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# Physical quantities and SI units

## Physical quantities and SI units

Physical quantities have a numerical value (a number) and a unit of measurement (e.g. two kilograms, one metre, ten newtons).

In "2 kilograms of bananas", "2" is the numerical value and "kilograms" is the unit.

A unit of measurement is a specific magnitude of a physical quantity that has been adopted by convention.

Kilogram, degree Celsius and centimetre are different units.

The International System of Units, abbreviated as SI (in French), defines the set of units of measurement and their symbols that are most widely used by scientists.

The metre is the SI unit for distance. The kelvin is the SI unit for temperature.

The mass of some strawberries is measured. They have a mass of $146 \text{ g}.$

## SI units of physical quantities

There are seven SI base units in the International System of Units. They can be used to define all other SI units (the derived units).

The seven SI base units
Quantity SI unit Symbol
Mass Kilogram $\text{kg}$
Length Metre $\text{m}$
Time Second $\text{s}$
Electric current Ampere $\text{A}$
Temperature Kelvin $\text{K}$
Luminous intensity Candela $\text{cd}$
Amount of substance Mole $\text{mol}$

SI derived units are products and/or ratios of base units.

Speed is the change in distance per unit of time. The unit of speed is the unit of distance (same as length) divided by the unit of time ($\text{m/s}$).

Some derived units have special SI names and symbols. Force is assigned the SI unit newton $(\text{N})$, where $1\text{ N}=1\text{ kg m s}^{-2}$.

The symbols for the SI units are written in uppercase if they are named after a person (like Isaac Newton). Otherwise they are always written in lowercase (as in the metre).

$$\text{Length} = 1 \um \quad \text{Force} = 10 \text{ N}$$

## Unit prefixes

Unit prefixes are symbols placed before the symbol of a unit to specify the order of magnitude of a quantity. They make it easier to express very large or very small quantities.

"k" (kilo) is the unit prefix in the unit "kg" (kilogram).

A unit prefix stands for a specific positive or negative power of 10.

"k" stands for $1000 = 10^{3}$.

The kilogram is the only SI base unit that contains a prefix. Additional prefixes (e.g. millikilograms) are not allowed.

Prefix Symbol Multiple of unit
pico $\text{p}$ $10^{-12}$
nano $\text{n}$ $10^{-9}$
micro $\mu$ $10^{-6}$
milli $\text{m}$ $10^{-3}$
centi $\text{c}$ $10^{-2}$
deci $\text{d}$ $10^{-1}$
deca $\text{da}$ $10^{1}$
hecto $\text{h}$ $10^{2}$
kilo $\text{k}$ $10^{3}$
mega $\text{M}$ $10^{6}$
giga $\text{G}$ $10^{9}$
tera $\text{T}$ $10^{12}$

## Table labelling conventions

In a table, it is conventional to have the independent variable (the variable being controlled in an experiment) in the first column from the left.

The dependent variables (i.e. the variable(s) that are being observed or calculated depending on the independent variable) in the columns are reported on the right.

A student passes different values of electrical current through a resistor. For each value of current, he records the voltage across the resistor.

The independent variable is the current and the dependent variable is the voltage.

The headings of the table are ideally represented as:$$\frac{\text{name of quantity, symbol}}{\text{unit}}$$

A typical table of experimental measurements is given below:

$$\frac{\text{Time, }t}{s}$$ $$\frac{\text{Speed, }v}{\text{m/s}}$$
$10$ $2.12$
$20$ $4.23$
$30$ $6.55$
$40$ $8.71$

## Graph labelling conventions

In a graph, it is conventional to have the independent variable on the horizontal axis and the dependent variable as the vertical axis. The axis labels are represented, ideally, in the same manner as table headings.

The axis labels are ideally represented as:$$\frac{\text{name of quantity, symbol}}{\text{unit}}$$

In most cases, the name of the quantity is omitted from the axis labels (it is assumed that the reader understands the symbol).

Graph labelling conventions. The independent variable (time) is on x-axis while the dependent variable (speed) is on the y-axis.

## Estimation of common physical quantities

Estimation involves finding a value that is reasonably close to the true value of a physical quantity without any measurement. It is used to check any measured or reported values.

If you know that the mass of an adult is about $70\text{ kg}$, you can estimate that the mass of a bag of apples must be less than $70\text{ kg}$.

Some values of common physical quantities are as follows:

Physical quantity Value
Average mass of an adult $70\text{ kg}$
Room temperature $300\text{ K}$
Average height of an adult $1.7\text{ m}$
Household microwave power consumption $900\text{ W}$
Energy in a teaspoon of sugar $70\text{ kJ}$

Instead of giving a precise numerical value, it is often sufficient to estimate the order of magnitude of a quantity. This involves just stating the value of ten raised to the appropriate power.

The diameter of an atomic nucleus is around $10^{-12}\text{ m}.$

The sun has a mass of roughly $10^{30}\text{ kg}.$