Give a scientist a graph and the very first thing they hope to see is a straight line. Not because straight lines are pretty, but because they are the easiest shape in the world to read reliably. A straight line has exactly two numbers in it — a gradient and an intercept — and, as you saw when interpreting graphs, both of those numbers usually are the physics: the density, the acceleration, the spring constant, the decay rate. Read the slope and the intercept and you have measured a constant of nature.

A curve is a far harder thing to trust. Its steepness changes at every point, so there is no single gradient to read; you cannot tell a slightly-wrong curve from a right one by eye; and a stray anomalous point can hide happily on a bend. So physicists play a clever trick. They take the curved relationship they are actually studying and rearrange it until it has the shape of a straight line:

Then they plot the right combination of their measurements — not always the raw numbers — so that the messy curve comes out dead straight, and the gradient m and the intercept c hand them the physical constants directly. This move is called linearising, and it is one of the most useful habits in the whole of experimental physics.

The recipe: force the physics into y = mx + c

Linearising is not a law to memorise — it is a method, the same four steps every time. The whole art is deciding what to plot up the side (the role of y) and what to plot along the bottom (the role of x) so that the leftover constants line up as the gradient and the intercept.

1. Write the relationship between your two quantities, symbols and all.
2. Transform it — square a root, cube, take reciprocals, or take logarithms — until it reads Y = mX + c, where Y and X are things you can actually calculate from each row of your results table.
3. Plot Y against X and draw the best-fit straight line.
4. Read off the gradient and intercept, then match them term-by-term to the constants in your rearranged equation to get the physical quantity you were after.

Almost every relationship you meet at A-level falls into one of two families, and each has its own standard move:

• Power laws y = k\,x^{n} — plot y against x^{n} (a straight line of gradient k).
• Exponentials y = A\,e^{kx} — take natural logs and plot \ln y against x (a straight line of gradient k).

Power laws: plot y against x^{n}

Suppose two quantities obey a power law, y = k\,x^{n}, where the power n is known but the constant k is what you want. Plotting y against x gives a curve. But compare the law with y = mx + c: if you treat the whole of x^{n} as your horizontal variable, then

which is a straight line through the origin with gradient m = k and intercept c = 0. So plot y against x^{n} and the curve snaps straight.

A pendulum. The period of a simple pendulum depends on its length L:

A graph of T against L is a curve (a square root). Square both sides first to clear the root:

Now this is Y = mX with Y = T^{2}, X = L and gradient m = \dfrac{4\pi^{2}}{g}. So plot T^{2} (not T) against L, and the gradient of that straight line lets you find little-g:

A mass on a spring. The same trick works for the period of a mass m bouncing on a spring of stiffness k:

Plot T^{2} against m; the gradient is \dfrac{4\pi^{2}}{k}, so the spring constant is k = \dfrac{4\pi^{2}}{\text{gradient}}. Same shape of law, same move.

Worked example: measuring g from a pendulum

A student times a pendulum at several lengths, squares each period, and plots T^{2} (s²) against L (m). The points fall on a good straight line through the origin. Reading two far-apart points off the best-fit line, it passes through (0,\,0) and (1.00\ \text{m},\ 4.00\ \text{s}^2).

Step 1 — gradient from the axis values (rise ÷ run):

Step 2 — match the gradient to the theory. We showed m = \dfrac{4\pi^{2}}{g}, so rearrange for g — the gradient is the whole coefficient of L, and g is on the bottom of it:

Bang on the accepted value. Notice what the straight line bought us: one gradient, read from a big rise-over-run triangle, gave g — and because the line goes through the origin, we also confirmed there was no systematic offset spoiling the timing. A single stray point would have leapt off the line and been spotted at once.

See it: the curve made straight

Here is real pendulum data plotted two ways. With the switch on Raw you get T against L — a bending square-root curve that no straight edge will ever fit. Flip to Linearised and the very same measurements, plotted as T^{2} against L, fall onto a dead-straight line through the origin. The gradient readout then hands you g = 4\pi^{2}/\text{gradient}. Same data, same pendulum — only the combination we chose to plot has changed.

Exponentials: take logs to straighten them

Squaring can straighten a power law, but it does nothing for an exponential, where the variable sits up in the power:

These are everywhere in physics — radioactive decay N = N_0 e^{-\lambda t} and the discharge of a capacitor V = V_0 e^{-t/RC} are two you must know. The move that flattens an exponential is to take the natural logarithm of both sides, because \ln is the exact inverse of e and turns a product into a sum and a power into a multiplier:

Line this up against y = mx + c. The horizontal variable is still plain x, but the vertical variable is now \ln y:

So plot \ln y against x: a straight line whose gradient is the exponential constant k and whose y-intercept is \ln A (so A = e^{\text{intercept}}). For radioactive decay, \ln N = \ln N_0 - \lambda t: a graph of \ln N against t is a straight line of gradient -\lambda and intercept \ln N_0. And for the capacitor, \ln V = \ln V_0 - \dfrac{1}{RC}\,t, so the gradient is -\dfrac{1}{RC} and the discharge time constant RC drops out of it.

Worked example: the decay constant from a log graph

A sample's activity is measured over time and the natural log of the count N is plotted against time t. The points give a straight, falling line. Two well-separated points on the best-fit line are (0\ \text{s},\ \ln N = 4.60) and (20\ \text{s},\ \ln N = 2.60).

Step 1 — gradient of the \ln N–t line:

Step 2 — match to the theory. The gradient equals -\lambda, so

Step 3 — read the intercept too. At t = 0 the line hits \ln N_0 = 4.60, so N_0 = e^{4.60} \approx 100 counts. And from the decay constant the half-life follows, t_{1/2} = \dfrac{\ln 2}{\lambda} = \dfrac{0.693}{0.100} \approx 6.9\ \text{s}. One straight line, and the whole decay is pinned down — a feat no eyeballed exponential curve could manage.