Show that β=3 α1 by calculating the infinitesimal change in volume d V of a cube with sides of l

Show that β=3 α1 by calculating the infinitesimal change in...

Key Concepts

Thermal Expansion

Thermal expansion describes the tendency of a material to change its dimensions in response to a change in temperature. This phenomenon is quantified using expansion coefficients that relate changes in length or volume directly to temperature variations.

Linear Thermal Expansion Coefficient

The linear thermal expansion coefficient (often denoted as ??) measures the fractional change in a material’s length per unit temperature change. It serves as a first-order approximation of how much a single dimension of an object will increase or decrease with temperature variations.

Volumetric Thermal Expansion Coefficient

The volumetric thermal expansion coefficient (typically denoted as ?) quantifies the fractional change in a material’s volume per unit temperature change. In isotropic materials, where expansion is uniform in all directions, ? is directly related to the linear expansion coefficient.

Infinitesimal Change and Differential Analysis

Utilizing infinitesimal calculus, one can calculate the differential change in a physical quantity such as volume (dV) when a small temperature change (dT) occurs. By expanding the expression for the new volume and neglecting higher-order terms, it becomes evident that for isotropic materials, the volumetric expansion coefficient is three times the linear expansion coefficient, leading to the relation ? = 3??.

Transcript

00:01 All righty.

00:01 So today we're on a mission to show that beta, the volumetric thermal expansion coefficient, is equal to three times alpha.

00:13 All right.

00:14 And we'll be using the fact that the d, dl equals l not times linear cofantle linear thermal expansion coefficient times delta t, which is essentially the same as delta l equals l0.

00:36 Times alpha times delta t and on this end i was meant to say d t d t meaning the same stuff pretty much let me start with this statement here this is how i'm going to start my derivation to to demonstrate this volume i know i know is going to equal length initial plus a change in length cubed because this is essentially going to be some new volume cubed, essentially.

01:18 So i'm going to expand that out, though.

01:24 So mathematically, we'll just expand that out.

01:26 We get l0 cubed plus 3, l0 squared times delta l, plus 3 delta l times delta l times l.

01:43 And as when i least on the end, we have the delta l cubed.

01:49 All right.

01:50 Now, this term right here on the end, these two terms actually, as a matter of fact, are negligible.

01:59 So we can pretty much put them roughly equal to zero in our stuff here.

02:03 Why, because these delta ls are typically sometimes on the magnitude of, you know, 10 to the minus six, and even smaller a lot of times.

02:12 And then even if you're going to square it on top of that or cubic, these values are very small.

02:16 So they're negligible.

02:17 We're not going to look at them.

02:18 We're going to look at these.