use a = –ω2x and recall and use, as a solution to this equation, x = x0 sin ωt

Cambridge IGCSE / A‑Level Physics (9702) – Comprehensive Revision Notes

Assessment Objectives (AO)

• AO1 – Knowledge & Understanding: recall definitions, formulas and concepts.
• AO2 – Application: use knowledge to solve quantitative problems and interpret diagrams.
• AO3 – Practical & Experimental Skills: plan investigations, analyse data, evaluate uncertainties (Paper 3/5).

Common Symbols & Units

Symbol	Quantity	Unit	Typical Formula
x	Displacement from equilibrium	m	‑
x_0	Amplitude	m	‑
t	Time	s	‑
v	Velocity	m s⁻¹	v = dx/dt
a	Acceleration	m s⁻²	a = dv/dt = d²x/dt²
ω	Angular frequency	rad s⁻¹	ω = √(k/m) = √(g/L) …
f	Frequency	Hz	f = ω/2π
T	Period	s	T = 1/f = 2π/ω
φ	Phase constant	rad	‑
k	Spring constant	N m⁻¹	‑
m	Mass	kg	‑
g	Acceleration due to gravity	m s⁻²	≈9.81
I	Moment of inertia	kg m²	‑
Q	Electric charge	C	‑
V	Potential difference	V	‑
R	Resistance	Ω	‑
C	Capacitance	F	‑
μ₀, ε₀	Permeability & permittivity of free space	H m⁻¹, F m⁻¹	‑

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1. AS‑Level Topics (1‑11)

1.1 Quantities, Units & Prefixes

• SI base units and derived units.
• Common prefixes (k, M, m, µ, n) and conversion practice.

1.2 Kinematics (Straight‑line Motion)

Equation	When to use
v = u + at	constant acceleration
s = ut + ½at²	displacement with known u, a
v² = u² + 2as	no time needed

Example: A car accelerates from rest at 2 m s⁻² for 5 s. Find its final speed and distance travelled.

1. v = 0 + (2)(5) = 10 m s⁻¹
2. s = 0·5 + ½(2)(5)² = 25 m

1.3 Forces & Dynamics

• Newton’s 1st, 2nd, 3rd laws.
• Weight, normal reaction, tension, friction (μs, μk).
• Resultant force: ΣF = ma.

Example: A 3 kg block slides down a 30° incline with μk = 0.15. Find acceleration.

Component of weight down the plane: mg sin 30° = 3·9.81·0.5 = 14.7 N.
Friction: μk N = 0.15·mg cos 30° = 0.15·3·9.81·0.866 ≈ 3.8 N.
Net force = 14.7 – 3.8 = 10.9 N ⇒ a = F/m = 10.9/3 ≈ 3.6 m s⁻².

1.4 Energy, Work & Power

Quantity	Formula	Units
Work	W = F·s·cosθ	J
Kinetic energy	K = ½mv²	J
Gravitational potential energy	U = mgh	J
Power	P = W/t = Fv	W

Example: A 0.5 kg ball is dropped from 2 m. Find speed just before impact (ignore air resistance).

mgh = ½mv² ⇒ v = √(2gh) = √(2·9.81·2) ≈ 6.26 m s⁻¹.

1.5 Momentum & Collisions

• Linear momentum p = mv.
• Impulse J = Δp = FΔt.
• Conservation of momentum for isolated systems (elastic & inelastic).

Example (elastic): 0.2 kg bullet (v = 400 m s⁻¹) embeds in 0.8 kg block initially at rest. Find final speed.

p_initial = 0.2·400 = 80 kg m s⁻¹.
p_final = (0.2+0.8)v ⇒ v = 80/1.0 = 80 m s⁻¹.

1.6 Material Properties & Deformation

Quantity	Formula	Units
Stress	σ = F/A	Pa
Strain	ε = ΔL/L	‑ (dimensionless)
Young’s modulus	E = σ/ε	Pa
Hooke’s law	F = –k x	N

Example: A steel wire (A = 2 mm², E = 2×10¹¹ Pa) is stretched by 1 mm. Find the force.

ΔL/L = 0.001/0.5 m = 2×10⁻³ (assuming original length 0.5 m).
σ = E·ε = 2×10¹¹·2×10⁻³ = 4×10⁸ Pa.
F = σA = 4×10⁸·2×10⁻⁶ = 800 N.

1.7 Waves

• Wave speed v = fλ.
• Longitudinal vs transverse.
• Superposition & standing waves.

Example: A string fixed at both ends vibrates at its fundamental frequency of 120 Hz with wavelength 2 m. Find wave speed.

v = fλ = 120·2 = 240 m s⁻¹.

1.8 Electricity (Static)

Quantity	Formula
Charge	Q = It
Current density	J = σE
Coulomb’s law	F = k_e Q₁Q₂/r²

Example: A current of 3 A flows for 5 s. Find charge transferred.

