GCE A/L Physics cheat sheet

SI base units: metre, kilogram, second, ampere, kelvin, mole, candela; derived units are products of these, e.g. newton = kg·m·s⁻², joule = kg·m²·s⁻².

Homogeneity of units: every term in a valid physical equation must reduce to the same base-unit combination; this is used to check or derive equations.

Prefixes: p = 10⁻¹², n = 10⁻⁹, µ = 10⁻⁶, m = 10⁻³, k = 10³, M = 10⁶, G = 10⁹, T = 10¹².

Random error scatters readings about a mean and is reduced by repetition and averaging; systematic error shifts all readings by a constant and is corrected by calibration or zeroing.

Precision is the resolution of the instrument; accuracy is closeness to the true value; a precise instrument can still be inaccurate if systematically offset.

Combining uncertainties: for y = a ± b add absolute uncertainties; for y = ab or a/b add fractional uncertainties; for y = aⁿ the fractional uncertainty is |n| times that of a.

Percentage difference or uncertainty = (uncertainty / value) × 100%; percentage difference between two results = (difference / mean) × 100%.

A micrometer screw gauge reads to 0.01 mm and a vernier caliper to 0.1 mm (or 0.05 mm); both need a zero-error check before use.

On a graph, the gradient is Δy/Δx with uncertainty found from the steepest and shallowest lines of best fit through the error bars.

Uniform acceleration: v = u + at, s = ut + ½at², v² = u² + 2as, s = ½(u + v)t; valid only when acceleration is constant.

Newton's second law: F = ma = Δp/Δt; the resultant force acts in the direction of acceleration.

Momentum p = mv is conserved in any collision or explosion provided no external resultant force acts; kinetic energy is conserved only in elastic collisions.

Impulse = FΔt = change in momentum; the area under a force–time graph equals impulse.

For equilibrium of a rigid body: resultant force is zero and resultant moment about any point is zero; moment = force × perpendicular distance from pivot.

Friction: limiting static friction F ≤ µsN and kinetic friction F = µkN, where N is the normal reaction.

Projectile motion: horizontal velocity constant; vertical motion uses a = g; time of flight from vertical component, range = horizontal velocity × time.

Work done W = Fs cos θ and power P = Fv; efficiency = useful output power / total input power.

Centre of mass of a uniform body is at its geometric centre; a suspended body hangs with its centre of mass vertically below the point of suspension.

Density ρ = m/V; for a mixture or composite use total mass over total volume.

Hooke's law: extension is proportional to applied force up to the limit of proportionality; F = kx with k the spring constant in N m⁻¹.

Elastic deformation is recoverable and returns to original length; plastic deformation is permanent once the elastic limit is exceeded.

Stress σ = F/A (Pa) and strain ε = ΔL/L (dimensionless); the stress–strain graph's linear region gives Young modulus E = σ/ε.

Young modulus E = (F/A)/(ΔL/L) = FL/(AΔL); a steeper stress–strain line means a stiffer material.

Elastic potential energy stored in a stretched spring = ½Fx = ½kx²; the area under a force–extension graph gives the energy.

Ultimate tensile strength is the maximum stress before fracture; ductile materials show large plastic strain, brittle materials fracture with little plastic strain.

Energy stored per unit volume = ½ × stress × strain in the linear region.

Wave equation v = fλ; for a stretched string v = √(T/µ) where µ is mass per unit length.

Transverse waves oscillate perpendicular to propagation (light, waves on a string); longitudinal waves oscillate parallel (sound).

Superposition: displacements of overlapping waves add algebraically; constructive interference gives reinforcement, destructive gives cancellation.

Path difference for constructive interference is nλ, for destructive is (n + ½)λ, for coherent sources of the same frequency.

Two-source interference (Young's slits): fringe spacing w = λD/s, where D is slit-to-screen distance and s is slit separation.

Diffraction grating: d sin θ = nλ; the maximum observable order satisfies n ≤ d/λ.

Stationary waves form when two identical waves of opposite direction superpose; nodes are points of zero amplitude and adjacent nodes are λ/2 apart.

Fundamental frequency of a string fixed at both ends: f = (1/2L)√(T/µ); harmonics are integer multiples of this.

For a pipe open at both ends f = nv/2L; closed at one end f = (2n − 1)v/4L, giving only odd harmonics.

Current I = ΔQ/Δt; potential difference V = W/Q; resistance R = V/I and power P = VI = I²R = V²/R.

Ohm's law applies only to ohmic conductors at constant temperature; a filament lamp's resistance rises as temperature rises.

Resistors in series add: R = R₁ + R₂; in parallel: 1/R = 1/R₁ + 1/R₂.

Kirchhoff's first law: sum of currents into a junction equals sum out (charge conservation); second law: sum of emfs equals sum of p.d.s around a loop (energy conservation).

