6.15 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

Step 1: Identify the independent and dependent variables

• • Independent variable = blade angle / °
  • Dependent variable = voltage / V

Step 2: Choose an appropriate scale

• • The range of the blade angle is 0 – 90°
  • Ideally, every small square represents 10°

• • The range of the voltage is 0 – 2.2 V
  • Ideally, each small square represents 0.5 V

• • Both axes should occupy at least 50% of the grid

Step 3: Label the axes

• • The dependent variable (voltage / V) goes on the y-axis
  • The independent variable (blade angle / °) goes on the x-axis
  • Both axes should be labelled with a quantity and a unit

Step 4: Plot the points

• • Each point should be accurate within half a small square

Step 5: Draw a curve of best fit

• • The curve should be smooth with a roughly equal distribution of points on either side of the curve
  • It must start at (0,0) and peak at (20, 2.2)

• 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