an object will expand or contract when subjected to a temperature change, and it forms the basis for further calculations in higher-dimensional thermal expansions like area or volume.

Area Expansion

Area expansion describes the change in a two-dimensional measurement of an object (such as cross-sectional area) due to temperature changes. For isotropic materials, the fractional change in area is approximately twice the coefficient of linear expansion multiplied by the temperature change. This concept is instrumental when determining how features like holes or slots in materials expand with increasing temperatures.

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