Physical Quantities, Units & Dimensions – Cambridge International AS & A Level Physics (9702)
A physical quantity is a property of a system that can be measured. It must always be expressed as a numerical value + unit and any equation used to relate quantities must be dimensionally homogeneous (the units on both sides are identical).
1. SI Base Quantities (AO1)
| Base Quantity | Symbol | SI Unit (name) | Unit Symbol |
|---|
| Length | ℓ | metre | m |
| Mass | m | kilogram | kg |
| Time | t | second | s |
| Electric current | I | ampere | A |
| Thermodynamic temperature | T | kelvin | K |
| Amount of substance | n | mole | mol |
| Luminous intensity | Iv | candela | cd |
2. Derived Quantities (AO1)
Derived quantities are obtained by combining base quantities. The unit of a derived quantity follows from algebraic combination of the base‑unit symbols.
| Quantity | Symbol | Definition (formula) | Derived unit |
|---|
| Speed | v | v = ℓ/t | m s⁻¹ |
| Acceleration | a | a = Δv/Δt | m s⁻² |
| Force | F | F = ma | kg m s⁻² = N |
| Pressure | P | P = F/A | kg m⁻¹ s⁻² = Pa |
| Energy / Work | E, W | E = Fℓ = ½ mv² | kg m² s⁻² = J |
| Power | P | P = E/t = Fv | kg m² s⁻³ = W |
| Electric charge | Q | Q = It | A s = C |
| Potential difference | V | V = W/Q | kg m² s⁻³ A⁻¹ = V |
| Resistance | R | R = V/I | kg m² s⁻³ A⁻² = Ω |
| Capacitance | C | C = Q/V | A² s⁴ kg⁻¹ m⁻² = F |
| Magnetic flux density | B | B = F/(IL sinθ) | kg s⁻² A⁻¹ = T |
3. SI Prefixes – Order‑of‑Magnitude Scaling (AO1)
| Prefix | Symbol | Factor |
|---|
| giga | G | 10⁹ |
| mega | M | 10⁶ |
| kilo | k | 10³ |
| hecto | h | 10² |
| deca | da | 10¹ |
| (none) | | 10⁰ |
| deci | d | 10⁻¹ |
| centi | c | 10⁻² |
| milli | m | 10⁻³ |
| micro | µ | 10⁻⁶ |
| nano | n | 10⁻⁹ |
| pico | p | 10⁻¹² |
4. Scalars, Vectors & Dimensional Checks (Section 1.4, AO1)
- Scalar: magnitude only (e.g. mass, temperature, speed).
- Vector: magnitude + direction (e.g. displacement, velocity, force).
Key vector operations required for the syllabus:
- Graphical addition/subtraction (tip‑to‑tail).
- Resolution into perpendicular components:
\[
\mathbf{A}=A_{x}\hat{i}+A_{y}\hat{j},\qquad
A_{x}=A\cos\theta,\;A_{y}=A\sin\theta
\]
- Resultant magnitude: \(|\mathbf{A}|=\sqrt{A_{x}^{2}+A_{y}^{2}}\).
Example of dimensional check: In the equation \(v^{2}=u^{2}+2as\), the units of each term are \((\text{m s}^{-1})^{2}= \text{m}^{2}\text{s}^{-2}\); therefore the equation is homogeneous.
5. Significant Figures, Uncertainty & Error Analysis (Section 1.3, AO2)
5.1 Significant Figures
- Report measured quantities with the number of significant figures justified by the instrument.
- Multiplication / division → result has as many sf as the factor with the fewest sf.
- Addition / subtraction → round to the same decimal place as the least‑precise term.
5.2 Types of Uncertainty
- Random (statistical) error – varies from trial to trial; reduced by repeated measurements.
- Systematic error – shifts all measurements in the same direction (zero‑offset, calibration).
- Precision – closeness of repeated measurements (related to random error).
- Accuracy – closeness of the mean value to the true value (affected by systematic error).
