Singapore-Cambridge O-Level Combined Science (Physics/Chemistry)

Physical quantities and units

A physical quantity is something that can be measured, and it is always written as a numerical magnitude together with a unit. A length of 5 means nothing until it is given as 5 m, 5 cm or 5 km, and the number changes with the unit chosen although the length does not.

Every physical quantity can be built from a small set of base quantities, each with an SI base unit.

All other quantities have derived units made from these. Speed is length divided by time, so its unit is $\mathrm{m\,s^{-1}}$ (written m/s in this course). Force is measured in newtons, where $1\ \mathrm{N} = 1\ \mathrm{kg\,m\,s^{-2}}$, and energy in joules, where $1\ \mathrm{J} = 1\ \mathrm{N\,m}$.

Prefixes

Very large and very small values are written with a prefix in front of the unit. Each prefix is a power of ten.

To convert, replace the prefix by its power of ten. A wavelength of 650 nm is $650 \times 10^{-9}\ \mathrm{m} = 6.5 \times 10^{-7}\ \mathrm{m}$, and a frequency of 2.4 GHz is $2.4 \times 10^{9}\ \mathrm{Hz}$. Areas and volumes need the conversion squared or cubed, because $1\ \mathrm{cm} = 10^{-2}\ \mathrm{m}$ gives $1\ \mathrm{cm^2} = 10^{-4}\ \mathrm{m^2}$ and $1\ \mathrm{cm^3} = 10^{-6}\ \mathrm{m^3}$.

Orders of magnitude

The sizes of objects in physics span a very wide range, so it helps to know roughly how big common things are in powers of ten of a metre.

A typical atom is therefore about $10^{7}$ times smaller than a grain of sand, since $10^{-3} / 10^{-10} = 10^{7}$. Comparing values by the difference in their powers of ten gives a quick estimate of how many times larger one is than another.

Scalars and vectors

A scalar quantity has magnitude only. A vector quantity has both magnitude and direction.

The difference matters when quantities are combined. Scalars add as ordinary numbers: 3 kg plus 2 kg is 5 kg. Vectors must take direction into account. Two forces of 3 N and 4 N acting in the same direction give a resultant of 7 N, the same forces in opposite directions give 1 N, and at right angles they give 5 N, since $\sqrt{3^2 + 4^2} = 5$. A vector is drawn as an arrow, with its length to scale and its head showing the direction.

A runner who completes a 400 m lap of a track and ends where she began has travelled a distance of 400 m but her displacement, the straight-line change in position from start to finish, is zero. Her average speed is not zero, yet her average velocity over the whole lap is, because velocity is displacement divided by time.