Sec 3 Physics Physical Quantities & Measurement notes

A physical quantity is anything you can measure, and stating it always takes two parts: a numerical magnitude and a unit. Writing "the current is 0.35" is incomplete — 0.35 A and 0.35 mA differ by a factor of a thousand. Examiners withhold the mark for a missing or wrong unit on any calculated answer, so build the habit now: every numerical answer ends with its unit.

Every other unit in the course is a derived unit assembled from these: speed is metres per second (m/s), force is the newton (N), which is kg·m/s², and energy is the joule (J), which is N·m. You do not need to derive them yet — you just need to recognise that the newton and the joule are not base units, while the ampere is.

The syllabus prefixes attach a power of ten to any unit, so the same measurement can be written at a comfortable size whether you are describing a bacterium or a power station.

Converting is multiplying or dividing by the prefix factor: 98.4 MHz = 98.4 × 10⁶ Hz = 9.84 × 10⁷ Hz, and 450 nm = 450 × 10⁻⁹ m = 4.5 × 10⁻⁷ m. Keep the magnitude and the power of ten separate in your working until the final line — most prefix slips are lost in the arithmetic, not the concept.

An order of magnitude is the nearest power of ten. Estimating them trains the physicist’s most useful reflex: the sanity check. The syllabus spans from a typical atom (about 10⁻¹⁰ m) to the Earth (radius about 6.4 × 10⁶ m, so order 10⁷ m). Between them: a molecule 10⁻¹⁰–10⁻⁹ m, a virus 10⁻⁷ m, a red blood cell 10⁻⁵ m, a sheet of paper 10⁻⁴ m thick, your height 10⁰ m, a school field 10² m, Mount Everest 10³–10⁴ m, and Singapore's length about 5 × 10⁴ m.

Two questions pick the right instrument. Range: does the instrument cover the whole quantity — a 30 cm rule cannot measure a 1.8 m jump. Precision: how small is one division, and is that fine enough for the job — timing a 100 m sprint to 0.01 s is wasted precision if your finger is 0.2 s late.

Three reading errors account for most lost marks. A zero error is a constant offset: close the micrometer jaws gently and read the scale — if it shows 0.03 mm with nothing between the jaws, every reading is 0.03 mm too big and must be corrected by subtraction. Parallax happens when your eye is not level with the mark, so the reading shifts; read scales squarely from above. With liquids, read the bottom of the meniscus (the curved surface) for water and most wetting liquids.

Choosing is also about the job, not only the size. The internal diameter of a test tube needs the inside jaws of vernier calipers; the external diameter of a wire needs a micrometer; the volume of a pebble needs a displacement (measuring) cylinder, not any rule. When a question asks you to "select and explain", name the instrument and the property that makes it suitable — range or precision — or the mark is not earned.

A digital stopwatch displays 0.01 s, but your finger does not obey the display: between seeing the event and pressing the button, about 0.2–0.3 s of reaction time passes, and it is different every trial. For a single short event this swamps the display precision — timing one pendulum swing of about 1.8 s with a personal error of ±0.25 s is a ±14% mistake hiding behind a hundredth-of-a-second display.

The fix is to time N oscillations and divide by N. The same ±0.25 s reaction error now spreads across 20 swings: the uncertainty in the period T = total time ÷ 20 shrinks to about ±0.013 s. Counting begins and ends when the bob passes a fixed fiducial marker (a line behind the string at the centre of the swing), where the bob moves fastest and the timing judgement is sharpest — at the turning points it lingers and the start is ambiguous.

The same trick measures anything that repeats: the period of a bouncing spring, the rotation of a turntable, the time for a card to fall. Whenever a question asks "how would you improve the measurement of a short time interval", the expected answer is the pair: time many cycles and divide, and use a fiducial marker.

A scalar is fully specified by a magnitude and a unit: 2.4 kg of rice, 12 s, 350 J. A vector also needs a direction: a 5.0 N pull *upwards* is a different physical situation from a 5.0 N pull *to the left*. Knowing which quantities are which decides how they combine — scalars add arithmetically, vectors add geometrically.

At this level, currents and temperatures are treated as scalars even though field directions exist; the syllabus’s vector list that matters for calculations is forces, velocities, accelerations and displacements. If two forces act along the *same line*, they still add like scalars — but with signs: +8 N east and −5 N east (i.e. 5 N west) give a resultant of 3 N east.

When two vectors act at an angle, the resultant is the single vector that would have the same effect. The syllabus method is graphical — a carefully drawn scale diagram:

1. 1Choose a scale

   e.g. 1 cm : 1 N, big enough to fill at least half the page — small diagrams magnify drawing errors.

2. 2Draw the first vector

   an arrow, correct length and direction, starting at the point.

3. 3Draw the second from its tip

   the second arrow starts where the first ends (tip-to-tail), correct length and direction.

4. 4Close the triangle

   the resultant runs from the start of the first to the tip of the second; mark its arrowhead.

5. 5Measure and convert back

   measure the resultant’s length with a ruler and its angle with a protractor, then convert with the scale.

The maximum possible resultant of two forces is their sum (they help each other, acting the same way) and the minimum is their difference (they partly cancel, acting opposed); any angle in between gives a value between these limits, found by the diagram. A question that offers "which resultant is impossible" is testing exactly this: 3 N and 4 N can never give 8 N.