Physics · Class 9-10 · Chapter 1 · Physical quantities and measurement

Physical quantities and their units.

Anything in the world that can be measured is a quantity. The surprise is that the entire endless list of them can be built out of seven. Everything else is those seven multiplied and divided.

1. Quantities, and why a number is not enough

The definition is one line: anything that can be measured is a quantity. Length can be measured, so length is a quantity. So are mass, time, temperature and velocity. Love and beauty are not quantities, because there is no way to measure them. “I love you very much” may well be true, but it is not a measurement.

And what does measuring mean? The measure of a quantity always comes in two pieces, and you need both of them.

measure = number × unit

Say “12” and nobody knows anything. Twelve metres, twelve kilograms, twelve seconds? Only the unit gives the number a meaning. Pick a number and a unit below and see what you get, then pick one without the other and see what you do not get.

Notice what happens: a bigger unit makes the number smaller, a smaller unit makes it bigger, and the product stays put. 2 metres and 200 centimetres are not two lengths, they are one length written twice. And more digits after the decimal point does not mean more precise. It means a smaller unit was used.

2. Base quantities and derived quantities

There are countless quantities, but they fall into two classes. A base quantity is one that stands on its own, does not depend on any other quantity, and can be used to express the others. There are exactly seven of them, no more and no fewer.

The seven are length, mass, time, electric current, temperature, luminous intensity and amount of substance. A derived quantity is one that depends on the base ones, which is to say one you get out of them. Area, volume, velocity, acceleration, force, work, power, density, pressure and momentum are all derived.

Twelve quantities below. Sort them yourself, and count how many end up in the base box.

It takes a moment to accept that area is not a base quantity. But area is length times length, so it leans on length, and its unit is m². Going the other way, luminous intensity and amount of substance do not sound basic at all, and yet both are on the list. You have to judge by the definition, not by how the name sounds.

3. What is inside a derived quantity

A derived quantity is base quantities multiplied and divided, and nothing more than that. Velocity is displacement over time, so m/s. Acceleration is change of velocity over time, so m/s². Force is mass times acceleration, so kg·m/s², which gets the name newton (N). Work is force times displacement, so N·m, which gets the name joule (J). Power is work over time, so J/s, the watt (W).

The names look different, but there is nothing inside them except base units. The grid below just counts: how many times length, mass and time are multiplied in (positive) or divided out (negative). Tap any row.

The grid counts how many times each base quantity is needed. The tree below shows the relationship instead: which derived quantity came out of which. In a later note this counting gets a name of its own: dimension.

One thing to notice about the tree: nearly all of mechanics hangs off just three branches, length, mass and time. The other four base quantities each look after their own corner.

4. Units, and the SI system

What is a unit? To measure a quantity you fix on a definite amount of that same kind of thing and treat it as the standard, and that standard amount is the unit. To measure a length you need an agreed length; to measure a mass you need an agreed mass. A unit has to be four things: well defined, unchanging, available to everyone, and accepted by everyone.

For a long time different countries used different systems: CGS (centimetre, gram, second), FPS (foot, pound, second), MKS (metre, kilogram, second). In 1960 the General Conference on Weights and Measures adopted a single system called SI, short for the French Système International d'Unités. The seven SI units for the seven base quantities are below.

There is a design hiding inside those definitions. The metre comes from the speed of light, the second from the vibration of a caesium atom. Both are constants of nature, so any laboratory anywhere can build its own standard instead of borrowing one. The kilogram was the only one that leaned on an object, and objects can lose a little of themselves over time. That is why in 2019 the international definition of the kilogram was moved onto the Planck constant. For an exam, though, write the book's definition: the cylinder at Sevres.

5. When the units do not match

There is a good illustration of why units are taken so seriously. In 1999 NASA's Mars Climate Orbiter was lost as it tried to enter orbit around Mars. The cause was simple and awful: one team had worked in pound-seconds and the other in newton-seconds. The numbers were fine. Nobody had checked that the units agreed.

In an exercise book the mistake is cheaper, but it still costs marks. At the end of any calculation, look at the unit that came out and ask whether it is the unit that quantity is supposed to have.

One last thing to notice: some derived units have names of their own (newton, joule, watt, pascal, hertz, coulomb) and some never got one (m/s², kg/m³). Nobody teaches that difference on purpose, and it turns up in questions anyway. Turn the cards and go over them.

This chapter's board questions, with answers and reasons →

Source: Physics, Classes 9-10 (NCTB), Chapter 1 “Physical quantities and measurement”, the section on physical quantities and their measurement.