Year 3 physics: measurements and figures

Mid-September. At the liceo artistico (the Italian art-focused high school) physics in Year 3 is a subject you have never seen before — it is not a revision, it starts from zero — and the teacher these days is doing physical quantities, the International System of Units, scientific notation, significant figures. With 2 hours a week instead of the 3 at the science high school, the syllabus runs: the first test arrives in five to eight weeks, so between the end of October and the beginning of November. Nothing serious has happened yet, but it is the moment to understand exactly what will be asked, because in November there is no more time to go back.

Significant figures and conversions: what comes up in the first physics test

I say it right away because it is the most useful thing in this post: the first physics test in Year 3 at the art high school is not theory. They will not ask you to repeat by heart what the fundamental quantities are. They will give you numbers to convert, an expression in scientific notation to simplify, a datum to write with the right significant figures, perhaps a graph to read. It is a calculation test, not a rote-learning one — and it is precisely the one with which, among my art-high-school students, an unexpected 4 most often turns up: not because physics is hard in itself, but because it arrives while the 12 weekly hours of specialist subjects — drawing boards, projects, deadlines — eat up your time, and physics ends up being the subject studied last, the night before. With this kind of test, the night before is not enough: you need automatisms that the right method builds in half the time.

Physical quantities and the International System

A physical quantity is anything that can be measured with an instrument and expressed with a number and a unit of measurement. The International System (SI) fixes seven base ones, from which all the others derive:

Quantity	SI unit	Symbol
Length	metre	m
Mass	kilogram	kg
Time	second	s
Electric current	ampere	A
Temperature	kelvin	K
Amount of substance	mole	mol
Luminous intensity	candela	cd

All the other quantities — speed, density, force, pressure — are derived: they are obtained by combining the base ones. Speed is $m/s$ , density is $kg/ m^{3}$ , area is $m^{2}$ . A practical rule for the test: a number without a unit of measurement is worth zero marks, even if the numerical value is right. The teacher does not always write it in the text of the test, but it is almost always in the marking scheme.

Equivalences: the method that doesn’t fail

The commonest way of getting an equivalence wrong is “moving the decimal point from memory”. It works as long as the conversion factor is 10, 100, 1000 — it stops working (and systematically) with squared or cubed units, or with factors like 3.6. The method that doesn’t fail is different: substitute the unit with its expression in SI units inside the calculation, as if it were a number.

km/h ↔ m/s

$1 km/h$ means $1000 m$ every $3600 s$ . So to convert from km/h to m/s you divide by 3.6:

72 \frac{km}{h} = 72 \cdot \frac{1000 m}{3600 s} = \frac{72000}{3600} \frac{m}{s} = 20 \frac{m}{s}

To go back from m/s to km/h you multiply by 3.6 — it is the same fraction turned upside down, not a different rule to remember by heart.

You will find the complete calculation, in the opposite direction, in the first worked example below.

Squared and cubed units

This is where the most frequent mistake of all is concentrated: converting $cm^{3}$ to $m^{3}$ by dividing by 100 instead of by a million. The reason is simple if you write it out in full: if $1 cm = 1 0^{- 2} m$ , then $1 cm^{2} = (1 0^{- 2} m)^{2} = 1 0^{- 4} m^{2}$ , and $1 cm^{3} = (1 0^{- 2} m)^{3} = 1 0^{- 6} m^{3}$ . The exponent also applies to the power of ten, not just to the unit.

Conversion	Factor
$1 cm^{2}$	$1 0^{- 4} m^{2}$
$1 cm^{3}$	$1 0^{- 6} m^{3} = 1 mL$
$1 dm^{3}$	$1 L = 1 0^{- 3} m^{3}$

g/cm³ ↔ kg/m³

Same principle. $1 g/c m^{3} = \frac{1 0 ^{- 3} kg}{1 0 ^{- 6} m ^{3}} = 1 0^{3} kg/ m^{3}$ , that is, to go from g/cm³ to kg/m³ you multiply by 1000. It is why the density of water is written both as $1 g/c m^{3}$ and as $1000 kg/ m^{3}$ : it is the same physical value, in two different units.

Scientific notation

A number in scientific notation is written as $a \times 1 0^{n}$ , with $1 \leq a < 10$ and $n$ an integer (positive, negative or zero). It serves to avoid writing strings of zeros and, in a test, to handle significant figures without ambiguity.

Products and quotients: you multiply/divide the coefficients and add/subtract the exponents. $(2 \times 1 0^{3}) \cdot (3 \times 1 0^{- 5}) = 6 \times 1 0^{- 2}$ .
Sums and differences: the exponents must be equal before adding the coefficients. $2 \times 1 0^{3} + 5 \times 1 0^{2}$ cannot be added directly: it is rewritten as $2 \times 1 0^{3} + 0.5 \times 1 0^{3} = 2.5 \times 1 0^{3}$ .

Significant figures: the complete rule

Significant figures (sig figs) are the digits of a number that carry real information about the precision of the measurement. The rules the teacher applies in marking:

Non-zero digits are always significant.
Zeros between two significant digits are significant ( $305$ → 3 figures).
Zeros to the left of the first non-zero digit are never significant ( $0.0042$ → 2 figures: they are only positional).
Zeros to the right, after the decimal point, are significant ( $3.20$ → 3 figures — that zero declares the precision of the instrument).
Trailing zeros of a whole number with no decimal point are ambiguous: scientific notation resolves them ( $200$ written as $2 \times 1 0^{2}$ has 1 figure, as $2.00 \times 1 0^{2}$ it has 3).

