A machinist bores a hole of diameter 1.30 cm in a steel plate that is at 26.5^∘ C. What is the

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A machinist bores a hole of diameter 1.30 cm in a steel...

A machinist bores a hole of diameter $1.30 \mathrm{~cm}$ in a steel plate that is at $26.5^{\circ} \mathrm{C}$. What is the cross-sectional area of the hole (a) at $26.5^{\circ} \mathrm{C}$ and $(\mathrm{b})$ when the temperature of the plate is increased to $170^{\circ} \mathrm{C} ?$ Assume that the coefficient of linear expansion remains constant over this temperature range.

Key Concepts

Hole Expansion

Hole expansion is the concept that, when a material expands due to a temperature change, the holes within the material also expand proportionally. This counterintuitive idea means that the relative dimensions of the holes remain consistent with the rest of the material, following the same thermal expansion rules.

Area Thermal Expansion

Area thermal expansion describes how the area of a two-dimensional object changes with temperature. For isotropic materials, where expansion is uniform in all directions, the area scaling factor is approximately the square of the linear expansion factor (i.e., (1 + ??T)²). This relationship allows calculation of the new area after a temperature change.

Coefficient of Linear Expansion

The coefficient of linear expansion is a material-specific constant that quantifies the fractional change in length per unit temperature change. It is essential in predicting how much a dimension (such as a rod or a diameter) of a material will increase or decrease when the temperature is altered.

Thermal Expansion

Thermal expansion refers to the tendency of matter to change its dimensions—length, area, or volume—as a function of temperature. When a material is heated, its molecules vibrate more intensely and tend to occupy more space, causing the material to expand in all directions if it is free to move.

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