Terminal p.d. V = E − Ir, where E is emf, r internal resistance; maximum power transfer to a load occurs when R = r.

Capacitance C = Q/V; energy stored = ½QV = ½CV² = ½Q²/C.

Capacitors in parallel add (C = C₁ + C₂); in series 1/C = 1/C₁ + 1/C₂.

Capacitor discharge: Q = Q₀e^(−t/RC); time constant τ = RC is the time to fall to 1/e of the initial value.

Potential divider: V_out = V_in × R₂/(R₁ + R₂); a potentiometer balances when the p.d. across the balance length equals the unknown emf.

Newton's law of gravitation: F = GMm/r²; gravitational field strength g = GM/r², directed toward the mass.

Coulomb's law: F = Q₁Q₂/(4πε₀r²); electric field strength E = F/Q = Q/(4πε₀r²) for a point charge.

Uniform field between parallel plates: E = V/d, with field lines from positive to negative plate.

Magnetic flux density B is defined by F = BIL sin θ on a current-carrying wire; on a moving charge F = BQv sin θ.

Magnetic flux Φ = BA cos θ and flux linkage = NΦ; Faraday's law: induced emf = −d(NΦ)/dt.

Lenz's law: the induced current opposes the change producing it, a consequence of energy conservation.

A straight wire's field B = µ₀I/(2πr); a solenoid's field B = µ₀nI.

Gravitational potential energy change = mΔφ with φ = −GM/r; escape velocity v = √(2GM/r).

Absolute temperature in kelvin: T/K = θ/°C + 273.15; thermodynamic temperature is proportional to the mean kinetic energy of particles.

Ideal gas law: pV = nRT with R = 8.31 J mol⁻¹ K⁻¹; alternatively pV = NkT with k = 1.38 × 10⁻²³ J K⁻¹.

Mean translational kinetic energy per molecule = (3/2)kT; total internal energy of a monatomic ideal gas = (3/2)nRT.

Specific heat capacity: E = mcΔθ; specific latent heat: E = mL, used at constant temperature during a phase change.

Internal energy is the sum of random kinetic and potential energies of the particles; for an ideal gas potential energy is zero.

First law of thermodynamics: ΔU = Q − W, where Q is heat added to the system and W is work done by the system.

Isothermal change: pV = constant; adiabatic change: no heat exchange, pV^γ = constant.

A thermodynamic cycle's enclosed area on a p–V diagram equals the net work done per cycle.

Notation ᴬ_Z X: Z protons, A − Z neutrons; isotopes have the same Z but different A.

Alpha decay reduces A by 4 and Z by 2; beta-minus decay raises Z by 1 with A unchanged; gamma emission changes neither.

Decay law: N = N₀e^(−λt); activity A = λN; half-life t½ = ln 2/λ ≈ 0.693/λ.

Mass defect Δm converts to binding energy via E = Δmc²; 1 u = 931.5 MeV.

Binding energy per nucleon peaks near A ≈ 56 (iron); fusion releases energy for light nuclei, fission for heavy nuclei.

Quark content: proton = uud (charge +1), neutron = udd (charge 0); hadrons are baryons (3 quarks) or mesons (quark–antiquark).

Beta-minus decay at quark level: d → u + e⁻ + ν̄ₑ; beta-plus: u → d + e⁺ + νₑ.

Fundamental forces: strong, electromagnetic, weak and gravitational, mediated by gluons, photons, W/Z bosons and (hypothetical) gravitons respectively.

Conservation laws in particle reactions: charge, baryon number, lepton number and energy must be conserved.

Photon energy E = hf = hc/λ, with h = 6.63 × 10⁻³⁴ J s; 1 eV = 1.60 × 10⁻¹⁹ J.

Photoelectric equation: hf = φ + E_k(max), where φ is the work function; below the threshold frequency f₀ = φ/h no emission occurs regardless of intensity.

Maximum kinetic energy is found from the stopping potential: eV_s = E_k(max).

de Broglie wavelength λ = h/p = h/(mv); electrons accelerated through p.d. V have λ ≈ h/√(2meV).

Atomic line spectra arise from electron transitions between discrete energy levels: hf = E₂ − E₁.

The ionisation energy of hydrogen from the ground state is 13.6 eV; the ground-state energy is −13.6 eV.

Wave–particle duality: electrons diffract through crystal lattices (wave behaviour) yet arrive as discrete dots (particle behaviour).

Photon momentum p = E/c = h/λ; radiation pressure arises from momentum transfer.

Circular motion: angular velocity ω = 2π/T; v = rω; centripetal acceleration a = v²/r = rω², directed toward the centre.

Centripetal force F = mv²/r = mrω²; it is provided by tension, gravity, friction or electrostatic force depending on the system.