The two operating rules that really count in a test are different from each other, and that is where marks are lost when the same rule is applied to both:

Multiplication and division: the result has as many significant figures as the datum with the fewest significant figures among those used in the calculation.
Addition and subtraction: you look at the decimal places, not the significant figures. The result has as many decimal places as the datum with the fewest decimal places.

And one rule that always holds: you round only at the end. Rounding in the intermediate steps introduces an error that accumulates and sometimes changes the last digit of the final result.

Absolute error, relative error, and how to write a measurement

Every real measurement has an uncertainty. If you measure a length $l$ with an uncertainty $Δ l$ , the measurement is written $(l \pm Δ l)$ , with the uncertainty rounded to a single significant figure (and the central value rounded to the same order of magnitude as the uncertainty). The relative error is $Δ l / l$ and is often expressed as a percentage, $100 \cdot Δ l / l$ : it says how “large” the error is compared with the measurement itself, not just in absolute value.

Graphs and direct proportionality

In the first physics graphs in Year 3, the independent quantity (the one you decide or control) goes on the $x$ -axis, the dependent one (the one you measure as a consequence) on the $y$ -axis. If the graph is a straight line through the origin, the two quantities are directly proportional: doubling $x$ doubles $y$ . The slope of the line is not a pure number: it has a unit of measurement, that of $y$ divided by that of $x$ — if time in seconds is on the abscissa and distance in metres on the ordinate, the slope is in m/s and is the speed. An inverse proportionality, instead, gives a curve (a hyperbola), not a line: $y$ decreases when $x$ increases, but not linearly.

Four worked examples

1. Conversion km/h ↔ m/s

Convert $72 km/h$ to m/s:

72 \frac{km}{h} = 72 \cdot \frac{1000 m}{3600 s} = 20 \frac{m}{s}

Convert $15 m/s$ to km/h (opposite direction, you multiply by 3.6):

15 \frac{m}{s} = 15 \cdot 3.6 \frac{km}{h} = 54 \frac{km}{h}

If you want to check for yourself whether you have understood the method before going on, in the Year 1 physics test — measurements, significant figures and errors there are more conversions of this kind among the 15 questions: the correct answer is free, the guided step-by-step solution unlocks with the first lesson.

2. Volume of a cube, from cm³ to m³

A cube has side $l = 2.5 cm$ . The volume is:

V = l^{3} = (2.5 cm)^{3} = 15.625 cm^{3}

The starting datum, $2.5 cm$ , has 2 significant figures: cubing it (a product), the result must also have 2 significant figures, so $V = 16 cm^{3}$ . To go to m³ you use the factor $1 0^{- 6}$ (not $1 0^{- 2}$ , because the exponent 3 of the volume also applies to the power of ten):

16 cm^{3} = 16 \times 1 0^{- 6} m^{3} = 1.6 \times 1 0^{- 5} m^{3}

3. Density, with a final conversion

An object has mass $m = 47.3 g$ and volume $V = 12.4 cm^{3}$ . The density is:

ρ = \frac{m}{V} = \frac{47.3 g}{12.4 cm ^{3}} = 3.8145... g/c m^{3}

Both data have 3 significant figures, so the result is rounded to 3 significant figures: $ρ = 3.81 g/c m^{3}$ . Converting to kg/m³ (you multiply by 1000):

3.81 g/c m^{3} = 3.81 \times 1 0^{3} kg/ m^{3}

4. The two rules for significant figures, in the same exercise

One last example to see why product/quotient and sum/difference are not treated the same way. First a sum: $12.5 m + 3.42 m$ . Here you do not count significant figures but decimal places: the first datum has 1 decimal place, the second has 2, so the result is rounded to 1 decimal place:

12.5 m + 3.42 m = 15.92 m \to 15.9 m

Now a product with those same two numbers: $12.5 m \times 3.42 m$ . Here you count significant figures: the first datum has 3, the second has 3, so the result has 3 significant figures:

12.5 m \times 3.42 m = 42.75 m^{2} \to 42.8 m^{2}

The same two starting numbers, two different rounding rules, two results with a different number of final digits. It is exactly the point where an otherwise correct test loses half a mark.

How we work on these topics

In my online lessons on measurements and significant figures I do not redo the textbook theory: I have you do the conversions one after another, with the method of substitution inside the calculation, until it becomes automatic — because in a test there is no time to think it through from zero. For those arriving in Year 3 at the art high school and seeing physics for the first time, a free 30-minute introductory call is often enough to see whether the method makes a difference before committing to a whole package (rates and packages on the pricing page — payment offline: bank transfer, Satispay, PayPal, card, Apple Pay or Revolut).

If you want to train on your own beforehand, you will find more questions on measurements and errors in the Year 1 physics test — it is called “Year 1” because at the science high school this same syllabus is done in Year 1, but at the art high school physics starts in Year 3 from zero: it is the right simulation for you, the label only indicates where it sits in the science high school’s path, not your year. The next stumbling block in the syllabus, straight afterwards, is the resolution of vectors: I talk about it in resolving forces and vectors in Year 3 without trigonometry. If instead you are following the same path at the science high school, where the first test is on kinematics and not on measurements, you will find the twin post in the first physics test in Year 3. How we work on physics, from measurements onwards, is described on the physics page.

I dealt with measurement errors (propagation, half-range spread) more calmly in measurement errors and error propagation: it is the natural continuation of this chapter.

If you want a hand before the October-November test, write to me via contact.

Year 3 physics: measurements and significant figures