Quantities, units and measurements

A physical question needs a quantity

A description such as “the container is large” allows many things to be imagined, but does not allow results to be compared. We can ask about its height, internal volume or mass. These are different quantities: height describes an extent, volume describes occupied space and mass characterizes the body's inertia. First decide which property you want to know; only then choose the instrument.

A physical quantity is a property to which we can assign a value through measurement or a defined relationship with other quantities. We write a result as the product of a number and a unit: 1,35 m. The number 1,35 alone does not tell us whether we measured length, time or mass. The unit is part of the information, not decoration added at the end.

Distinguish direct measurement, calculation and estimation. Reading a ruler is direct measurement; calculating an area from two lengths is indirect measurement; judging that area visually is an estimate. All can be useful if the method and expected precision are clear. A property such as beauty does not gain a physical unit merely because we assign it a score.

Base units and constructed units

The International System of Units, SI, allows results from different places to be compared. It has seven base units: metre, m, for length; kilogram, kg, for mass; second, s, for time; ampere, A, for electric current; kelvin, K, for thermodynamic temperature; mole, mol, for amount of substance; and candela, cd, for luminous intensity. You need not master all these quantities yet to recognize their symbols.

Other units arise from combinations. Area is length times length and is measured in m²; volume in m³. Average speed divides distance by duration and is measured in m/s. Density divides mass by volume and is measured in kg/m³. Later you will encounter the newton, N, equivalent to kg·m/s², and the joule, J, equivalent to kg·m²/s².

Symbols take no plural or their own full stop: write 5 kg, not “5 kgs”. Leave a space between number and symbol and respect capitalization: m means metre, whereas M is the prefix mega. The kilogram already contains a prefix in its name, but remains the base unit of mass.

Convert without guessing which way the decimal point moves

A prefix changes the size of the unit, not the object. Kilo means 10³; centi, 10⁻²; milli, 10⁻³; micro, 10⁻⁶. Thus 1 km = 1000 m and 1 cm = 0,01 m. The same table can measure 1,2 m or 120 cm: the number increases when you choose a smaller unit.

The safest way to convert is to multiply by a ratio equal to one. To convert 240 cm to metres, use 240 cm × (1 m/100 cm) = 2,40 m. The cm units cancel. If the chosen factor leaves cm²/m, you inverted it. Writing cancellation avoids memorizing contradictory “multiply or divide” rules.

For areas and volumes, the factor is raised to the power too. Since 1 cm = 10⁻² m, then 1 cm² = 10⁻⁴ m² and 1 cm³ = 10⁻⁶ m³. A litre corresponds to 10⁻³ m³, but is not an SI base unit. For durations, 1 min = 60 s and 1 h = 3600 s; these factors are not powers of ten. Write the equivalence before converting.

Scientific notation and the scale of the result

Scientific notation is written a × 10ⁿ, with 1 ≤ |a| < 10 e n inteiro. O comprimento 0,000042 m torna-se 4,2 × 10⁻⁵ m. O expoente negativo indica uma quantidade menor que uma unidade, não um comprimento com sentido negativo. Já −4,2 m pode representar uma coordenada num eixo: são significados diferentes para o sinal.

To multiply powers of ten, add exponents; to divide, subtract them. For example, (6 × 10³)/(2 × 10²) = 3 × 10¹. Keep units during these operations. A calculator may display 4.2E−5; this means 4,2 × 10⁻⁵, not an operation involving a variable E.

A scale estimate detects absurd results before all decimal places are calculated. If a bottle contains about one litre, a calculation giving thousands of cubic metres has failed. When measuring a room, expecting lengths of a few metres is reasonable. Estimation does not replace measurement: it is an independent check and should be based on an explained comparison.

Scales of the Universe: a unit is not an object

Units help compare scales beyond everyday experience. Earth is a planet orbiting the Sun; the Sun is a star in the Solar System. The Solar System belongs to our galaxy, the Milky Way. A galaxy is not another name for a planetary system, and the Universe is not limited to our galaxy. This distinction organizes questions before numbers are introduced.

For Solar System distances, the astronomical unit, AU, is useful: approximately 1,50 × 10¹¹ m, near the scale of the average Earth–Sun distance. For distances between stars, the light-year is often used: distance travelled by light in vacuum in one year. Although it contains the word year, it is not a unit of time.

Taking a year as 365,25 days gives a duration of approximately 3,156 × 10⁷ s. Multiplying by c ≈ 3,00 × 10⁸ m/s gives about 9,47 × 10¹⁵ m per light-year, or 9,47 × 10¹² km. If a light signal travels one AU in this model, duration is approximately (1,50 × 10¹¹)/(3,00 × 10⁸) = 500 s, about 8 min and 20 s.

Observing distant light corresponds to information emitted earlier because propagation takes time. Simple calculations of distance = speed × duration require stating the model; studying the Universe at cosmological scales also introduces the evolution and expansion of space. Here we practice units and scales, while orbits and gravitation are developed in the lesson on circular motion and gravitation.

A measurement has limits

Choose an instrument suited to the property and scale: a ruler for length, stopwatch for duration, balance for mass. Check the origin, spacing between marks and positioning. Looking obliquely at a ruler shifts the apparent reading: this is parallax. An instrument that does not show zero unloaded may introduce a systematic offset.

Resolution is the smallest distinguishable change under the reading conditions. Uncertainty describes doubt associated with the result and also depends on method. On an analog ruler, half the smallest division can serve as an initial estimate when the reading is clear; it is not a universal law. With a hand-operated stopwatch, reaction time may exceed the last displayed digit.

Repeating measurements helps reveal spread. Averaging can reduce random effects, but does not eliminate a miscalibrated instrument. Accuracy means closeness to a reference value; precision concerns a small spread of repeated results. Very close readings may all be shifted by a zero error.

Results that can be discussed

Writing L = (12,4 ± 0,1) cm communicates the value and an uncertainty estimate. In this elementary presentation, the associated interval is 12,3 to 12,5 cm; it is not an absolute guarantee of containing the true value. The estimation method must be stated. Relative uncertainty divides uncertainty by the measured value; multiplying by 100 gives a percentage.

Significant figures show the numerical information supported by the method. In 0,00450 m, initial zeros position the decimal point, while the final zero communicates resolution: there are three significant figures. Avoid rounding at every step; retain working digits and round only at the end consistently with the measurements.

Check dimensions. In A = length × width, units give m × m = m². Adding 3 m to 2 s has no physical meaning. An equality with incompatible units is wrong; a dimensionally correct equality can still have the wrong factor or model. Dimensional checking is necessary but not sufficient.

Worked examples, without skipping steps

Follow the reasoning first. Then cover the solution and try to reconstruct it in your own words.

Now try it yourself

There is no timer. Write what you know, choose a relationship and justify each step. Hints and solutions are available whenever you need them.

Mistakes worth avoiding

A number without a unit is a complete measurement.

The result needs a unit and context; uncertainty and method indicate quality.

Converting cm² to m² means dividing by one hundred.

The length factor is squared: divide by 10 000.

Every calculator digit is significant.

Physical information remains limited by measurements.