An aluminum block measures 1.000 cm by 2.000 cm by 3.000 cm . Find its volume after a 100^∘ C

step 1 The given sphere is of aluminum. So for the given sphere diameter diameter is given 20 centimete

An aluminum block measures 1.000 cm by 2.000 cm by 3.000...

An aluminum block measures $1.000 \mathrm{~cm}$ by $2.000 \mathrm{~cm}$ by $3.000 \mathrm{~cm} .$ Find its volume after a $100^{\circ} \mathrm{C}$ temperature increase in two ways: (a) using aluminum's linear expansion coefficient on each side and then determining the new volume and (b) using the volume expansion coefficient. Your results should verify the rela$\operatorname{tion} \beta=3 \alpha$.

Key Concepts

Thermal Expansion

Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature. Materials typically increase in size as temperature rises due to increased molecular motion, which requires space, and decrease when cooled. This concept is fundamental in understanding how objects behave under temperature variations and is applicable in various scientific and engineering applications.

Linear Expansion Coefficient

The linear expansion coefficient, often denoted as alpha (?), quantifies the fractional change in length per unit temperature change for a material. It is used to predict how the dimensions of a solid object change in each linear direction when the temperature is altered. This concept is crucial for calculating expansions in objects with well-defined lengths, such as rods, beams, or any object where one-dimensional expansion is considered.

Volume Expansion Coefficient

The volume expansion coefficient, denoted as beta (?), describes the fractional change in volume per unit temperature change. It provides a measure of how the overall volume of an object or substance changes as the temperature varies. This coefficient is important for applications where changes in volume need to be predicted, such as in liquids and gases, or solids where volumetric changes are of interest.

Relation between Linear and Volume Expansion

For isotropic materials that expand uniformly in all directions, the volume expansion coefficient is approximately three times the linear expansion coefficient (? = 3?). This relationship arises because the expansion in each of the three spatial dimensions contributes to the overall volume change. Understanding this relation helps in cross-verifying calculations of thermal expansion whether approached through linear dimensions or directly via volume changes.

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