Physical Quantities and Units — A-Level Physics Study Guide

For: A-Level Physics candidates sitting A-Level Physics.

Covers: base SI and derived units, scalar vs vector quantity classification, order of magnitude estimation, dimensional homogeneity analysis, and random/systematic error calculation rules.

You should already know: IGCSE Physics, basic algebra and trigonometry.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the A-Level Physics style for educational use. They are not reproductions of past Cambridge International examination papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official Cambridge mark schemes for grading conventions.

1. What Is Physical Quantities and Units?

Physical quantities are measurable properties of objects or natural phenomena, expressed as a numerical magnitude multiplied by a globally standardized unit of measurement. This topic is the foundational framework for all calculations, experimental analysis, and cross-study result comparison across the entire A-Level Physics syllabus, with explicit questions appearing on every theory and practical exam paper to test your ability to apply consistent measurement conventions.

2. Base SI units and derived units

The Système International (SI) is the global standard for scientific measurement, built on 7 independent base quantities with corresponding base units. Only the first 6 are regularly tested in A-Level Physics:

Derived units are formed by multiplying or dividing base units according to the defining formula of a derived quantity. Many derived units have special names for convenience, for example the newton ($\text{N}$) for force, the joule ($\text{J}$) for energy, and the pascal ($\text{Pa}$) for pressure.

Worked Example

Derive the SI base unit for power, defined as the rate of work done:

1. Start with the defining formula: $P=\frac{W}{t}$, where $W$ is work done, $t$ is time
2. Work done $W=F\times d$, where $F$ is force, $d$ is distance
3. Force $F=m\times a$, where $m$ is mass, $a$ is acceleration
4. Acceleration $a=\frac{\Delta v}{t}$, units ${\text{mcdotps}}^{-2}$
5. Substitute back: $W=\text{kg}\times {\text{mcdotps}}^{-2}\times \text{m}={\text{kgcdotpm}}^2{\text{cdotps}}^{-2}$
6. Power units: $\frac{{\text{kgcdotpm}}^2{\text{cdotps}}^{-2}}{\text{s}}={\text{kgcdotpm}}^2{\text{cdotps}}^{-3}$ (the special name for this unit is the watt, $\text{W}$)

Exam tip: Examiners award marks for each step of unit derivation, so always start from the defining formula even if you remember the final base unit.

3. Scalars vs vectors

Physical quantities are split into two categories based on whether they include directional information:

• Scalar: A quantity with magnitude only, no spatial direction. Scalars follow standard arithmetic rules for addition, subtraction, multiplication and division. Common examples include mass, time, temperature, energy, speed, distance, charge, and density.
• Vector: A quantity with both magnitude and spatial direction, which follows vector addition (triangle/parallelogram) rules. Common examples include force, velocity, displacement, acceleration, momentum, and electric field strength.

A common trap here is assuming any quantity with a positive/negative sign is a vector: signs for scalars (e.g. negative charge, negative work done) only indicate a value relative to a reference point, not a direction in space.

Worked Example

State whether electric potential difference (voltage) is a scalar or vector quantity, and justify your answer: Electric potential difference is a scalar. It describes the energy per unit charge between two points in a circuit, with positive/negative values only indicating whether energy is gained or lost by charges, not a spatial direction of flow.

4. Estimation — order of magnitude

Order of magnitude estimation is the process of approximating a physical quantity to the nearest power of 10, used to quickly check if calculation results are physically sensible. The standard convention for rounding is:

• If the leading digit of the value is 3 or higher, round up to the next power of 10
• If the leading digit is less than 3, round down to the lower power of 10

You should memorize these common approximate values for A-Level Physics exam questions:

• Mass of an adult human: ~70 kg
• Height of a standard room: ~3 m
• Speed of a car on a highway: ~30 ${\text{mcdotps}}^{-1}$
• Mass of an apple: ~0.1 kg
• Current in a domestic light bulb: ~0.1 A
• Room temperature: ~300 K

Worked Example

Estimate the order of magnitude of the number of air molecules in a standard bedroom. State your assumptions:

1. Assumptions: Room dimensions = 4 m × 3 m × 3 m = 36 ${\text{m}}^3$, molar volume of air at room temperature = $0.024{\text{ m}}^3{\text{cdotpmol}}^{-1}$, Avogadro constant $N_A=6\times 10^{23}{\text{ mol}}^{-1}$
2. Number of moles of air = $\frac{36}{0.024}=1.5\times 10^3\text{ mol}$
3. Number of molecules = $1.5\times 10^3\times 6\times 10^{23}=9\times 10^{26}$
4. Leading digit is 9 ≥3, so round up to order of magnitude $10^{27}$

Exam tip: Examiners accept any order of magnitude within ±1 power of 10 as correct, as long as you clearly state your working assumptions.

