All🏃 Motion🧲 Force & Laws⚡ Work, Energy & Power🪐 Gravitation💧 Pressure & Fluids🔥 Heat & Thermodynamics🌊 Waves & Sound🔦 Light & Optics🔌 Electricity🧭 Magnetism⚛️ Modern Physics📏 Measurement & Errors🌀 Rotation & Rigid Bodies🔔 Oscillations & Vibration🚿 Fluid Flow🌡️ Heat Transfer🔋 Fields & Capacitors〰️ AC Circuits & Induction📡 Electromagnetic Waves🌈 Wave Optics🚀 Relativity🔬 Quantum & Atoms🔩 Semiconductors & Electronics🏗️ Stress, Strain & Structures🎛️ Signals & Control🔭 Astrophysics & Cosmology

🏃MOTION

Speed, acceleration and the equations of motion

Speedv = s / tMiddle

Accelerationa = (v − u) / tMiddle

First equation of motionv = u + atHigh

Second equation of motions = ut + ½at²High

Third equation of motionv² = u² + 2asHigh

Centripetal accelerationa = v² / rAdvanced

🧲FORCE & LAWS

Newton's laws, momentum and friction

Newton's second lawF = m aMiddle

WeightW = m gMiddle

Momentump = m vHigh

ImpulseF t = m v − m uHigh

Friction forcef = μ NHigh

⚡WORK, ENERGY & POWER

Doing work, storing energy, using it fast

Work doneW = F sMiddle

Kinetic energyKE = ½ m v²High

Gravitational potential energyPE = m g hHigh

PowerP = W / tMiddle

Efficiencyη = (useful output / total input) × 100%High

🪐GRAVITATION

Universal gravity and orbits

Newton's law of gravitationF = G m₁m₂ / r²High

Escape velocityvₑ = √(2GM / R)Advanced

Orbital speedv = √(GM / r)Advanced

💧PRESSURE & FLUIDS

Density, pressure and floating

Densityρ = m / VMiddle

PressureP = F / AMiddle

Pressure in a liquidP = ρ g hHigh

Archimedes' principleUpthrust = ρ V gHigh

🔥HEAT & THERMODYNAMICS

Temperature, heat energy and gases

Heat energyQ = m c ΔTHigh

Latent heatQ = m LHigh

Ideal gas equationP V = n R TAdvanced

First law of thermodynamicsΔU = Q − WAdvanced

🌊WAVES & SOUND

Wave speed, frequency and pitch

Wave equationv = f λMiddle

Time periodT = 1 / fMiddle

Simple pendulumT = 2π √(l / g)High

🔦LIGHT & OPTICS

Mirrors, lenses and refraction

Lens / mirror equation1/f = 1/v − 1/uHigh

Magnificationm = v / u = hᵢ / hₒHigh

Snell's lawn₁ sin θ₁ = n₂ sin θ₂High

Power of a lensP = 1 / f (in metres)High

🔌ELECTRICITY

Current, voltage, resistance and bills

Ohm's lawV = I RMiddle

Charge flowQ = I tMiddle

Electrical powerP = V I = I²RHigh

Resistors in seriesR = R₁ + R₂ + …High

Resistors in parallel1/R = 1/R₁ + 1/R₂ + …High

Coulomb's lawF = k q₁q₂ / r²Advanced

🧭MAGNETISM

Fields, motors and transformers

Force on a current-carrying wireF = B I LHigh

Force on a moving chargeF = q v B sin θAdvanced

Transformer equationVₛ / Vₚ = Nₛ / NₚHigh

⚛️MODERN PHYSICS

Photons, relativity and the nucleus

Photon energyE = h fAdvanced

Photoelectric equationhf = φ + KEₘₐₓAdvanced

Mass–energy equivalenceE = m c²Advanced

Radioactive decayN = N₀ (½)^(t / t½)Advanced

de Broglie wavelengthλ = h / pAdvanced

📏MEASUREMENT & ERRORS

Units, dimensions and how good a reading is

Dimensional formula[Q] = Mᵃ Lᵇ TᶜHigh

Mean and spreadx̄ = Σxᵢ / n ; s = √[Σ(xᵢ − x̄)² / (n − 1)]High

Uncertainty of the meanu(x̄) = s / √nAdvanced

Percentage error% error = (measured − true) / true × 100Middle

🌀ROTATION & RIGID BODIES

Torque, inertia and angular momentum

Torqueτ = r F sin θHigh

Newton's law for rotationτ = I αAdvanced

Common moments of inertiaDisc ½MR² · Ring MR² · Rod (centre) ML²/12Advanced

Angular momentumL = I ωAdvanced

Rotational kinetic energyKE = ½ I ω²Advanced

Parallel-axis theoremI = I_cm + M d²Advanced

🔔OSCILLATIONS & VIBRATION

Simple harmonic motion, damping and resonance

Simple harmonic motionx = A cos(ωt + φ) , a = −ω²xAdvanced

Mass on a springT = 2π √(m / k)High

Hooke's lawF = k xMiddle

Energy in SHME = ½ k A²Advanced

Damped oscillationx = A e^(−bt/2m) cos(ω_d t)Engineering

Resonancef_driver = f_naturalHigh

