make reasonable estimates of physical quantities included within the syllabus

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)

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.

3. SI Prefixes – Order‑of‑Magnitude Scaling (AO1)

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:

1. Graphical addition/subtraction (tip‑to‑tail).
2. 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

   \]

3. 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)

7. Reasonable Estimates – Typical A‑Level Quantities (AO1)

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): \(Ki+Ui = Kf+Uf\).
• 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.