Graphs and Analysis

Label each axis with the quantity and unit, in the form "length / m". The independent variable, the one you control, goes on the $x$-axis.

Choose scales so that the points fill at least half the grid in each direction, using sensible increments such as 1, 2 or 5 per square. Do not force the origin onto the graph unless it is needed.

Plot points precisely as small crosses, then draw a single smooth $\textbf{line of best fit}$ with roughly equal numbers of points on either side. Do not join the dots. Identify and ignore anomalous points, but say that you have done so.

Compare the equation with $y = mx + c$. Whatever you plot on the $y$-axis must match $y$; whatever multiplies the $x$ variable is the gradient; any remaining constant term is the intercept.

For a pendulum, $T = 2\pi\sqrt{l/g}$ is not linear in $l$. Squaring gives $T^{2} = \dfrac{4\pi^{2}}{g}l$, so plotting $T^{2}$ against $l$ gives a straight line through the origin with gradient $4\pi^{2}/g$.

For e.m.f. and internal resistance, $V = \varepsilon - rI$ is already linear: plotting $V$ against $I$ gives gradient $-r$ and $y$-intercept $\varepsilon$.

For a $\textbf{power law}$ $y = kx^{n}$, taking logs gives $\log y = n\log x + \log k$. A graph of $\log y$ against $\log x$ is a straight line with gradient $n$ and intercept $\log k$.

For an $\textbf{exponential}$ $y = y_{0}e^{-\lambda t}$, taking natural logs gives $\ln y = -\lambda t + \ln y_{0}$. A graph of $\ln y$ against $t$ is a straight line with gradient $-\lambda$.

Choosing which to use is itself a test: if $\ln y$ against $t$ is straight, the relationship is exponential; if $\log y$ against $\log x$ is straight, it is a power law.

$\textbf{Error bars}$ extend above and below (or either side of) each point by its absolute uncertainty. A line of best fit should pass through all or most of them.

To find the uncertainty in the gradient, draw the $\textbf{worst acceptable line}$ — the steepest or shallowest straight line that still passes through all the error bars.

The uncertainty in the gradient is the difference between the best-fit and worst-fit gradients. Its percentage uncertainty then feeds into the final result.

If a line expected to pass through the origin does not, even allowing for error bars, that indicates a systematic error — such as measuring pendulum length to the top of the bob instead of its centre.