🚿FLUID FLOW

Continuity, Bernoulli, viscosity and pipe losses

Continuity equationA₁v₁ = A₂v₂High

Bernoulli's equationP + ½ρv² + ρgh = constantAdvanced

Reynolds numberRe = ρ v D / μEngineering

Stokes' dragF = 6π μ r vEngineering

Pipe friction lossh_f = f (L/D)(v²/2g)Engineering

🌡️HEAT TRANSFER

Conduction, convection and radiation

Conduction (Fourier's law)Q/t = k A ΔT / LAdvanced

Convection (Newton's cooling)Q/t = h A (T_s − T_∞)Engineering

Stefan–Boltzmann lawP = ε σ A T⁴Advanced

Wien's displacement lawλ_max T = 2.898×10⁻³ m·KAdvanced

Thermal expansionΔL = α L₀ ΔTHigh

Carnot efficiencyη = 1 − T_c / T_hAdvanced

Entropy changeΔS = Q / TAdvanced

🔋FIELDS & CAPACITORS

Electric fields, potential and stored charge

Electric fieldE = F / q = V / dHigh

Potential of a point chargeV = k q / rAdvanced

CapacitanceC = Q / VHigh

Energy in a capacitorE = ½ C V²High

RC chargingV = V₀(1 − e^(−t/RC))Engineering

Gauss's lawΦ_E = Q_enclosed / ε₀Advanced

〰️AC CIRCUITS & INDUCTION

RMS values, reactance, resonance and induced emf

RMS valuesV_rms = V₀ / √2High

Faraday's lawε = −N dΦ/dtAdvanced

ReactanceX_L = 2πfL , X_C = 1/(2πfC)Engineering

Impedance of RLCZ = √[R² + (X_L − X_C)²]Engineering

Resonant frequencyf₀ = 1 / (2π√(LC))Engineering

AC power and power factorP = V_rms I_rms cos φEngineering

Energy in an inductorE = ½ L I²Engineering

📡ELECTROMAGNETIC WAVES

Light as a field, intensity and the spectrum

Speed of light from constantsc = 1 / √(μ₀ε₀)Advanced

Inverse-square intensityI = P / (4πr²)High

Field amplitudes in a waveE = c BAdvanced

Doppler shiftf' = f (v ± v_o) / (v ∓ v_s)Advanced

🌈WAVE OPTICS

Interference, diffraction and polarization

Young's double slitw = λ D / sAdvanced

Diffraction gratingd sin θ = n λAdvanced

Rayleigh resolutionθ ≈ 1.22 λ / DAdvanced

Malus's lawI = I₀ cos²θAdvanced

Critical anglesin C = n₂ / n₁High

🚀RELATIVITY

Time dilation, length contraction and energy

Lorentz factorγ = 1 / √(1 − v²/c²)Advanced

Time dilationΔt = γ Δt₀Advanced

Length contractionL = L₀ / γAdvanced

Relativistic energyE² = (pc)² + (mc²)²Advanced

🔬QUANTUM & ATOMS

Energy levels, uncertainty and spectra

Bohr energy levelsEₙ = −13.6 / n² eVAdvanced

Spectral line energyΔE = h c / λAdvanced

Heisenberg uncertaintyΔx Δp ≥ ħ / 2Advanced

Particle in a boxEₙ = n²h² / (8mL²)Advanced

Nuclear binding energyE_b = Δm c²Advanced

Activity and decay constantA = λN , t½ = ln2 / λAdvanced

🔩SEMICONDUCTORS & ELECTRONICS

Diodes, transistors and amplifiers

Diode equationI = I_s (e^(V/nV_T) − 1)Engineering

Transistor current gainI_C = β I_BEngineering

Semiconductor conductivityσ = q(n μₑ + p μₕ)Engineering

Inverting amplifier gainA_v = −R_f / R_inEngineering

Gain in decibelsdB = 20 log₁₀(V_out / V_in)Engineering

ResistivityR = ρ L / AHigh

🏗️STRESS, STRAIN & STRUCTURES

Elasticity, beams, torsion and buckling

Stress and strainσ = F / A , ε = ΔL / L₀High

Young's modulusE = σ / εHigh

Bending stress in a beamσ = M y / IEngineering

Torsion of a shaftτ = T r / JEngineering

Euler buckling loadP_cr = π²EI / (KL)²Engineering

Elastic strain energyU = ½ F x = ½ σ ε VAdvanced

🎛️SIGNALS & CONTROL

Sampling, filters and feedback systems

Nyquist samplingf_s > 2 f_maxEngineering

RC filter cut-offf_c = 1 / (2πRC)Engineering

Closed-loop gainG_cl = G / (1 + G H)Engineering

PID controlleru(t) = K_p e + K_i ∫e dt + K_d de/dtEngineering

Second-order responseω_d = ω_n √(1 − ζ²)Engineering

🔭ASTROPHYSICS & COSMOLOGY

Stars, orbits and the expanding universe

Kepler's third lawT² = 4π²r³ / (G M)Advanced

Stellar luminosityL = 4π R² σ T⁴Advanced

Hubble's lawv = H₀ dAdvanced

Schwarzschild radiusr_s = 2GM / c²Advanced

Redshiftz = Δλ / λ₀ ≈ v / cAdvanced