# Linearising Data

## Introduction

In general it is best for graphs to show a linear relationship ie. relationships where the y values are directly proportional the x values. This makes it much easier to see deviations from a straight line. It also makes it possible to express the relationship between the experimental quantities and those predicted by theory. For data which is non-linear it is first necessary to linearise it. This is illustrated in figure 1 and 2.

The data is clearly non-linear, but it is difficult to determine what the relation is.

^{2}in Earth's gravitional field.

By plotting time^{2} against distance, shown in figure 2, we can clearly see a linear trend. Therefore we can determine that the trend is an *x ^{2}* power law.

## Using Excel

*"Consider the data shown in this spreadsheet. It shows data from an object accelerating as a result of Earth's gravitational field (errors on each individual point have been ignored)."*

*"Use this data to compute g, acceleration due to gravity, and its associated error."*

How do we go about doing this? Well, firstly we have to consider a suitable model.*

The object is falling a small distance relative to the radius of the Earth. Therefore, it is a reasonable approximation to assume that *g* is a constant. From the equations of motion for constant acceleration (or "SUVAT" equations) we know that

Where all the symbols have their usual meaning (*s* is distance, *u* is the initial velocity etc).

Given that *u* = 0 in this case, we can reduce this to

This is now in a suitable form to linearise. Compare this to the equation for a straight line

If we set *y* to be our dependent variable *s* and *x* to be *t ^{2}*, the result is a gradient whose value corresponds to half the acceleration,

*g*, and the intercept

*c*= 0. I.e.

So how do we do this in Excel?

1.Create a new column on the spreadsheet for the time^{2} variable. Then in the top row, type in the command to square the time data point on that row, i.e. in this case "**=B7^2**". This is shown in figure 3. Press enter to accept the function.

2.Double click the small square circled in red on figure 4 so that it repeats the calculation across all the time data points

3. Apply the linest function to the data, where the *y* variable is *s* and the *x* variable is *t ^{2}* (not just

*t*).

If you are unsure how to do this, follow the guide for using linest available **[link required]**

You should end up with the data shown in figure 6.

4. Using equation 4, we know that the gradient, *m*, is half of the acceleration due to gravity *g*. Therefore multiply *m*, and its error, by 2 to obtain our measured value for *g*.This should give you a final answer of

(9.8 ± 0.3) m s^{-2}