Seven Units Are Enough to Measure Everything Else

Because five what? Five centimetres, five metres and five kilometres are all five long, and they differ by a factor of a hundred thousand.

Every measurement therefore has exactly two parts: a number and a unit. Leave the unit off and the number carries no information at all — which is why a physics answer without a unit scores nothing, however correct the arithmetic.

This page covers the first part of the ICSE Class 8 Physics chapter on physical quantities and measurement: what a physical quantity is, the seven base units everything else is built from, and how to measure area, volume and density in practice.

A physical quantity is anything that can be measured — length, mass, time, temperature, area, volume, density, speed, force.

Things that cannot be measured — beauty, honesty, taste, anger — are not physical quantities, however real they are.

A measurement is written as

$$\text{measurement}=\text{numerical value}\times \text{unit}$$

so $5\text{m}$ means five times the standard length called the metre.

A good standard unit must be:

- well defined, so everyone means the same thing by it
- of convenient size for what is being measured
- unchanging — the same today as yesterday, in every place
- reproducible, so a copy can be made anywhere
- universally accepted, so results can be compared

This is exactly why older units failed. A cubit was the length of a forearm and a foot the length of a foot — both differ from person to person, so they are neither unchanging nor reproducible. Two honest traders using cubits would disagree about the same piece of cloth.

The SI system is the internationally agreed set of units. It replaced the older CGS system, which used the centimetre, gram and second, and the FPS system, which used the foot, pound and second. Those units still appear, but only SI is standard in physics.

A fundamental quantity is one that does not depend on any other. There are seven, and every other quantity is built from them.

The seven fundamental quantities, with their SI units and symbols:

- Length — metre, $\text{m}$
- Mass — kilogram, $\text{kg}$
- Time — second, $\text{s}$
- Temperature — kelvin, $\text{K}$
- Electric current — ampere, $\text{A}$
- Amount of substance — mole, $\text{mol}$
- Luminous intensity — candela, $\text{cd}$

A derived quantity is defined in terms of fundamental ones, and its unit is built the same way:

- Area $=$ length $\times$ length, so the unit is ${\text{m}}^2$
- Volume $=$ length $\times$ length $\times$ length, so ${\text{m}}^3$
- Density $=$ mass $÷$ volume, so ${\text{kg/m}}^3$
- Speed $=$ distance $÷$ time, so $\text{m/s}$
- Force $=$ mass $\times$ acceleration, so $\text{kg}{\text{m/s}}^2$, which is given the name newton, $\text{N}$

Reading a derived unit backwards. A unit tells you how the quantity was built. Seeing ${\text{kg/m}}^3$, you know a mass was divided by a volume; seeing $\text{m/s}$, a distance was divided by a time. This is a genuinely useful check — if your working produces $\text{kg}\times {\text{m}}^3$ when the answer should be a density, you have multiplied where you should have divided.

Writing units correctly. Unit symbols are never made plural — $5\text{kg}$, not $5\text{kgs}$ — and take no full stop. A unit named after a person has a capital symbol but a lower-case name: the newton, written $\text{N}$; the kelvin, written $\text{K}$. Leave a space between the number and the unit.

For a regular shape, use a formula. For an irregular solid, use water displacement.

Volume units and their conversions. These trip up more answers than the formulas do:

$$1{\text{m}}^3=100\times 100\times 100=10^6{\text{cm}}^3$$

$$1\text{L}=1000{\text{cm}}^3=1000\text{mL},\text{so }1{\text{cm}}^3=1\text{mL}$$

$$1{\text{m}}^3=1000\text{L}$$

The cube of the length conversion is the point: a metre is $100$ centimetres, so a cubic metre is $100^3$ cubic centimetres, not $100$.

Worked example. A rectangular tank measures $2\text{m}\times 1.5\text{m}\times 1\text{m}$:

$$V=2\times 1.5\times 1=3{\text{m}}^3=3000\text{L}$$

An irregular solid, by displacement. A solid dropped into water pushes aside a volume of water exactly equal to its own volume.

Pour water into a measuring cylinder and read the level at the bottom of the meniscus, with your eye level with it to avoid parallax error. Say it reads $50.0\text{mL}$. Lower a stone in on a thread; the level rises to $68.0\text{mL}$. Then

$$V_{\text{stone}}=68.0-50.0=18.0\text{mL}=18.0{\text{cm}}^3$$

The method has conditions. The solid must sink, must not dissolve, and must not absorb water. A cork floats, sugar dissolves and a dry sponge soaks up water, so none of them can be measured this way — a cork needs a sinker tied to it.

For a solid too large for a cylinder, use a eureka can: fill it to the spout, lower the solid in, and collect and measure the water that overflows.