SHM defining condition: acceleration is proportional to displacement and directed toward equilibrium, a = −ω²x.

SHM solutions: x = A cos ωt, v = ±ω√(A² − x²), v_max = ωA, a_max = ω²A.

Period of a mass–spring system T = 2π√(m/k); of a simple pendulum T = 2π√(l/g).

Energy in SHM: total energy = ½mω²A², shared between kinetic and potential energy and constant if undamped.

Resonance occurs when driving frequency equals natural frequency; damping reduces amplitude and shifts the peak to a slightly lower frequency.

Orbital motion: GMm/r² = mv²/r gives v = √(GM/r) and T² = (4π²/GM)r³ (Kepler's third law).

Electric potential V = Q/(4πε₀r) and field E = −dV/dr; field strength is the negative potential gradient.

Luminosity L = 4πd²b, where b is the observed intensity (flux); the inverse-square law links apparent brightness to distance.

Absolute magnitude M and apparent magnitude m: m − M = 5 log₁₀(d/10 pc); a smaller magnitude means a brighter object.

Parallax distance d (pc) = 1/p (arcsec); 1 pc ≈ 3.09 × 10¹⁶ m ≈ 3.26 light-years.

Wien's law: λ_max T = 2.90 × 10⁻³ m K; Stefan's law: L = 4πr²σT⁴ with σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴.

Stellar spectral classes in order of decreasing temperature: O, B, A, F, G, K, M.

Hertzsprung–Russell diagram: main sequence runs from hot, luminous to cool, dim; giants and white dwarfs lie off the main sequence.

Hubble's law: v = H₀d with H₀ ≈ 2.2 × 10⁻¹⁸ s⁻¹; the reciprocal 1/H₀ estimates the age of the universe.

Redshift z = Δλ/λ ≈ v/c for small v; cosmological redshift indicates recession and an expanding universe.

A star's equilibrium balances gravitational collapse against radiation pressure from fusion in its core.

X-ray attenuation: I = I₀e^(−µx); the half-value thickness x½ = ln 2/µ.

The linear attenuation coefficient µ depends on photon energy and material density; higher density and higher atomic number give greater attenuation.

Ultrasound intensity reflection coefficient at a boundary: R = ((Z₂ − Z₁)/(Z₂ + Z₁))², where Z = ρc is the acoustic impedance.

Ultrasound frequency is typically 1–15 MHz; higher frequency gives better resolution but less penetration depth.

Doppler ultrasound measures blood flow speed from the frequency shift: Δf/f = 2v cos θ/c.

PET uses positron-emitting tracers (e.g. fluorine-18); annihilation of a positron with an electron produces two 511 keV gamma photons emitted back-to-back.

Gamma camera detection uses a scintillator (NaI) coupled to photomultiplier tubes to convert gamma photons into light pulses.

Radiation dose: absorbed dose in gray (J kg⁻¹); equivalent dose in sievert = absorbed dose × quality factor.

Rotational dynamics: torque τ = Iα; angular momentum L = Iω is conserved when no external torque acts.

Moment of inertia of a point mass I = mr²; for a uniform rod about its centre I = mL²/12, about one end I = mL²/3.

Rotational kinetic energy = ½Iω²; a rolling body's total kinetic energy = ½mv² + ½Iω².

A body rolls without slipping when v = ωr; the acceleration down an incline is a = g sin θ/(1 + I/mr²).

Stress–strain behaviour of engineering materials: elastic region, yield point, plastic region and fracture; the area under the curve is toughness.

The Young modulus of a material is measured from the gradient of the linear stress–strain region, not from the whole curve.

Resonance in structures occurs when a driving frequency matches a natural frequency; damping and stiffening raise or lower the resonant frequency to avoid failure.

Thermal expansion: ΔL = αL₀ΔT; bimetallic strips bend because the two metals have different expansion coefficients.

Michelson–Morley experiment found no ether drift, leading to the conclusion that the speed of light is the same in all inertial frames.

Einstein's postulates: physical laws are the same in all inertial frames, and the speed of light in vacuum is invariant.

Time dilation: t = t₀/√(1 − v²/c²); length contraction: L = L₀√(1 − v²/c²).

Relativistic mass increase: m = m₀/√(1 − v²/c²); rest energy E₀ = m₀c².

The photoelectric effect established the photon model: emission depends on frequency, not intensity, and occurs instantaneously above threshold.

Millikan's oil-drop experiment determined the elementary charge e by balancing gravitational and electric forces on charged droplets.

Thomson's cathode-ray experiments measured e/m for the electron; the value of e came from Millikan.

Wave–particle duality was confirmed by electron diffraction (Davisson–Germer) and by the double-slit interference of single particles.