Measurement

NUST NET Physics · Physics · Measurement, vectors and equilibrium. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Every physical quantity is a number multiplied by a unit. The seven SI base units are the metre, kilogram, second, ampere, kelvin, mole and candela. Every other unit, such as the newton, joule, pascal or coulomb, is built from these.

The dimensions of a quantity show how it depends on mass [M], length [L] and time [T]. Both sides of a correct equation must have the same dimensions, so you can test a formula, or find the dimensions of a constant by making it the subject.

No measurement is exact. The uncertainty in a result comes from the uncertainties in the readings, combined by two simple rules given below.

Rules to remember

Prefixes: giga 10⁹, mega 10⁶, kilo 10³, centi 10⁻², milli 10⁻³, micro 10⁻⁶, nano 10⁻⁹, pico 10⁻¹²
Force [MLT⁻²]; work, energy and torque [ML²T⁻²]; power [ML²T⁻³]; pressure [ML⁻¹T⁻²]
Momentum and impulse [MLT⁻¹]; angular momentum and Planck's constant [ML²T⁻¹]
Pure numbers, angles, sin θ, logs and constants such as 2π have no dimensions
Adding or subtracting quantities: add the absolute uncertainties
Multiplying or dividing quantities: add the percentage uncertainties; for a power xⁿ, multiply the percentage uncertainty in x by n
Significant figures: zeros at the front do not count (0.0320 has three); zeros between digits, and zeros at the end after a decimal point, do count

Common mistake

Converting a squared or cubed unit only once. 1 cm² = 10⁻⁴ m² and 1 cm³ = 10⁻⁶ m³, so apply the prefix factor once for each power.

Worked example

The energy of a photon is E = hf, where f is frequency. What are the dimensions of Planck's constant h, and which mechanical quantity shares them?

Make h the subject: h = E ÷ f.
Energy has dimensions [ML²T⁻²] and frequency has [T⁻¹].
[h] = [ML²T⁻²] ÷ [T⁻¹] = [ML²T⁻¹].
Angular momentum = mvr has [M][LT⁻¹][L] = [ML²T⁻¹], the same.

Answer: [ML²T⁻¹], the same as angular momentum (unit J s).

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