Q = It = 3·5 = 15 C.

1.9 D.C. Circuits

• Ohm’s law: V = IR.
• Series: R_eq = ΣR, V_total = ΣV, I same.
• Parallel: 1/R_eq = Σ1/R, I_total = ΣI, V same.
• Kirchhoff’s rules (junction & loop).

Example (parallel): Two resistors, 4 Ω and 6 Ω, are connected across a 12 V battery. Find total current.

1/R_eq = 1/4 + 1/6 = (3+2)/12 = 5/12 ⇒ R_eq = 12/5 = 2.4 Ω.
I = V/R_eq = 12/2.4 = 5 A.

1.10 Particle Physics

• Charge, mass, and spin of common particles (e, p, n, α, β).
• Radioactive decay types: α, β⁻, β⁺, γ.
• Half‑life law: N = N₀e^{-λt}, λ = ln2 / t_{½}.

Example: A sample contains 1.0 g of ^{60}Co (t_{½}=5.27 yr). How many atoms remain after 10 yr?

Number initially N₀ = (1.0 g / 60 g mol⁻¹)·6.02×10²³ ≈ 1.00×10²².
λ = ln2 / 5.27 yr = 0.131 yr⁻¹.
N = N₀ e^{-λ·10} ≈ 1.00×10²²·e^{-1.31} ≈ 2.7×10²¹ atoms.

1.11 Simple Harmonic Motion (SHM)

Fundamental Equation

$$\frac{d^{2}x}{dt^{2}} = -\omega^{2}x \qquad\text{(acceleration opposite to displacement)}

General Solution

$$x(t)=x_{0}\sin(\omega t+\phi)$$

• Amplitude x₀ – maximum displacement.
• Angular frequency ω – determines period T = 2π/ω.
• Phase constant φ – set by initial conditions.

Deriving ω for Common Systems

1. Mass‑spring system (Hooke’s law, F = –kx):
   • ma = –kx ⇒ a = –(k/m)x.
   • Compare with a = –ω²x ⇒ ω = √(k/m).
2. Simple pendulum (small‑angle):
   • Torque τ = –mgL sinθ ≈ –mgLθ.
   • Iα = τ with I = mL² ⇒ mL² θ̈ = –mgLθ.
   • θ̈ = –(g/L)θ ⇒ ω = √(g/L).

Velocity & Acceleration

$$v(t)=\frac{dx}{dt}= \omega x_{0}\cos(\omega t+\phi)$$

$$a(t)=\frac{d^{2}x}{dt^{2}}= -\omega^{2}x_{0}\sin(\omega t+\phi)= -\omega^{2}x(t)$$

Energy in SHM

Quantity	Expression
Maximum kinetic energy	K_{max}=½mω²x₀²
Maximum potential (elastic) energy	U_{max}=½k x₀² = ½mω²x₀²
Total mechanical energy	E = K + U = ½k x₀² (constant)

Worked Example (Mass‑spring)

Problem: A 0.5 kg mass attached to a spring (k = 200 N m⁻¹) is pulled 0.10 m from equilibrium and released from rest. Find its displacement after 0.05 s.

1. ω = √(k/m) = √(200/0.5) = 20 rad s⁻¹.
2. Initial conditions: x(0)=0.10 m, v(0)=0 ⇒ φ = –π/2, so use cosine form: $$x(t)=x_{0}\cos(\omega t)$$
3. x(0.05)=0.10 cos(20·0.05)=0.10 cos 1.0 ≈ 0.054 m.

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2. A‑Level Extensions (Topics 12‑25)

2.1 Circular Motion

• Uniform circular motion: a_c = v²/r = ω²r.
• Period T = 2πr/v = 2π/ω.

Example: A car rounds a curve of radius 50 m at 20 m s⁻¹. Find centripetal acceleration.

a_c = v²/r = 20²/50 = 8 m s⁻².

2.2 Gravitation

Quantity	Formula
Gravitational force	F = G M m / r²
Field strength	g = GM / r²
Orbital speed (circular)	v = √(GM/r)
Period of satellite	T = 2π√(r³/GM)

2.3 Thermodynamics – Ideal Gas & First Law

• Ideal‑gas equation: pV = nRT.
• First law: ΔU = Q – W.
• For a monatomic ideal gas: U = (3/2)nRT.

Example: 1 mol of an ideal gas is heated from 300 K to 400 K at constant volume. Find ΔU.

ΔU = (3/2)nRΔT = 1.5·8.31·(400–300) ≈ 1245 J.

2.4 Kinetic Theory of Gases

• Mean kinetic energy: ½ m ⟨v²⟩ = (3/2)k_B T.
• Pressure from molecular impacts: p = (1/3) N m ⟨v²⟩/V.