5.3 Propagation of Uncertainties (independent random errors)
- Addition / subtraction: \(\displaystyle \Delta R = \sqrt{(\Delta A)^{2}+(\Delta B)^{2}}\)
- Multiplication / division: \(\displaystyle \frac{\Delta R}{|R|}= \sqrt{\left(\frac{\Delta A}{A}\right)^{2}+\left(\frac{\Delta B}{B}\right)^{2}}\)
- Powers & roots: \(\displaystyle \frac{\Delta R}{|R|}=|n|\,\frac{\Delta A}{A}\) for \(R=A^{n}\).
Final answers should be written with the appropriate number of significant figures and the combined uncertainty, e.g. \(2.34\pm0.05\;\text{m}\).
6. Standard Physical Constants (Allowed for Use – AO1)
| Constant | Symbol | Value (SI) |
|---|
| Acceleration due to gravity (Earth) | g | 9.8 m s⁻² (≈10 m s⁻² for estimates) |
| Speed of light in vacuum | c | 3.00 × 10⁸ m s⁻¹ |
| Elementary charge | e | 1.60 × 10⁻¹⁹ C |
| Mass of a proton | mp | 1.67 × 10⁻²⁷ kg |
| Planck’s constant | h | 6.63 × 10⁻³⁴ J s |
| Permittivity of free space | ε₀ | 8.85 × 10⁻¹² F m⁻¹ |
| Permeability of free space | μ₀ | 4π × 10⁻⁷ N A⁻² |
| Density of water (4 °C) | ρwater | 1.0 × 10³ kg m⁻³ |
| Atmospheric pressure (sea level) | Patm | 1.01 × 10⁵ Pa |
7. Reasonable Estimates – Typical A‑Level Quantities (AO1)
| Quantity | Symbol | Typical Value (SI) | Tip / Estimation Aid |
|---|
| Gravitational field strength | g | 9.8 N kg⁻¹ | Same number as acceleration due to gravity |
| Typical laboratory voltage | V | 1–10 V (cells) ; 230 V (mains, UK) | Battery ≈1.5 V, mains ≈10² V |
| Typical current in a circuit | I | 10⁻³–10⁰ A | milli‑ampere for LED circuits, ampere for mains devices |
| Typical resistance of a copper wire (1 m, 1 mm²) | R | ≈1.7 × 10⁻⁸ Ω m · (1 m/1 mm²) ≈ 0.017 Ω | Use ρCu≈1.7 × 10⁻⁸ Ω m |
| Wavelength of visible light | λ | 4 × 10⁻⁷ – 7 × 10⁻⁷ m | ≈5 × 10⁻⁷ m (green) |
| Frequency of a typical radio station | f | 10⁸ – 10⁹ Hz | ≈100 MHz |
| Mass of a typical adult human | m | ≈70 kg | ≈7 × 10¹ kg |
| Energy stored in a 1.5 V AA battery | E | ≈2 × 10³ J | ≈3 Wh ≈ 1.1 × 10⁴ J (use 1 Wh = 3600 J) |
AS‑Level Core Topics (Blocks 1‑11)
8. Kinematics (Section 2.1, AO1)
- Displacement (s) – vector, SI unit = m.
- Distance – scalar, same unit.
- Speed (v) – scalar, \(v = \dfrac{ℓ}{t}\).
- Velocity (u, v) – vector, \( \mathbf{v} = \dfrac{\Delta\mathbf{s}}{\Delta t}\).
- Acceleration (a) – vector, \( \mathbf{a} = \dfrac{\Delta\mathbf{v}}{\Delta t}\).
Equations of uniformly accelerated motion (UAM) (use when acceleration is constant):
\[
\begin{aligned}
v &= u + at\\[2mm]
s &= ut + \tfrac12 at^{2}\\[2mm]
v^{2} &= u^{2}+2as\\[2mm]
s &= \tfrac12 (u+v)t\\[2mm]
s &= vt - \tfrac12 at^{2}
\end{aligned}
\]
Graphical interpretation:
- Gradient of a \(s\)‑vs‑\(t\) graph = speed.
- Gradient of a \(v\)‑vs‑\(t\) graph = acceleration.