5. Dimensional analysis and homogeneity

Each base quantity has a corresponding dimension: mass $[M]$, length $[L]$, time $[T]$, current $[I]$, temperature $[\theta ]$. Derived quantities have dimensions formed by combining base dimensions according to their defining formula.

A physical equation is dimensionally homogeneous if every term in the equation has the same dimensions. If dimensions do not match across terms, the equation is definitely incorrect. If dimensions do match, the equation may be correct, but dimensionless constants (e.g. $\frac{1}{2}$, $\pi$, 2) cannot be verified with dimensional analysis alone.

Worked Example

Check if the kinematic equation $v^2=u^2+2as$ is dimensionally homogeneous, where $v$ and $u$ are velocities, $a$ is acceleration, and $s$ is displacement:

1. Left-hand side (LHS): $v^2$ has dimensions $[L{T}^{-1}{]}^2=[L^2{T}^{-2}]$
2. Right-hand side (RHS) term 1: $u^2$ has the same dimensions as $v^2$: $[L^2{T}^{-2}]$
3. RHS term 2: $2as$ has dimensions $[L{T}^{-2}]\times [L]=[L^2{T}^{-2}]$ (the constant 2 is dimensionless)
4. All terms have identical dimensions, so the equation is homogeneous.

6. Errors — random, systematic; absolute, fractional, percentage

All experimental measurements contain error, which is categorized into two types:

• Random error: Causes readings to scatter randomly around the true value, reducing measurement precision. It arises from uncontrolled variables in the experiment (e.g. reaction time when using a stopwatch, small fluctuations in ambient temperature). It can be reduced by repeating measurements and calculating an average.
• Systematic error: Causes all readings to differ from the true value by a consistent fixed amount or factor, reducing measurement accuracy. It arises from faulty equipment or unaccounted experimental design flaws (e.g. zero error on a ruler, a stopwatch that runs 10% slow, unaccounted heat loss in calorimetry). It cannot be reduced by repeating readings, but can be eliminated by calibrating equipment or adjusting the experimental method.

Error calculation rules

For a measured value $x$ with absolute error $\Delta x$ (half the range of repeated readings, or the smallest division of the measuring instrument):

• Fractional error: $\frac{\Delta x}{|x|}$
• Percentage error: $\left(\frac{\Delta x}{|x|}\right)\times 100\%$

When combining errors for calculated quantities:

1. For addition/subtraction ($z=x+y$ or $z=x-y$): add absolute errors: $\Delta z=\Delta x+\Delta y$
2. For multiplication/division ($z=x\times y$ or $z=\frac{x}{y}$): add percentage errors: $\%\Delta z=\%\Delta x+\%\Delta y$
3. For powers ($z=x^n$): multiply the percentage error of $x$ by the absolute value of the power: $\%\Delta z=|n|\times \%\Delta x$

Worked Example

A student measures the mass of a metal block as $2.5\pm 0.1\text{ kg}$ and its volume as $1.2\pm 0.05{\text{ m}}^3$. Calculate the density of the block and its percentage uncertainty:

1. Density formula: $\rho =\frac{m}{V}=\frac{2.5}{1.2}\approx 2.08{\text{ kgcdotpm}}^{-3}$
2. Percentage error in mass: $\left(\frac{0.1}{2.5}\right)\times 100=4\%$
3. Percentage error in volume: $\left(\frac{0.05}{1.2}\right)\times 100\approx 4.17\%$
4. Total percentage error in density: $4\%+4.17\%\approx 8.2\%$
5. Final value: $\rho =2.1\pm 0.2{\text{ kgcdotpm}}^{-3}$ (absolute error rounded to 1 significant figure, measured value rounded to the same decimal place as the error, per mark scheme convention)

7. Common Pitfalls (and how to avoid them)

• Wrong move: Treating signed scalars (e.g. charge, work done, temperature) as vectors. Why: Students associate all positive/negative values with spatial direction. Correct move: Only classify a quantity as a vector if its sign refers to a direction in 2D/3D space; signs for scalars only indicate a value relative to a reference point.
• Wrong move: Using grams instead of kilograms for mass when deriving SI base units. Why: The kilogram is the only base unit with a built-in prefix, which causes confusion. Correct move: Always convert mass values to kilograms before deriving base units, unless explicitly asked to use other units.
• Wrong move: Assuming dimensional homogeneity proves an equation is fully correct. Why: Students overlook the limitation of dimensional analysis. Correct move: Dimensional analysis only proves an equation is not definitely wrong; it cannot verify the presence of dimensionless constants like $\frac{1}{2}$ or $\pi$.
• Wrong move: Adding absolute errors for multiplication/division operations. Why: Mixing up error combination rules. Correct move: Use absolute errors only for addition/subtraction; use percentage/fractional errors for all other operations.
• Wrong move: Rounding absolute errors to 2 or more significant figures. Why: Students use standard arithmetic rounding instead of exam convention. Correct move: Always round absolute errors to 1 significant figure, and round the measured value to the same decimal place as the rounded error.

