The big idea: A straight line is easy to read — its gradient is a single number.
So when a law is curved (like P getting bigger as V gets smaller), we re-plot it as a straight line: pick what to put on each axis so the graph comes out straight. This is linearizing.
Once it is straight you can do two things: read a physics quantity off the gradient, and test whether the data really obey the law.
What 'directly proportional' looks like: Two quantities are directly proportional when their graph is a straight line that passes through the origin (0, 0).
So to test a 'proportional' claim, you check both: is the line straight, and does it go through the origin?
Here is a curved law made straight. The gas law PV = constant is a curve if you plot P against V — but if you instead plot P against 1/V (one over the volume), the points fall on a straight line through the origin:
[Diagram: phys-best-fit] - Available in full study mode
Read it like y = mx + c: Compare your plotted variables to the line equation Y = mX + c:
- the thing on the up axis is Y - the thing on the across axis is X - the gradient m and intercept c are then physics quantities.
Method: rearrange the law into the form Y = mX + c. Whatever multiplies the variable becomes the gradient. For example, the gas law PV = k rearranges to P = k × (1/V) — so plotting P (up) against 1/V (across) gives a straight line whose gradient is k.
- the quantity plotted on the vertical axis (chosen so the graph is straight)
- the quantity plotted on the horizontal axis
- the gradient — equals a physics constant you are trying to find
- the vertical intercept — usually 0 for a 'directly proportional' law
The gradient is a physics quantity: After linearizing, the gradient is never 'just a number' — it equals a constant in the law (a spring constant, a refractive index, a gas constant…). Always say what the gradient represents and give its units.
IB-style question — decide what to plot
A student measures the time period T of a pendulum for several lengths L. Theory says T = k√L (T is proportional to the square root of L). The student wants a straight-line graph through the origin so they can find k from the gradient. (a) State what to plot on each axis. (b) State what the gradient represents.
Solution
- (a) Match the law to Y = mX + c. Write T = k√L, so compare:
T is the Y quantity and √L is the X quantity.
Plot T (up axis) against √L (across axis). - Because there is no '+ c' term, the line should pass through the origin — that is what 'proportional' means.
- (b) The gradient m multiplies X, and here that multiplier is k:
Final answer
(a) Plot T against √L (T up, √L across). (b) The gradient equals the constant k.
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How this is tested: Paper 1B almost always builds a whole question around one experiment, and linearizing is the part that ties it together.
- What they ask: state what to plot to get a straight line · find a missing point's coordinates so you can plot it · explain how the graph decides whether a proposed law holds · show (with two data rows) that two quantities are not directly proportional. - The classic trap: assuming any straight-ish line means 'proportional'. It only counts as proportional if the line also passes through the origin.
An experiment measures how the depth d that a marker sinks depends on the water pressure P pushing on it. Theory predicts d = k√P, so the student plans to plot depth against root-pressure (√P). The graph below is straight and passes through the origin, supporting the law:
[Diagram: phys-best-fit] - Available in full study mode
IB-style question — find a missing point to plot
One table row reads pressure P = 9.0 kPa and depth d = 4.6 cm. For the depth-versus-root-pressure graph, determine the coordinates of the point this row should be plotted at.
Solution
- The across axis is √P, so process the pressure into a root first:
- Work it out — keep the unit:
- The up axis is the depth, plotted as measured: d = 4.6 cm.
So the coordinates are (√P, d):
Final answer
Plot the point at (3.0 kPa0.5, 4.6 cm).
IB-style question — show it is NOT proportional
Suppose a classmate instead claims depth is directly proportional to pressure itself (d ∝ P). Using these two rows — P = 4.0 kPa with d = 3.1 cm, and P = 9.0 kPa with d = 4.6 cm — show that depth and pressure are not directly proportional.
Solution
- If d ∝ P then the ratio d/P must be the same for every row. Check the first row:
- Now the second row:
- The two ratios are clearly different (0.78 vs 0.51), so d/P is not constant.
Final answer
The ratio d/P is not constant (0.78 ≠ 0.51 cm kPa⁻¹), so depth is not directly proportional to pressure. (It fits d ∝ √P instead.)