- Area under a \(v\)‑vs‑\(t\) graph = displacement.
9. Dynamics (Section 2.2, AO1‑AO2)
- Newton’s First Law – a body remains at rest or in uniform motion unless acted on by a net external force.
- Second Law – \(\mathbf{F}=m\mathbf{a}\) (vector form).
- Third Law – For every action there is an equal and opposite reaction.
- Momentum – \(\mathbf{p}=m\mathbf{v}\); SI unit = kg m s⁻¹.
- Impulse – \(\mathbf{J}= \Delta\mathbf{p}= \mathbf{F}\Delta t\).
- Conservation of linear momentum in isolated systems (elastic & inelastic collisions).
- Friction:
\(\displaystyle F_{\!f}= \mu N\) (static \(\mu_{\!s}\), kinetic \(\mu_{\!k}\)).
- Air‑drag (quadratic approximation): \(F_{\!d}= \tfrac12 C_{\!d}\rho A v^{2}\).
10. Forces, Density & Pressure (Section 2.3, AO1‑AO2)
- Density – \(\rho = \dfrac{m}{V}\) (kg m⁻³).
- Pressure – \(P = \dfrac{F}{A}\) (Pa). Hydrostatic pressure: \(P = \rho g h\).
- Archimedes’ principle – Upthrust = weight of displaced fluid.
- Torque (moment) – \(\tau = rF\sin\theta\); equilibrium when \(\sum\tau = 0\).
- Conditions for static equilibrium: \(\sum\mathbf{F}=0\) and \(\sum\tau =0\).
11. Work, Energy & Power (Section 2.4, AO1‑AO2)
- Work – \(W = \mathbf{F}\cdot\mathbf{s}=Fs\cos\theta\) (J).
- Work‑energy theorem – Net work = change in kinetic energy, \(\Delta K = \tfrac12 m(v^{2}-u^{2})\).
- Gravitational potential energy – \(U = mgh\) (near Earth).
- Elastic potential energy – \(U = \tfrac12 kx^{2}\) (spring).
- Conservation of mechanical energy (no non‑conservative forces): \(K_i+U_i = K_f+U_f\).
- Power – \(P = \dfrac{W}{t}=Fv\) (W). Efficiency \(\eta = \dfrac{\text{useful output}}{\text{input}}\times100\%.\)
12. Deformation of Solids (Section 2.5, AO1‑AO2)
- Stress – \(\sigma = \dfrac{F}{A}\) (Pa).
- Strain – \(\varepsilon = \dfrac{\Delta L}{L_{0}}\) (dimensionless).
- Hooke’s law – \(\sigma = E\varepsilon\) where \(E\) is Young’s modulus (Pa).
- Elastic limit – beyond which permanent deformation occurs.
- Energy stored in a stretched/compressed spring: \(U = \tfrac12 kx^{2}\).
13. Waves (Section 3.1, AO1‑AO2)
- Wave definition – periodic disturbance that transfers energy without permanent transport of matter.
- Transverse vs longitudinal – particle motion ⟂ or ∥ direction of propagation.
- Wave speed – \(v = f\lambda\) (m s⁻¹).
- Wave equation – \(\dfrac{\partial^{2}y}{\partial x^{2}} = \dfrac{1}{v^{2}}\dfrac{\partial^{2}y}{\partial t^{2}}\).
- Intensity \(I = \dfrac{P}{A}\) (W m⁻²); for sound \(I \propto A^{2}\).
- Doppler effect – \(f' = f\frac{v\pm v_{O}}{v\pm v_{S}}\).
- Electromagnetic spectrum – order of decreasing wavelength (γ, X‑ray, UV, visible, IR, microwave, radio).
- Polarisation – only transverse waves can be polarised.
14. Superposition (Stationary Waves, Diffraction & Interference) (Section 3.2, AO1‑AO2)
- Principle of superposition – resultant displacement is algebraic sum of individual displacements.
- Standing waves – formed by two waves of same frequency travelling in opposite directions.
- Condition for nodes in a string fixed at both ends: \(L = n\frac{\lambda}{2}\) (n = 1,2,3…).
