Physical quantities and measurement

Physics is a science of measurement. Every physical law is a relation between quantities, and to test it you must be able to express a quantity as a number with a unit. In the exam this topic appears both as a separate question and in every calculation, as correct unit conversion. Unit errors are among the most common mistakes in problems, so mastering this topic pays off across the whole physics course.

§1. Physical quantities and measurement

A physical quantity is a measurable property of a body or phenomenon: length, mass, time, speed, force, temperature. The value of a quantity = number × unit: m = 3 kg, t = 20 s. A number without a unit means nothing in physics: is “a length of 5” five metres or five centimetres?

Measurement means comparing a quantity with a quantity of the same kind taken as the unit. A measurement is direct (length with a ruler, mass with a balance) or indirect (via a formula: density ρ = m/V, speed v = s/t).

• A scalar quantity is given by a number alone: mass, time, energy, temperature, work.
• A vector quantity has a size and a direction: displacement, velocity, acceleration, force, momentum.

Mass and weight are different quantities: mass is measured in kilograms, while weight is a force, measured in newtons.

§2. The International System of Units (SI)

Since 1960 the whole world has used the SI. It has seven base units:

Since 2019 all base units have been defined through fundamental constants: for example, the metre is the distance light travels in vacuum in 1/299 792 458 s, and the kilogram is fixed through the Planck constant. So the standards no longer depend on physical objects.
All other units are derived units, obtained from the base units through formulas:

• 1 N = 1 kg·m/s² (force, from F = ma);
• 1 J = 1 N·m (work), 1 W = 1 J/s (power), 1 Pa = 1 N/m² (pressure);
• 1 C = 1 A·s (charge), 1 V = 1 J/C (voltage), 1 Ω = 1 V/A (resistance);
• 1 Hz = 1 s⁻¹ (frequency).

Non-SI units are also used: the litre, tonne, minute, hour, km/h, atmosphere, electronvolt. Convert them to SI before putting them into formulas.

§3. Prefixes and unit conversion

Very large or very small numbers are written in standard form: 0.00052 = 5.2·10⁻⁴, 340 000 = 3.4·10⁵.

For units of area and volume the prefix is squared or cubed too:

• 1 cm² = (10⁻² m)² = 10⁻⁴ m², 1 mm² = 10⁻⁶ m²;
• 1 cm³ = 10⁻⁶ m³, 1 l = 1 dm³ = 10⁻³ m³.

Speed: 1 km/h = 1000 m / 3600 s = 1/3.6 m/s. So 72 km/h = 20 m/s and 36 km/h = 10 m/s; to go from m/s to km/h, multiply by 3.6.
Method: replace each unit by its value in SI and work out the numbers.

Worked example

Convert 0.5 g/cm³ to kg/m³.

1. 1 g = 10⁻³ kg, 1 cm³ = 10⁻⁶ m³.
2. 0.5·10⁻³/10⁻⁶ = 0.5·10³ = 500.
3. Answer: 500 kg/m³ (it helps to remember 1 g/cm³ = 1000 kg/m³).

§4. Using measuring instruments

On a scale, first find the scale interval: divide the difference between two neighbouring numbered marks by the number of divisions between them, e.g. (20 − 10)/5 = 2 units.

• The instrument uncertainty is usually taken as half the scale interval.
• Read the scale looking straight at it, otherwise you get a parallax error; in a measuring cylinder read the bottom of the meniscus.
• The range is the largest value the instrument can measure; exceeding it can damage the instrument.

The stacking method measures very small quantities: measure many identical items together and divide by their number. The diameter of a wire comes from a tight coil on a pencil, the thickness of a sheet from the thickness of a book, a pendulum’s period from the time of many swings. The method reduces the uncertainty by the number of items.

Worked example

20 turns of wire are wound tightly on a pencil and cover 1.2 cm. What is the wire’s diameter?

1. d = l/N = 1.2 cm/20.
2. d = 0.06 cm = 0.6 mm.

§5. Measurement uncertainty

No measurement is perfectly exact. A result is written as

• The absolute uncertainty Δx is how far the measured value may differ from the true value.
• The relative uncertainty shows the quality of the measurement:

• Random errors differ each time; repeating the measurement and averaging reduces them.
• Systematic errors always push the same way (a badly adjusted instrument, a wrong zero); averaging does not help, so find and remove the cause.

The number of digits in a result must match its precision: round the result to the same decimal place as the uncertainty.

Worked example

A rope has length l = (17.2 ± 0.005) m. Find the relative uncertainty.

1. ε = Δl/l = 0.005/17.2.
2. ε ≈ 0.00029.
3. As a percentage: ε ≈ 0.03 %.

§6. Indirect measurement and dimensional analysis

When a quantity is found from a formula, its uncertainty comes from the uncertainties of the measured quantities. For products and quotients the relative uncertainties add: ε_ρ = ε_m + ε_V; for sums and differences the absolute uncertainties add.

Dimensional analysis checks a formula: both sides of an equation must have the same units. For example, s = at²/2: m/s²·s² = m, correct; the pendulum period T = 2π√(l/g): √(m/(m/s²)) = s, correct. If the unit of your answer is not what you expect, the formula is wrong.

Worked example

A body has mass m = (200 ± 1) g and volume V = (80 ± 2) cm³. Find its density and the uncertainty.

1. ρ = m/V = 200/80 = 2.5 g/cm³ = 2500 kg/m³.
2. Relative uncertainty: ε = 1/200 + 2/80 = 0.005 + 0.025 = 0.03 = 3 %.
3. Δρ = 0.03·2500 = 75 kg/m³ → ρ = (2500 ± 80) kg/m³.

Check yourself

1. What is 54 km/h in m/s?

2. How many m³ are in 250 cm³?

3. A scale from 0 to 10 has 20 divisions. Scale interval?

4. Time t = (40 ± 0.2) s. Relative uncertainty?

5. How many newtons is 1 kPa·cm²?

6. Which quantity is a vector: mass, work, momentum, temperature?

7. A book of 100 sheets is 1 cm thick. Thickness of one sheet in millimetres?

8. How many base units does the SI have?

Common mistakes

• Using 10⁻² to convert cm² to m²: the correct factor is 10⁻⁴.
• Multiplying by 3.6 to convert km/h to m/s: you must divide.
• Confusing relative and absolute uncertainty, or not turning the relative one into a percentage.
• Calling mass “weight”: mass is in kg, weight is a force in N.
• Writing every digit from the calculator: digits beyond the precision are meaningless.
• Working with mixed units (one quantity in grams, another in kilograms): convert everything to SI first.