Singapore-Cambridge O-Level Physics

Physical quantities and SI units

A physical quantity is something that can be measured, and it is written as a numerical magnitude together with a unit. A length of 4.2 means nothing until it is given as 4.2 m or 4.2 cm, and the same quantity can be written in different units with different numbers.

The SI system is built on base quantities, each with its own unit.

Every other unit is derived from these. The newton, for example, is the force that gives a mass of 1 kg an acceleration of $1\,\mathrm{m\,s^{-2}}$, so $1\,\mathrm{N} = 1\,\mathrm{kg\,m\,s^{-2}}$.

Prefixes

Prefixes scale a unit by a power of ten, which keeps very large and very small numbers readable.

To convert, substitute the prefix for its factor. 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 $2.4 \times 10^{9}\,\mathrm{Hz}$.

Area and volume units need more care, because the conversion factor is squared or cubed. Since $1\,\mathrm{m} = 100\,\mathrm{cm}$,

so a volume of $250\,\mathrm{cm^3}$ is $250 \times 10^{-6} = 2.5 \times 10^{-4}\,\mathrm{m^3}$.

Orders of magnitude

The sizes of objects run over a huge range, and it helps to know roughly where common things sit.

The Earth is therefore about $10^{16}$ times wider than an atom. Comparing powers of ten like this is the quickest way to judge whether a calculated answer is reasonable.

Measuring instruments

An instrument is chosen for its range, which is the largest and smallest values it can measure, and its precision, which is the smallest division it can read. A metre rule reads to 1 mm, which is fine for a length of a desk and useless for the thickness of a wire.

A measuring cylinder is read with the eye level with the bottom of the meniscus, because looking from above or below gives a false reading. A rule is placed against the object with the scale touching it, and the reading is taken from directly above to avoid parallax, the apparent shift of a pointer against a scale when viewed at an angle.

Before a reading is trusted, the instrument is checked for a zero error. If an instrument reads something other than zero when it should read zero, there is a zero error, and every reading is shifted by that amount. The correction is subtracted from each reading.

Reducing the effect of error

The period of a pendulum is best found by timing many oscillations and dividing. Timing 20 swings in 28.4 s gives a period of $28.4 \div 20 = 1.42\,\mathrm{s}$. A reaction-time error of about 0.2 s is now shared over 20 swings, so it affects each period by about 0.01 s, instead of spoiling a single timed swing.

The same idea applies to a thin wire. Measuring the width of 20 turns wound side by side on a pencil and dividing by 20 gives the diameter of one turn with much less relative error than a single measurement.

Repeating a measurement and taking the average reduces the effect of random errors, which vary from one reading to the next. It does nothing for a systematic error such as a zero error, which is the same every time.

Scalars and vectors

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

Stating "30 m" for a displacement is incomplete; it must be "30 m north", or "30 m to the right". Two vectors with the same magnitude but opposite directions are different vectors, and cancel when added.

Adding vectors graphically

Two vectors are added by drawing them to scale, head to tail. Draw the first as an arrow, then draw the second starting from the head of the first. The resultant is the straight line from the tail of the first to the head of the second, and its length and angle are measured from the scale drawing.

For example, a force of 3 N east and a force of 4 N north are drawn at right angles with a scale of 1 cm to 1 N, so the arrows are 3 cm and 4 cm. The resultant is a 5 cm line, so the resultant force is 5 N. The angle it makes with the east direction is measured with a protractor as about $53^\circ$ north of east.

Parallel vectors are simply added if they point the same way and subtracted if they point opposite ways: 6 N to the right and 2 N to the left give a resultant of 4 N to the right. The order in which the vectors are added does not change the resultant.