Measurement & tools

The big picture

A number without a unit or an uncertainty tells only part of the story. Begin by reading what an instrument actually resolves, then decide which digits survive the calculation. This is the foundation of experimental physics, not just a rounding exercise.

The mathematical tools in this route let you translate the same situation between words, arrows, graphs and equations. If the algebra feels difficult, return to the quantity being measured: a gradient is a rate; an area adds many small contributions.

What to read, and why

Start at the first stage if the subject is new. Otherwise, use the descriptions to find the missing step in your understanding. All chapter explanations are open; the labels identify chapters whose solutions and answer checks are also free.

1. STAGE 1

   Measure before calculating

   Read scales and distinguish resolution, offsets and uncertainty before combining measurements.

   Significant figures →What the digits in a measurement are allowed to claim5 worked examples · 9 checks · Complete free chapterUnits: keep the quantity, change the number →Use units as part of the calculation, including powers, rates and experimental scales.4 worked examples · 5 checks · Open explanation · member practiceMeasurement uncertainty →What a reading tells you—and what it leaves unresolved6 worked examples · 8 checks · Open explanation · member practicePropagation of uncertainty →Follow the measurement model before choosing a formula8 worked examples · 8 checks · Open explanation · member practiceRead a vernier, understand the measurement →Derive the fine reading from two scales, correct the zero, and judge what the instrument can actually establish.4 worked examples · 5 checks · Open explanation · member practiceMicrometer screw gauge →Turn rotation into length, then turn a reading into a defensible measurement.6 worked examples · 6 checks · Open explanation · member practice

2. STAGE 2

   Make a graph say something

   Use units and gradients to identify a model and interpret the intercept. Connect changing slopes to differentiation.

   Make an experiment speak through a graph →Read slopes, intercepts and uncertainty as physical information rather than as a line-drawing exercise.4 worked examples · 6 checks · Open explanation · member practiceRates and accumulation: calculus for physics →Connect tangents, areas and initial conditions so that derivatives and integrals describe real changes.7 worked examples · 7 checks · Open explanation · member practiceExponential change, logarithms and controlled approximations →Recognise fractional change, solve for time and decide when a small-angle model is accurate enough.7 worked examples · 7 checks · Open explanation · member practice

3. STAGE 3

   Keep direction in the calculation

   Choose axes deliberately. Resolve components before using products for work or torque.

   Vectors and their components →Turn directions into equations without losing the geometry4 worked examples · 7 checks · Open explanation · member practiceThe cross product: perpendicular effects and direction →Build the geometry, signs and component method behind torque and magnetic force.4 worked examples · 7 checks · Open explanation · member practice

Choose a method before an equation

Use the physical conditions to narrow the choices. A formula becomes useful after you know why its model applies.

Try a complete free problem

These problems include the hints, worked reasoning and answer feedback. Attempt one before opening the solution.