Valencian Community PAU Physics cheat sheet

Newton's law of gravitation: F = G·m₁·m₂/r², always attractive and directed along the line joining the centres; G = 6.67×10⁻¹¹ N·m²·kg⁻².

Field strength at a point: g = F/m = G·M/r², pointing toward the source mass; on Earth's surface g ≈ 9.8 m·s⁻².

Gravitational potential V = −G·M/r (zero at infinity, always negative); potential energy U = −G·M·m/r.

Escape velocity from a body of mass M and radius R: v_e = √(2GM/R); for Earth ≈ 11.2 km·s⁻¹.

Orbital speed for a circular orbit of radius r: v = √(GM/r); period follows Kepler's third law T² = 4π²r³/(GM).

Kepler's second law is conservation of angular momentum: equal areas swept in equal times, so r·v·sinθ is constant.

Uniform circular motion: a_c = v²/r = ω²r; centripetal force is the net radial force, never a separate force.

Work–energy theorem: W_net = ΔE_c = ½mv² − ½mv₀²; gravity is conservative, so work depends only on endpoints.

Simple harmonic motion (SHM): a = −ω²x; solution x = A·sin(ωt + φ₀) with ω = 2π/T = 2πf.

In SHM, v = ±ω√(A² − x²) (maximum ωA at x = 0) and a = −ω²x (maximum ω²A at x = ±A).

Spring–mass: T = 2π√(m/k); simple pendulum (small angles): T = 2π√(L/g); both independent of amplitude.

Total mechanical energy in SHM: E = ½kA² = ½mω²A²; kinetic and potential energies interchange with period T/2.

Wave equation: v = λ·f = λ/T; for a stretched string v = √(T/μ) with μ the linear mass density.

Harmonic wave: y(x,t) = A·sin(ωt − kx) for propagation in +x, with k = 2π/λ; phase difference Δφ = k·Δx.

Intensity of a spherical wave: I = P/(4πr²), so I falls as 1/r²; amplitude falls as 1/r.

Standing waves: nodes separated by λ/2; string fixed at both ends has λ_n = 2L/n, f_n = n·v/(2L).

Doppler effect (non-relativistic, source and observer along the line): f' = f·(v ± v_o)/(v ∓ v_s); use upper signs for approach.

Coulomb's law: F = k·q₁·q₂/r² with k = 1/(4πε₀) ≈ 9×10⁹ N·m²·C⁻²; force along the joining line, sign gives attraction or repulsion.

Electric field of a point charge: E = k·Q/r²; potential V = k·Q/r; work to move charge q is W = q·(V_A − V_B).

Gauss's law: Φ = ∮E·dA = Q_enc/ε₀; useful for spheres, infinite lines (E = λ/(2πε₀r)) and planes (E = σ/(2ε₀)).

Magnetic force on a moving charge: F = q·v×B, magnitude qvB·sinθ; on a current wire F = I·L×B.

Field of a long straight wire: B = μ₀I/(2πr); inside a solenoid B = μ₀·n·I with n turns per metre.

Charge entering B perpendicular to v moves in a circle of radius r = mv/(qB); period T = 2πm/(qB) is independent of speed.

Faraday–Lenz law: ε = −dΦ/dt with Φ = B·A·cosθ; the induced current opposes the change in flux.

Motional emf for a rod of length L moving at v perpendicular to B: ε = B·L·v.

Lorentz force total: F = q·(E + v×B); a velocity selector needs v = E/B for zero deflection.

Snell's law: n₁·sinθ₁ = n₂·sinθ₂; refractive index n = c/v; frequency is unchanged on refraction, wavelength scales as λ/n.

Total internal reflection occurs going from denser to less dense medium when sinθ_c = n₂/n₁.

Thin lens and spherical mirror: 1/f = 1/s' − 1/s (lens, real-is-positive convention) and 1/f = 1/s + 1/s' (mirror); magnification m = y'/y = −s'/s.

Lens maker's equation: 1/f = (n − 1)·(1/R₁ − 1/R₂) with sign convention for radii.

Young's double slit: bright fringes at d·sinθ = m·λ; fringe spacing on a distant screen Δy = λ·D/d.

Single-slit diffraction minima: a·sinθ = m·λ (m ≠ 0); the central maximum is twice as wide as the secondary maxima.

Diffraction grating: d·sinθ = m·λ gives sharp principal maxima; resolving power R = m·N.

Photoelectric effect: E_photon = h·f = h·c/λ; stopping potential satisfies e·V₀ = h·f − W₀ (work function).

Threshold frequency f₀ = W₀/h; no emission below f₀ regardless of intensity; maximum kinetic energy depends on frequency, not intensity.

Time dilation Δt = γ·Δt₀ and length contraction L = L₀/γ with γ = 1/√(1 − v²/c²); proper time is measured in the rest frame.

Relativistic mass–energy: E = γ·m·c²; rest energy E₀ = m·c²; kinetic energy E_c = (γ − 1)·m·c².

de Broglie wavelength λ = h/p; for an electron accelerated through V volts, λ ≈ 1.23/√V nm.

Heisenberg uncertainty: Δx·Δp ≥ ħ/2 with ħ = h/(2π); also ΔE·Δt ≥ ħ/2.

Bohr model: E_n = −13.6 eV/n² for hydrogen; transition photon energy h·f = E_f − E_i; Rydberg 1/λ = R(1/n₁² − 1/n₂²).

Radioactive decay: N = N₀·e^(−λt); half-life T½ = ln2/λ; activity A = λ·N in becquerels.

Mass defect and binding energy: ΔE = Δm·c²; 1 u = 931.5 MeV/c²; binding energy per nucleon peaks near A ≈ 56 (iron).

Nuclear reactions conserve charge (Z) and nucleon number (A); alpha decay reduces A by 4 and Z by 2, beta-minus raises Z by 1.

Photon energy handy value: E(eV) ≈ 1240/λ(nm).

Absolute uncertainty of a single reading is usually half the smallest scale division; record the value with the same precision as the instrument.

For sums and differences, absolute uncertainties add: if z = x ± y then Δz = Δx + Δy.

For products and quotients, relative uncertainties add: if z = x·y or z = x/y then Δz/z = Δx/x + Δy/y.

For powers, z = xⁿ gives Δz/z = |n|·Δx/x; for logarithms, Δ(ln x) = Δx/x.

Report a result as (value ± uncertainty) with consistent units and the uncertainty rounded to one or two significant figures.

Graphical analysis: slope of the best-fit line gives the derived quantity; uncertainty from the steepest and shallowest acceptable lines.

Linearise data before plotting (e.g. plot T² against L for a pendulum) so the slope yields g = 4π²/slope.

Systematic errors shift all readings in one direction (zero error, calibration); random errors scatter and reduce by averaging repeated measurements.

Significant figures: keep the number justified by the instrument's resolution; do not quote more digits than the data support.