Physical quantities

Structure of a Physical Quantity

Every measurement in physics is a physical quantity, which always consists of two inseparable parts: a numerical magnitude and a unit. The magnitude tells you how much, and the unit tells you of what. Writing one without the other is meaningless in physics.

For example, stating a length as “5” tells you nothing useful, because 5 metres, 5 millimetres, and 5 kilometres describe vastly different physical sizes. The complete quantity must be written as $5 m$, pairing the magnitude with the SI unit.

This rule applies to every quantity in the syllabus, scalar or vector. Mass, time, force, current, and energy must each be expressed with both a numerical value and a recognised unit. In Cambridge 9702 examinations, omitting the unit in a final answer typically loses a mark, even when the arithmetic is correct.

MisconceptionStudents often treat the unit as optional decoration rather than part of the quantity itself. A “bare number” answer is incomplete. The unit defines the physical meaning of the magnitude, so the two must always be written together. Exam cue: always check that every numerical answer carries the correct SI unit.

Examiner InsightCambridge mark schemes routinely penalise missing or wrong units in calculation answers. A correct numerical value with a missing unit often scores zero on that mark. Exam cue: write the unit immediately after the number, even in intermediate steps.

Reasonable Estimates of Physical Quantities

A key skill is the ability to make reasonable estimates of everyday physical quantities to the correct order of magnitude. Examiners test whether you have a realistic feel for the typical size of common physical values, not exact figures.

Estimation requires you to relate unfamiliar quantities to familiar reference values. The mass of an apple is approximately $0.1 kg$, so a bag of ten apples is roughly $1 kg$. A person walking at a steady pace moves at about $1.5 m {s}^{-1}$, which gives a useful benchmark for human-scale speeds.

The table below lists representative estimates that appear in 9702 examinations. Magnitudes within a factor of 2–3 of these values are normally accepted as reasonable.

When estimating an unfamiliar quantity, break the problem into smaller, familiar parts. To estimate the mass of air in a classroom, multiply the room’s volume by the density of air ($kg {m}^{-3}$). This step-by-step decomposition is the technique examiners reward.

Worked Example: Estimating the Mass of Air in a Classroom

Scenario

A typical classroom measures approximately $8 m$ by $6 m$ by $3 m$. Estimate the mass of air it contains, taking the density of air as $1.2 kg {m}^{-3}$.

Step 1 — Calculate the volume of the room

Equation used

$$V=lwh$$

Given

$$l=8 m$$

$$w=6 m$$

$$h=3 m$$

Substitution:

$$V=8\times 6\times 3$$

$$V=144 {m}^{3}$$

Step 2 — Calculate the mass of air

Equation used: density formula, rearranged for mass.

$$m=\rho V$$

Given

$$\rho =1.2 kg {m}^{-3}$$

$$V=144 {m}^{3}$$

Substitution:

$$m=1.2\times 144$$

$$m=172.8 kg$$

Answer

$$m\approx 170 kg (2 s.f.)$$

Interpretation

A typical classroom contains roughly $170 kg$ of air, which is heavier than two adults. This shows that air, despite being invisible, has significant mass on a room-sized scale.

Examiner InsightEstimation questions are marked on order-of-magnitude correctness, not precise values. Answers within a factor of two or three of the expected value usually score full marks. Exam cue: state assumptions clearly, and always check whether the final magnitude looks physically sensible.