2.5 Damped, Forced & Resonant Oscillations

Equation of motion (damped):

$$m\ddot{x}+b\dot{x}+kx=0$$

Natural frequency ω₀ = √(k/m); damping ratio ζ = b/(2√{mk}).

Resonance occurs when driving frequency ω ≈ ω₀ (small ζ).

2.6 Electric & Magnetic Fields

Quantity	Formula
Electric field	E = F/q = V/d (for uniform field)
Magnetic field (Biot‑Savart)	B = μ₀I/2πr (long straight wire)
Force on moving charge	F = q(v×B)

2.7 A.C. Circuits

• rms values: V_{rms}=V_{max}/√2, I_{rms}=I_{max}/√2.
• Impedance: Z = √(R²+(X_L−X_C)²).
• Resonance in series LCR: ω₀ = 1/√(LC).

2.8 Quantum Physics

• Photoelectric equation: hf = Φ + ½mv_{max}².
• De Broglie wavelength: λ = h/p.
• Energy levels of hydrogen: E_n = –13.6 eV / n².

2.9 Nuclear Physics

• Binding energy per nucleon, mass–energy equivalence (E=Δmc²).
• Fission: heavy nucleus → lighter fragments + neutrons + energy.
• Fusion: light nuclei combine, releasing energy.

2.10 Medical Physics (Imaging)

• X‑ray production: bremsstrahlung & characteristic radiation.
• Attenuation: I = I₀e^{-μx} (μ = linear attenuation coefficient).
• CT number (Hounsfield unit): HU = 1000·(μ−μ_{water})/μ_{water}.

2.11 Astronomy

• Hubble’s law: v = H₀d.
• Luminosity distance, apparent & absolute magnitude relation.
• Kepler’s laws (derived from gravitation).

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3. Practical & Experimental Toolbox (AO3)

3.1 Planning an Investigation

1. State a clear, testable hypothesis.
2. Identify independent, dependent and controlled variables.
3. Choose appropriate apparatus (ensure safety & calibration).
4. Sketch a labelled diagram of the set‑up.
5. Write a step‑by‑step method (include repeatability).

3.2 Data Collection & Presentation

• Record raw data in a table with units and uncertainties.
• Use digital timers, multimeters, or Vernier sensors where possible.
• Plot graphs with correctly labelled axes, error bars, and best‑fit lines.

3.3 Uncertainty Analysis

1. Identify sources of random error (reading, timing) and systematic error (zero‑offset, friction).
2. Calculate absolute and relative uncertainties (Δx, Δx/x).
3. Propagate uncertainties using:
   • Addition/subtraction: Δz = √(Δa²+Δb²).
   • Multiplication/division: (Δz/z) = √((Δa/a)²+(Δb/b)²).
4. State final result to the appropriate number of significant figures.

3.4 Sample Investigation – Period of a Simple Pendulum

Step	Action
1	Set up a light string of length L = 0.50 m with a small bob; attach a stopwatch.
2	Displace the bob by ≤ 5° and release; measure the time for 20 complete oscillations. Repeat three times.
3	Average the three total times, then divide by 20 to obtain T.
4	Calculate g using g = 4π²L/T² and propagate uncertainties from L (±0.001 m) and T (stopwatch ±0.2 s).

Typical result: T ≈ 1.42 s → g ≈ 9.8 m s⁻² (within experimental uncertainty).

3.5 Mark‑scheme Tips for Paper 3/5

• Show a clear hypothesis, method, and risk assessment (5 marks).
• Present data in a well‑labelled table and graph (5 marks).
• Include a quantitative analysis with uncertainties (5 marks).
• Discuss limitations, improvements and evaluate the conclusion (5 marks).

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4. Quick Revision Checklist (Cambridge 9702)

• Can you write the differential equation ¨x = –ω²x and explain its physical meaning?
• Do you know how to obtain ω for a mass‑spring system and for a simple pendulum?
• Can you express the general SHM solution x(t)=x₀ sin(ωt+φ) and determine φ from any pair of initial conditions?
• Are you able to convert between f, ω, T and use the correct units?
• Can you calculate kinetic, potential and total energy in SHM and show energy conservation?
• Do you understand the derivations and key formulas for all AS‑level topics (1‑11) and A‑level extensions (12‑25)?
• Are you comfortable planning a practical, analysing uncertainties and presenting data as required for AO3?
• Can you apply a = –ω²x directly to find displacement, velocity or acceleration at a given time?

5. Suggested Diagram (Insert when printing)

Mass‑spring system: a block of mass m attached to a horizontal spring (constant k) displaced a distance x from equilibrium. Arrows indicate the restoring force F = –kx and the resulting acceleration a = –ω²x toward the equilibrium position.