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1.2.3Development of Practical Skills in Physics

Plotting and Interpreting Graphs

5 min read 9 parts Lesson 10 of 29 1 quiz

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Lesson summary

1.2.3 Plotting and Interpreting Graphs — OCR A Level Physics A notes: Plot a clear graph with error bars and a best-fit line, and use its gradient and intercept (with uncertainties) to find physical quantities.A graph averages out random scatter across many points and shows a rel…

  • straight line
  • Line of best fit
  • y-intercept
  • large triangle
  • worst acceptable line
  • area under a graph

Section 1 of 9Why graphs are central to physics

A graph averages out random scatter across many points and shows a relationship at a glance.

A single calculation from one pair of readings can be wrong by a lot; a gradient taken from a good best-fit line is far more reliable.
Straight line
Where possible, design an experiment so that the physics law becomes a straight line, because lines are easy to judge by eye and their gradient and intercept have physical meaning.

Section 2 of 9Plotting a good graph

Axes
Axes independent variable on the x-axis, dependent on the y-axis. Label each with quantity and unit, such as T2 / s2.
Scale
Scale choose simple scales (1, 2 or 5 units per large square, never 3 or 7).
  • The plotted points should fill more than half the grid in both directions.
  • The origin only needs to be shown if you need the intercept or the line passes through it.
Points
Points plot with a small cross or a dot in a circle, accurate to within half a small square.
Error bars
Error bars draw vertical and/or horizontal bars of length equal to the absolute uncertainty in each direction (for example ±0.2 s becomes a bar 0.4 s long).
Line of best fit
Line of best fit a single thin line, drawn with a clear ruler, with roughly equal numbers of points above and below it, passing through every error bar if possible. For a curve, draw one smooth continuous curve, not a join-the-dots.

Section 3 of 9Straight-line graphs and linearising

Any straight line obeys y = mx + c, where m is the gradient and c the y-intercept. A physics equation is matched to this form by choosing what to plot.

RelationshipPlot y against xGradient
T = 2π√(l/g)T2 against l4π2/g
s = ut + ½at2s/t against t½a (intercept u)
V = E − IrV against I−r (intercept E)
N = N0e−λtln N against t−λ

For power laws y = axn, plotting lg y against lg x gives a straight line of gradient n and intercept lg a.

Section 4 of 9Gradient and intercept with uncertainty

To find a gradient use a large triangle that uses more than half the length of the line, with the two points read from the line (not from data points) and well apart. Gradient = Δy / Δx, with units. Show the coordinates you used.

The uncertainty in the gradient comes from the worst acceptable line: a second straight line, steeper or shallower than the best fit, that still passes through all of the error bars. Then

absolute uncertainty in gradient = (best gradient − worst gradient) in magnitude, and the percentage uncertainty is this divided by the best gradient. The same approach gives the uncertainty in the intercept.

The area under a graph can also carry meaning, for instance area under a force–extension graph is work done, and the area under a velocity–time graph is displacement.

Section 5 of 9Interpreting what a graph shows

State whether the line passes through the origin (proportionality) or has a non-zero intercept (offset, such as a zero error or a constant term).

Interpreting what a graph shows
A curve that flattens suggests saturation or a limit; a curve that steepens suggests a power law or exponential.
Anomalous points should lie clearly off the line and be commented on.
A negative intercept where a positive one is expected often indicates a systematic error, which connects to Precision, Accuracy and Experimental Limitations (1.2.9).

Section 6 of 9Checking a graph before you use it

Before trusting a gradient, run through a quick checklist.

1
Are the axes labelled with quantity and unit?
2
Do the points occupy most of the grid?
3
Are the error bars drawn to scale?
4
Does the line of best fit go through the centre of the points rather than favouring the first or last?
5
Is the triangle large and are its corners read from the line?
6
A graph that passes this check will usually give a gradient with an uncertainty of a few percent.

Section 7 of 9Worked examples

  1. For a pendulum, a graph of T2 / s2 against l / m is a straight line. Two points on the best-fit line are (0.20, 0.82) and (1.00, 4.10). Find g.

  2. A cell's terminal p.d. V is plotted against current I. The best-fit line has intercept 1.52 V and passes through (0.50 A, 1.30 V). Find the e.m.f. and internal resistance.

  3. The best-fit gradient of a graph is 2.40 N m−1. The steepest and shallowest acceptable lines have gradients 2.55 and 2.28 N m−1. Give the gradient with its uncertainty.

Section 8 of 9Key terms

TermMeaning
Line of best fitA single line passing as close as possible to all the plotted points, with scatter balanced above and below.
GradientChange in y divided by change in x for a straight line.
InterceptThe value of y where the line crosses the y-axis (x = 0).
Error barA bar showing the range of uncertainty in a plotted value.
LineariseRearrange an equation so that it plots as a straight line.
Worst acceptable lineThe steepest or shallowest straight line that still passes through all error bars.

Section 9 of 9Exam technique

1
Joining the data points dot to dot instead of drawing a smooth best-fit line or curve.
2
Using data points rather than points on the line when calculating the gradient.
3
Using a tiny triangle, which magnifies reading errors.
4
Forgetting that gradient has units (and that the sign matters).
5
Choosing awkward scales such as 3 or 7 squares to a unit, which causes plotting errors.
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