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Units and Measurements

Dimensions and errors on nearly every paper, and it has grown. Quick questions once the dimensional formulas of the common quantities are known by heart.

Questions in the bank
190
q/paper in 2025–26
1.72
Numeric answer
10%
Notes pages
8

Tier: Cornerstone

When you’ll see it

A quantity's dimensions, an unknown constant or power in an equation, a percentage error from measured values, or a vernier or screw-gauge reading.

How this chapter is tested

Dimensions carry the larger part of the chapter. Build each quantity's dimensions from the equation that defines it, then use them to find a constant, find unknown powers or test an equation. Match lists pair quantities that differ by one power, such as viscosity and pressure or flux and inductance.

The rest is errors and instruments, and the arithmetic there is short. The power rule gives the error of a product, absolute errors add in a sum or a difference, and a laboratory reading carries the least count of the instrument that took it.

This is a cornerstone chapter, and it has grown in recent papers. Marks are lost on a sign or a factor: a zero error added instead of subtracted, a power left out of an error sum, or a timing error divided by one period instead of the whole run.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Significant figures and large distances

    Round a sum by decimal places and a product by significant figures; order the AU, the light year and the parsec, and use radians in d = θD.

  • Mechanical and thermal dimensions

    Write each quantity's dimensions in M, L, T and K from its defining equation, and tell apart pairs such as surface tension and surface energy.

  • Electric and magnetic dimensions

    Add current A as a base quantity; flux is ML²T⁻²A⁻¹, inductance ML²T⁻²A⁻², and ½ε₀E² is an energy per volume.

  • Homogeneity and unknown constants

    A constant added to a variable takes its dimensions, and the argument of a sine, an exponential or a log is a pure number.

  • Deriving relations and new base quantities

    Equate the powers of M, L and T to find unknown powers; to change the base quantities, solve for M, L and T in the new ones first.

  • Propagation of errors

    For a product of powers, add each relative error times the size of its power; in a sum or a difference, add the absolute errors.

  • Errors in laboratory experiments

    Each reading's error is its least count; a time read over many oscillations divides the watch's error by the whole run.

  • Vernier callipers and screw gauge

    Least count is one MSD minus one VSD, or the pitch over the circular divisions; the true reading is the observed reading minus the zero error, with its sign.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • A positive zero error added

    A positive zero error means every reading is already too large. Subtract it; the added value is the commonest wrong option.

  • A power left out of the error

    In g = 4π²l/T² the period's relative error counts twice, and in an area a diameter's error counts twice. Dropping the factor gives a distractor.

  • The timing error over one period

    A watch read once over a run of many oscillations shares its error across the whole run. Divide by the total time, not by one period.

  • Dimensionally correct is not correct

    T = π√(l/g) passes the dimension test but is wrong by a factor 2. A dimension check can only rule an equation out.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Units and Measurements notes

Drill every Units and Measurements question

190 questions from the bank, across 8 subtopics.

Drill the 190 questions

Drill one subtopic at a time

The 8 subtopics, in teaching order.

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