Show that β≈3 α, by calculating the change in volume ΔV of a cube with sides of length L.

Name: Problem 21, Chapter 13: College Physics
Uploaded: 2019-12-20T06:04:34-08:00
Duration: 3 min 15 s
Channel: Sri Datta Vikas Buchemmavari
Description: Show that β≈3 α, by calculating the change in volume ΔV of a cube with sides of length L.

step 1 So we'll consider a cube of length L. So it's three sides are L, L, and L, which means the volum

Show that β≈3 α, by calculating the change in volume ΔV of a...

Key Concepts

Thermal Expansion

Thermal expansion refers to the tendency of matter to change its dimensions in response to a change in temperature. When materials are heated, their constituent particles move more vigorously, leading to an increase in distances between them and resulting in an overall expansion of the material. This concept is fundamental in understanding how dimensions and volume change with temperature fluctuations.

Linear Thermal Expansion Coefficient

The linear thermal expansion coefficient, commonly denoted as ?, quantifies the fractional change in length of a material per unit change in temperature. It is used when considering one-dimensional expansion, where each linear dimension increases by a factor proportional to both ? and the temperature change. This coefficient is crucial for predicting dimensional changes in structures and components as they are heated or cooled.

Volumetric Thermal Expansion Coefficient

The volumetric thermal expansion coefficient, denoted as ?, represents the fractional change in volume per unit temperature change. For many materials with isotropic expansion properties, the increase in volume can be expressed in terms of the changes in three orthogonal dimensions. Understanding ? is essential when dealing with systems where volume changes have significant practical implications, such as in fluid containers or structural applications.

Relationship Between Linear and Volumetric Expansion

For isotropic materials, where expansion is uniform in all directions, the volumetric expansion coefficient is approximately three times the linear expansion coefficient (? ? 3?). This relationship arises because a change in each of the three dimensions contributes to the overall volume change. This concept is typically demonstrated by considering a cube and analyzing how small changes in its length along each axis sum to produce a change in volume.

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