A machinist bores a hole of diameter 1.35 cm in a steel plate at a temperature of 25.0^∘ C . Wha

step 1 So firstly we wish to calculate the area of the hole at 25 degrees Celsius. So we'll call this a

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

A machinist bores a hole of diameter 1.35 $\mathrm{cm}$ in a steel plate at a temperature of $25.0^{\circ} \mathrm{C}$ . What is the cross-sectional area of the bole $(\mathrm{a})$ 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 17.26 .

Key Concepts

Thermal Expansion

Thermal expansion is the tendency of a material to change its dimensions with changes in temperature. In solids, the increase or decrease in linear dimensions (such as lengths, diameters, or thicknesses) is directly proportional to the temperature change, making it a central concept in understanding how structures and components behave under varying thermal conditions.

Coefficient of Linear Expansion

The coefficient of linear expansion quantifies how much a material’s linear dimensions change per degree change in temperature. It is a material-specific constant that, when multiplied by the original dimension and the temperature change, gives the change in length. This coefficient is vital in engineering designs where precise tolerances must be maintained over temperature variations.

Relationship Between Linear and Area Expansion

While linear expansion deals with changes in one-dimensional lengths, the expansion of areas involves changes in two dimensions. The cross-sectional area of a shape, such as a circle, is proportional to the square of its linear dimension (like its diameter). Therefore, an increase in temperature that changes a linear dimension by a certain factor will cause the area to change approximately by the square of that factor, making area expansion roughly dependent on twice the coefficient of linear expansion for small temperature ranges.

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