{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 6. Linearizing a Non-Linear Relationship" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 6.1 Introduction" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The method of linear regression discussed in Chapter 5 provides a powerful tool for extracting information from data when there is a linear relationships between the variables measured. However, in many cases, the relationship between the experimental variables is not linear. As we will see in the next section, a straightforward graph of such a nonlinear relationship does not tell us very much. It is also more difficult to determine the “best fit” to a nonlinear graph. This chapter will introduce you to methods for creating a linear graph of quantities that we expect to be related nonlinearly. If you can artificially create a linear graph, you can use linear regression to extract information about the relationship that would otherwise be hard to obtain." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 6.2 An Example of Linearization" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Imagine an experiment where we want to determine an object’s acceleration as it slides from rest down a nearly frictionless incline by measuring its displacement as a function of time. The object should experience a constant acceleration along the incline with a magnitude $a = g\\ sin\\theta$, where $\\theta$ is the angle the incline makes with the horizontal. If we define the _z_-axis to point down along the incline, then theoretical displacement $d$ of the object as a funciton of the time $t$ since its release is\n", "\n", "\\begin{equation}\n", "d = z - z_0 = \\frac{1}{2}at^2, \\tag{6.1}\n", "\\end{equation}\n", "\n", "where $z_0$ is the starting position. \n", "\n", "If we made a graph of $d$ vs. $t$, we would expect a parabola as shown in Figure 6.1. However, it is difficult to distinguish a graph of a $d \\propto t^2$ relationship from one showing $d \\propto t^3$ or $d \\propto t^4$ or other relationships. Since there are a large number of relationships between $d$ and $t$ that could produce similar-looking curves, it is difficult to verify by just looking at the graph that our assumption that the object has constant acceleration is reasonable. In addition, there is no simple way to compute the value of $a$ from this graph. \n", "\n", "