8. Practice Questions (A-Level Physics Style)

Question 1

(a) List the 6 base SI units regularly tested in A-Level Physics, and their corresponding base quantities. (2 marks) (b) Derive the SI base unit for capacitance, given the defining formula $C=\frac{Q}{V}$, where $Q$ is charge (units $\text{Acdotps}$) and $V$ is potential difference (units ${\text{kgcdotpm}}^2{\text{cdotps}}^{-3}{\text{cdotpA}}^{-1}$). (3 marks)

Solution

(a) Correct list: length (metre), mass (kilogram), time (second), electric current (ampere), thermodynamic temperature (kelvin), amount of substance (mole). 2 marks for all 6 correct, 1 mark for 3-5 correct. (b) Step 1: Substitute units into formula: $C=\frac{\text{Acdotps}}{{\text{kgcdotpm}}^2{\text{cdotps}}^{-3}{\text{cdotpA}}^{-1}}$ Step 2: Simplify exponents: $\text{Acdotps}\times {\text{kg}}^{-1}{\text{cdotpm}}^{-2}{\text{cdotps}}^3{\text{cdotpA}}^1={\text{kg}}^{-1}{\text{cdotpm}}^{-2}{\text{cdotps}}^4{\text{cdotpA}}^2$ 3 marks for correct final unit, 1 mark for each correct substitution step.

Question 2

Check if the formula for the time period of a mass-spring system $T=2\pi \sqrt{\frac{m}{k}}$ is dimensionally homogeneous, where $m$ is mass and $k$ is the spring constant with units ${\text{kgcdotps}}^{-2}$. (4 marks)

Solution

1. LHS: Period $T$ has dimensions $[T]$. (1 mark)
2. RHS: Units of $\frac{m}{k}=\frac{\text{kg}}{{\text{kgcdotps}}^{-2}}={\text{s}}^2$, dimensions $[T^2]$. (2 marks)
3. Square root of $\frac{m}{k}$ has dimensions $[T]$, matching the LHS. The constant $2\pi$ is dimensionless. (1 mark) The equation is dimensionally homogeneous.

Question 3

A student measures the radius of a sphere as $5.2\pm 0.1\text{ cm}$. Calculate the percentage uncertainty in the volume of the sphere, given $V=\frac{4}{3}\pi r^3$. (3 marks)

Solution

1. The constants $\frac{4}{3}$ and $\pi$ are dimensionless, so they contribute no uncertainty. (1 mark)
2. Percentage uncertainty in radius $r$: $\left(\frac{0.1}{5.2}\right)\times 100\approx 1.92\%$.
3. Volume depends on $r^3$, so percentage uncertainty in volume = $3\times 1.92\%\approx 5.8\%$. (2 marks for applying power rule and correct final value)

9. Quick Reference Cheatsheet

Key Rules

1. Order of magnitude: Round to nearest power of 10; leading digit ≥3 → round up, <3 → round down.
2. Dimensional homogeneity: All terms in a valid equation have matching dimensions; dimensionless constants cannot be verified with dimensional analysis.
3. Error types: Random error = scatter, reduce by averaging, affects precision; Systematic error = consistent offset, reduce by calibration, affects accuracy.
4. Error combination: Add/subtract → add absolute errors; Multiply/divide → add percentage errors; $x^n$ → multiply percentage error by $|n|$.
5. Error convention: Round absolute error to 1 sig fig, round measured value to match the error's decimal place.

10. What's Next

This topic underpins every module in the A-Level Physics syllabus. You will use unit derivation to verify electromagnetism formulas for capacitance and magnetic flux, vector addition to solve projectile motion and force equilibrium problems, order of magnitude estimation to sanity-check nuclear physics calculations, and error analysis for all practical assessments (Paper 3 and Paper 5) as well as theory questions on experimental methods. Mastering these fundamentals early will eliminate avoidable marks lost to unit mistakes or incorrect uncertainty calculations across all your exam papers.

If you have any questions about unit derivation, error combination, or any other content in this guide, you can ask Ollie, our AI tutor, any time on the homepage. Next, you can move on to our study guide for Kinematics, the first mechanics topic in the A-Level Physics syllabus, which builds directly on the scalar/vector distinction you learned here.

Aligned with the Cambridge International AS & A Level Physics 9702 syllabus. OwlsAi is not affiliated with Cambridge Assessment International Education.