New Zealand Scholarship Physics cheat sheet

Dimensional analysis: every valid equation must be dimensionally homogeneous; use it to check derivations and to guess scaling laws before solving (e.g. period of a pendulum must scale as sqrt(L/g)).

SI base units: kg, m, s, A, K, mol, cd. Derived units: N = kg·m·s⁻², J = N·m, W = J·s⁻¹, Pa = N·m⁻², C = A·s, V = J·C⁻¹, Ω = V·A⁻¹, T = kg·s⁻²·A⁻¹.

Uncertainty combination: for z = x ± y, δz = sqrt(δx² + δy²); for z = xⁿyᵐ, fractional uncertainty δz/z = sqrt((n·δx/x)² + (m·δy/y)²). Report final uncertainty to 1–2 significant figures.

Significant figures: the result of multiplication/division carries the smallest number of significant figures among inputs; addition/subtraction carries the smallest number of decimal places.

Vector decomposition: resolve into orthogonal components, sum components separately, then recombine with Pythagoras and atan2; never add magnitudes directly unless collinear.

Graphical analysis: gradient of a straight-line fit gives the rate constant; intercept gives the offset. Linearise power laws (y = axⁿ → log y = log a + n log x) before fitting.

Limiting cases: check a derived expression at extremes (m → 0, v → c, r → ∞); a correct formula must reduce to the physically sensible limit.

Order-of-magnitude estimation: anchor with known constants (g ≈ 9.81 m·s⁻², c ≈ 3.00×10⁸ m·s⁻¹, h ≈ 6.63×10⁻³⁴ J·s, k_B ≈ 1.38×10⁻²³ J·K⁻¹, N_A ≈ 6.02×10²³ mol⁻¹, e ≈ 1.60×10⁻¹⁹ C, G ≈ 6.67×10⁻¹¹ N·m²·kg⁻²).

Justify approximations explicitly: state what is neglected (air resistance, relativity, quantum effects) and show the dimensionless ratio that makes the neglect valid.

Newton's second law in vector form: ΣF = dp/dt = ma for constant mass; in non-inertial frames add a pseudo-force −ma_frame.

Projectile motion: horizontal velocity constant (no air resistance); vertical motion under a = −g. Range R = v² sin 2θ / g on level ground; maximum range at θ = 45°.

Circular motion: centripetal acceleration a = v²/r = ω²r directed toward the centre; centripetal force is the net inward force, never an extra force to add.

Work–energy theorem: W_net = ΔK = ½mv² − ½mu². Conservative forces have potential energy U with F = −dU/dx; mechanical energy conserved if only conservative forces act.

Momentum: p = mv; impulse J = ∫F dt = Δp. Total momentum conserved in any collision or explosion; kinetic energy conserved only in elastic collisions.

Rotational analogues: τ = Iα, L = Iω, K_rot = ½Iω². Parallel-axis theorem I = I_cm + Md². Rolling without slipping: v = ωr.

Gravitation: F = GMm/r²; g = GM/r²; gravitational potential energy U = −GMm/r (zero at infinity); orbital speed v = sqrt(GM/r); Kepler's third law T² = (4π²/GM) r³.

Escape velocity v_esc = sqrt(2GM/r) = sqrt(2gr); independent of projectile mass and launch angle (ignoring rotation and drag).

Simple harmonic motion: a = −ω²x; x = A cos(ωt + φ); T = 2π sqrt(m/k) for a spring, T = 2π sqrt(L/g) for a small-angle pendulum (θ < ~10°).

Wave speed v = fλ; for a stretched string v = sqrt(T/μ) where μ is linear mass density; for sound in a gas v = sqrt(γRT/M).

Superposition: resultant displacement is the algebraic sum of individual displacements. Constructive interference when path difference = nλ; destructive when (n + ½)λ.

Standing waves: nodes separated by λ/2. String fixed both ends: f_n = nv/(2L). Pipe open both ends: f_n = nv/(2L). Pipe closed one end: f_n = (2n − 1)v/(4L).

Doppler effect (source and observer on same line): f' = f (v ± v_o)/(v ∓ v_s), with signs chosen so approach increases observed frequency.

Snell's law: n₁ sin θ₁ = n₂ sin θ₂; refractive index n = c/v. Total internal reflection occurs when θ₁ exceeds the critical angle θ_c = arcsin(n₂/n₁) with n₁ > n₂.

Thin lens / mirror: 1/f = 1/v + 1/u with the real-is-positive convention; magnification m = −v/u. Power P = 1/f in dioptres (m⁻¹).

Two-source interference (Young's slits): fringe spacing Δy = λD/d for D ≫ d; maxima at d sin θ = nλ.

Thermal: Q = mcΔT; latent heat Q = mL. Ideal gas pV = nRT; average translational kinetic energy per molecule = (3/2)k_BT.

First law of thermodynamics: ΔU = Q − W where W is work done by the gas; W = ∫p dV. Adiabatic: pV^γ constant; isothermal: pV constant.

Entropy: ΔS = ∫dQ_rev/T; the second law states total entropy of an isolated system never decreases. Carnot efficiency η = 1 − T_c/T_h (absolute temperatures).

Coulomb's law: F = kQ₁Q₂/r² with k = 1/(4πε₀) ≈ 8.99×10⁹ N·m²·C⁻²; electric field E = F/q; uniform field between plates E = V/d.

Electric potential: V = kQ/r; potential energy U = qV. Field lines point from high to low potential and are perpendicular to equipotentials.

DC circuits: V = IR; resistors in series add, in parallel 1/R_total = Σ1/R_i. Power P = VI = I²R = V²/R. Kirchhoff's laws: ΣI at a node = 0; ΣV around a loop = 0.

Capacitance: C = Q/V; parallel-plate C = ε₀A/d; energy stored U = ½CV² = ½QV. Capacitors in parallel add; in series reciprocals add.

Magnetic force: F = qv × B (magnitude qvB sin θ); force on a current-carrying wire F = BIL sin θ. A moving charge in a uniform B moves in a circle of radius r = mv/(qB).

Electromagnetic induction: Faraday's law ε = −dΦ/dt, Φ = BA cos θ; Lenz's law gives the direction opposing the change in flux.

Photoelectric effect: photon energy E = hf = hc/λ; maximum kinetic energy K_max = hf − φ where φ is the work function; threshold frequency f₀ = φ/h.

Atomic energy levels: photon energy equals the energy difference between levels, hf = E₂ − E₁. Hydrogen levels E_n = −13.6 eV / n².

de Broglie wavelength λ = h/p; for an electron accelerated through potential V, λ ≈ h/sqrt(2meV).

Mass–energy equivalence E = mc²; nuclear binding energy is the mass defect times c²; radioactive decay N = N₀ e^(−λt), half-life t½ = ln 2 / λ.