Solve Problem 15.8 using the result of Problem 15.9 . We...

Solve Problem $15.8$ using the result of Problem $15.9 .$ We have $$ \Delta \rho \approx-\rho_{0} \beta \Delta T $$ Hence $\quad \Delta \rho \approx-\left(13600 \mathrm{~kg} / \mathrm{m}^{3}\right)\left(182 \times 10^{-6 \circ} \mathrm{C}^{-1}\right)\left(50.0^{\circ} \mathrm{C}\right)=-124 \mathrm{~kg} / \mathrm{m}^{3}$ and $\quad \rho_{50^{\circ} \mathrm{C}}=\rho_{0^{+} \mathrm{C}}-124 \mathrm{~kg} / \mathrm{m}^{3}=13.5 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$

Solve Problem $15.8$ using the result of Problem $15.9 .$
We have
$$
\Delta \rho \approx-\rho_{0} \beta \Delta T
$$
Hence $\quad \Delta \rho \approx-\left(13600 \mathrm{~kg} / \mathrm{m}^{3}\right)\left(182 \times 10^{-6 \circ} \mathrm{C}^{-1}\right)\left(50.0^{\circ} \mathrm{C}\right)=-124 \mathrm{~kg} / \mathrm{m}^{3}$
and $\quad \rho_{50^{\circ} \mathrm{C}}=\rho_{0^{+} \mathrm{C}}-124 \mathrm{~kg} / \mathrm{m}^{3}=13.5 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$

Key Concepts

Linear Approximation in Thermal Physics

When temperature changes are relatively small, it is common in thermal physics to use linear approximations to simplify complex relationships. For instance, the relationship between density change and temperature can be approximated by ?? ? -????T, where ?? is the original density, ? is the volumetric thermal expansion coefficient, and ?T is the temperature change. This approximation makes it easier to calculate the expected change in density without resorting to more complex nonlinear models.

Density Change Due to Temperature Variation

Density is defined as mass per unit volume. As temperature increases, the thermal expansion causes the volume of a material to increase, leading to a decrease in density. This inverse relationship is especially important in systems where temperature variations play a critical role, and it can often be approximated using linear relationships for small temperature changes.

Coefficient of Thermal Expansion

The coefficient of thermal expansion is a parameter that quantifies how much a material expands per unit temperature increase. When applied to volumetric expansion, it defines the fractional change in volume per degree change in temperature. This property helps in predicting how density and other related properties change with temperature, since density is inversely related to volume when mass is constant.

Thermal Expansion

Thermal expansion refers to the tendency of matter to change in volume in response to a change in temperature. This concept is fundamental in understanding how a material’s dimensions—including length, area, or volume—vary when heated or cooled. In many practical applications, the change in volume due to an increase in temperature can affect the performance and safety of the material or structure.

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