Thermal Expansion Coefficient Lab Report

Determination of the Thermal Expansion Coefficient Using a Second Order Lagrangian Polynomial

Jeffery Jep Brown1, Siti Nashirah binti Jeruis1, Mohammad Amir Luqman bin Zainal1, Mohd Nadzreen bin Chamar1

1Faculty of Mechanical and Manufacturing Engineering

Universiti Tun Hussein Onn Malaysia, 86400 Parit Raja, Batu Pahat, Johor, Malaysia.

Abstract

This journal is concerned with the application of interpolation in real life application. There are several methods involved in Interpolation method. For example, Lagrange interpolation, Newton’s Divided Difference, and Spline interpolation. This journal is about the determination of Second Order Lagrange interpolating polynomial. It is often used to estimate the value of the intermediate value

The table of the coefficient of thermal expansion vs. temperature data is given in Table 2.

Table 1 Thermal expansion coefficient as a function of temperature:

Temperature,

Thermal Expansion Coefficient,

90 6.47

60 6.00

30 5.58

0 4.72

-30 3.58

-60 2.45

If the coefficient of thermal expansion needs to be calculated at the average temperature of , determine the value of the coefficient of thermal expansion at using a second order Lagrangian polynomial. Find the absolute relative approximate error for the second order polynomial interpolation.

Solution

For second order Lagrange polynomial interpolation (also called quadratic interpolation), the coefficient of thermal expansion given by

Figure 3 Quadratic interpolation.

Since we want to find the coefficient of thermal expansion at , we need to choose data points that are closest to , that also bracket , to evaluate it. The three points are , and .

gives