- Fundamental frequency \(f_{1}= \dfrac{v}{2L}\).
- Double‑slit interference – path‑difference \(\delta = d\sin\theta\); constructive when \(\delta = m\lambda\).
- Diffraction grating – \(d\sin\theta = m\lambda\) (where \(d\) is grating spacing).
- Intensity pattern for N‑slit grating: \(I = I_{0}\left(\dfrac{\sin(N\beta)}{\sin\beta}\right)^{2}\) with \(\beta = \dfrac{\pi d\sin\theta}{\lambda}\).
15. Electricity (Section 4.1, AO1‑AO2)
- Charge – \(Q = It\) (C).
- Current – \(I = \dfrac{dQ}{dt}\) (A).
- Potential difference – \(V = \dfrac{W}{Q}\) (V).
- Resistance – \(R = \rho\frac{L}{A}\) (Ω); \(\rho\) is resistivity.
- Ohm’s law – \(V = IR\) (valid for ohmic conductors).
- Power in electrical circuits – \(P = VI = I^{2}R = \dfrac{V^{2}}{R}\).
- Temperature coefficient of resistance: \(R = R_{0}[1+\alpha(T-T_{0})]\).
- Series and parallel combinations:
\[
R_{\text{series}}=\sum R_{i},\qquad
\frac{1}{R_{\text{parallel}}}= \sum\frac{1}{R_{i}}.
\]
16. DC Circuits (Section 4.2, AO1‑AO2)
- Standard circuit symbols (battery, resistor, switch, ammeter, voltmeter, etc.).
- Kirchhoff’s Current Law (KCL) – algebraic sum of currents at a node = 0.
- Kirchhoff’s Voltage Law (KVL) – algebraic sum of potential differences round any closed loop = 0.
- Potential divider – \(V_{R} = V_{\text{total}}\frac{R}{R_{\text{total}}}\).
- Internal resistance of a cell: terminal voltage \(V = \mathcal{E} - Ir\).
- Power rating of resistors and fuses; safe design considerations.
17. Particle Physics & Radioactivity (Section 5.1, AO1‑AO2)
- Alpha (α) particles – \(^4_2\!He\); charge +2e, mass ≈ 4 u.
- Beta (β) particles – electrons (\(\beta^{-}\)) or positrons (\(\beta^{+}\)).
- Gamma (γ) rays – high‑energy photons; no charge, penetrate deeply.
- Nuclear notation – \(\,^{A}_{Z}\!X\) where \(A\) = mass number, \(Z\) = atomic number.
- Radioactive decay law: \(N = N_{0}e^{-\lambda t}\); half‑life \(t_{1/2} = \frac{\ln2}{\lambda}\).
- Binding energy per nucleon curve – explains stability of iron‑peak nuclei.
- Basic quark model (up, down, strange) – optional for A‑level extension.
A‑Level Extensions (Blocks 12‑25 – brief outline)
- Rotational dynamics – moment of inertia, torque, angular momentum, conservation of angular momentum, rotational kinetic energy.
- Gravitation – Newton’s law of universal gravitation, gravitational potential energy, satellite motion, escape velocity.
- Thermal physics – ideal gas law, specific heat capacities, latent heat, kinetic theory, first law of thermodynamics.
- Electric fields & potentials – field lines, equipotentials, capacitance of parallel‑plate capacitor, energy stored in a capacitor.
- Magnetic fields – Biot–Savart law, Ampère’s law, force on a moving charge, electromagnetic induction (Faraday’s law, Lenz’s law), AC circuits.
- Waves – advanced – standing waves on strings & air columns, resonance, quality factor, wave speed in different media.
- Optics – reflection, refraction, lenses, optical instruments, wave optics (interference, diffraction, polarisation).
- Modern physics – photoelectric effect, de Broglie wavelength, atomic models, nuclear binding energy, particle accelerators.
- Astrophysics (optional) – Hubble’s law, black‑body radiation, stellar lifetimes.
These extensions are examined at A‑Level (9702) and can be studied after mastering the AS core topics.