Plotting Graphs (Edexcel International A Level Physics)

Plotting Graphs

• When plotting graphs, it is important to consider the importance of the following factors:

  • Selecting appropriate scales

  • Labelling axes with quantities and units

  • Carefully plotting the points

Choice of Scale

• When choosing a scale, it must be big enough to accommodate all the collected values using as much of the graph paper as possible

• At least half of the graph grid should be occupied in both the x and y directions

• Scales should be clearly indicated and have suitable, sensible ranges that are easy to work with

  • For example, scales with multiples of 3 should be avoided

• The scales should increase outwards and upwards from the origin

• Each axis should be labelled with the quantity that is being plotted, along with the correct unit

Labelling the Axes

• Label each axis with the name of the quantity and its unit

  • For example, F / N means force measured in Newtons

• The convention is that a forward slash ( / ) is used to separate the quantity and the unit

• In general:

  • The independent variable goes on the x-axis

  • The dependent variable goes on the y-axis

  

Example of labelled axes with the name of the variable, its symbol and its unit

Plotting the Points

• Points should be plotted so that they all fit on the graph grid and not outside it

• All values should be plotted, and the points must be precise to within half a small square

• Points must be clear, and not obscured by the line of best fit, and they need to be plotted with a sharp pencil so that they are thin

• There should be at least six points plotted on the graph, with any major outliers identified

Line or Curve of Best Fit

• There should be equal numbers of points above and below the line of best fit

  • Using a clear plastic ruler will help with this

• Not all lines will pass through the origin and nor should they be forced to

• The line (or curve) of best fit should not be too thick or joined dot-to-dot like a frequency polygon

• Anomalous values that have not been identified during the implementation stage should be ignored if they are obviously incorrect

  • This is because they will have a large effect on the gradient of the line of best fit

Determining the y-intercept

• The y-intercept is the y value obtained where the line crosses the y-axis at x = 0

• Values should be read accurately from the graph, with the scale on the y-axis being interpreted correctly

Logarithmic Scales

• Graphs can be logarithmic in nature

• A logarithmic (log) scale is a non-linear scale often used for analysing a large range of quantities

  • The log of a number is always greater than 1, so all log values are only positive

  • Hence, when drawing a log-log graph, the graph will only have a positive quadrant

• Often, in practicals, if the log of a value is required, then a separate column is needed in the data table to calculate this, for example:

Table of Results Using ln

A separate column is often needed to calculate ln(V)

• In the above case, the potential difference V is determined from a voltmeter, but the ln(V) values are calculated using a calculator

• The most common example of this in A level physics is in:

  • Radioactive decay

  • capacitor charge and discharge equations

Using Natural logs (ln)

• Taking natural logs (ln) of an equation with an exponential function means the equation can become linear i.e. in the form y = mx + c

• Straight-line graphs tend to be more useful than curves for interpreting data

  • Gradients and intercepts are useful values that can be seen from a straight-line graph 

• Nuclei decay exponentially, therefore, to achieve a straight-line plot, logarithms can be used

• Take the exponential decay equation for the number of nuclei

N = N₀ e^–λt

• Taking the natural logs of both sides

ln N = ln (N₀e^–λt) = ln (N₀) + ln(e^–λt)

ln N = ln (N₀) − λt

• In this form, this equation can be compared to the equation of a straight line

y = mx + c

ln N = − λt + ln (N₀) 

• Where:

  • y = ln (N) is plotted on the y-axis

  • x = t is plotted on the x-axis

  • gradient, m = −λ

  • y-intercept = ln (N₀) is a constant

• The exponential decay version of the equation could produce a curve, whilst the ln(N) equation produces a straight line

Linear decay curve vs. a log graph