Physical quantities, units and measurement

Quantities, units and reliable measurement

Physical quantities, SI units and scale

A physical quantity is a measurable property written as a numerical magnitude and a unit. A bare number is incomplete when a question asks for a physical quantity: write 2.4 m, not just “2.4”.

SI base quantities and units

Prefixes

A prefix rescales a unit without changing the physical quantity: 3.2 km and 3200 m are the same length. Each prefix stands for one power of ten, so replace the prefix with its power and simplify.

For example, 450 nm = 450 × 10⁻⁹ m = 4.5 × 10⁻⁷ m, and 2.5 dm = 2.5 × 10⁻¹ m = 0.25 m. Check the direction: a smaller unit needs a larger number, so 0.25 m becomes 25 cm, not 0.0025 cm.

Converting area and volume units

An area unit contains two lengths and a volume unit contains three, so the length conversion is applied once for each length. Since 1 cm = 10⁻² m:

• 1 cm² = (10⁻² m)² = 10⁻⁴ m²

• 1 cm³ = (10⁻² m)³ = 10⁻⁶ m³

• 1 dm³ = (10 cm)³ = 1000 cm³, the volume also called one litre; since 1 dm = 0.1 m, 1 dm³ = 10⁻³ m³

Worked example. A floor tile measures 20 cm by 30 cm. Its area is 20 × 30 = 600 cm². In square metres this is 600 × 10⁻⁴ m² = 0.060 m². Converting the sides first gives the same answer: 0.20 m × 0.30 m = 0.060 m². Dividing 600 by 100 instead gives 6 m², which is bigger than a small room and cannot be the area of one tile.

In the same way, a 250 cm³ drink can holds 250 × 10⁻⁶ m³ = 2.5 × 10⁻⁴ m³.

Orders of magnitude

An order of magnitude is a power of ten. An atom is about 10⁻¹⁰ m, a person is about 10⁰ m and Earth’s diameter is about 10⁷ m. Use familiar scales to reject physically implausible answers.

Instrument range, resolution and reading

The range of an instrument runs from the smallest to the largest value it can measure. Its resolution is the smallest change it can show; on a scale this is the value of one smallest division. A finer resolution can make a reading more precise, but it cannot correct a poorly zeroed instrument. Choose a range that covers the quantity and a resolution fine enough for the decision you need to make.

• A measuring tape (range tens of metres, resolution 1 mm) suits the length of a room; a metre rule (range 1 m, resolution 1 mm) suits a book.

• Calipers (resolution 0.1 mm or better) suit the diameter of a pipe, and a micrometer (resolution 0.01 mm) suits the thickness of a thin sheet, where 1 mm divisions would be too coarse to tell values apart.

• A measuring cylinder measures the volume of a liquid, an electronic balance measures mass and a stopwatch measures time. A 10 cm³ cylinder with 0.2 cm³ divisions measures a small volume more precisely than a 100 cm³ cylinder with 1 cm³ divisions, but only if the volume fits.

Read the scale

A reading is only as good as the way it is taken. The end of a rule is often worn, so the rod below is lined up with a clear mark instead of the end.

The left end is at 2.0 cm and the right end is at 7.6 cm, so the rod is 7.6 − 2.0 = 5.6 cm long. The rule’s smallest division is 1 mm = 0.1 cm, so the length is recorded to 0.1 cm: 5.6 cm, or 56 mm. Reading only the right end would give 7.6 cm, which is too long by the 2.0 cm before the rod starts. For an analogue rule, estimate uncertainty in each end reading as half its smallest division, or ±0.05 cm here. Because the length subtracts two end readings, their uncertainties can add to ±0.10 cm. Look straight down at each mark; looking from one side makes the mark appear to move against the scale. This is parallax error.

In this cylinder, the labelled marks are 10 cm³ apart and there are five spaces between each pair, so one small division is 10 ÷ 5 = 2 cm³. Water curves up at the walls. Read the bottom of the curved surface, called the meniscus, with your eye level with it: here it is three small divisions above 30, so the volume is 30 + 3 × 2 = 36 cm³. Half a small division gives an estimated reading uncertainty of ±1 cm³ for this analogue scale.

Before using a balance or a meter, check that it reads zero with nothing being measured. If a balance shows 0.3 g when empty, reset it to zero or subtract 0.3 g from each reading.

Worked measurement

A metal rod has diameter 8.24 mm. Digital calipers fit the rod and have a suitable range and increment. Since 1 mm = 10⁻³ m, the reading is 8.24 × 10⁻³ m. Keep the unit and only the precision justified by the instrument.

Scalars and vectors

A scalar has magnitude and unit. A vector also needs direction. Mass, time, distance, speed, energy and temperature are scalars; displacement, velocity, acceleration, force and weight are vectors. “12 m/s” may be a speed, while a velocity needs a direction such as “12 m/s east”.

To decide, ask whether the direction changes what happens. A bag has a mass of 5 kg whichever way it is turned, so mass is a scalar. A 30 N push on a box has a different effect to the left than to the right, so “the force is 30 N” is incomplete for a force; write “30 N to the left”. Weight is a force, so it has a direction too: downwards, towards the centre of the Earth.