Balearic PAU Physics cheat sheet

Newton's law: F = G·m₁·m₂/r², attractive, G = 6.67×10⁻¹¹ N·m²/kg²; direction along the line joining the masses.

Field strength g = F/m = G·M/r², a vector pointing toward the source mass; on Earth's surface g ≈ 9.8 m/s².

Uniform-field approximation g ≈ constant is valid only for height differences much smaller than the planet's radius.

Gravitational potential V = −G·M/r, always negative with V(∞) = 0; potential energy U = m·V = −G·M·m/r.

Work done by the field equals −ΔU; moving a mass along an equipotential surface requires zero work.

Escape velocity from a body of mass M and radius R: v_e = √(2GM/R), independent of the escaping mass.

Kepler's third law: T² = (4π²/GM)·r³ for circular orbits; orbital speed v = √(GM/r).

Satellites in circular orbit: total mechanical energy E = −GMm/(2r), so binding energy is positive and equal to half |U|.

Coulomb's law: F = k·q₁·q₂/r², k = 1/(4πε₀) ≈ 9×10⁹ N·m²/C²; force is repulsive for like signs.

Electric field E = F/q (N/C or V/m); field lines start on positive and end on negative charge.

Potential V = k·q/r with V(∞) = 0; work by external agent to move charge q between points is q·ΔV.

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

Gauss's law: electric flux through a closed surface equals Q_enclosed/ε₀; use symmetry to choose the surface.

Magnetic force on a moving charge: F = q·v×B, magnitude qvB·sinθ; it does no work because F ⊥ v.

Force on a straight current wire: F = I·L×B; parallel currents attract, antiparallel currents repel.

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

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

Wave relation: v = λ·f = λ/T; the medium fixes v, and f is set by the source.

Simple harmonic motion: x = A·sin(ωt + φ), with ω = 2π/T = 2πf; maximum speed Aω, maximum acceleration Aω².

Mass–spring oscillator: ω = √(k/m), T = 2π√(m/k); simple pendulum: T = 2π√(L/g).

Mechanical waves need a material medium; electromagnetic waves propagate in vacuum at c = 3.00×10⁸ m/s.

Wave intensity is proportional to the square of the amplitude; for a point source in 3D, I ∝ 1/r².

Standing waves on a string fixed at both ends: λ_n = 2L/n, f_n = n·v/(2L), n = 1, 2, 3…

Doppler effect for sound: f' = f·(v ± v_o)/(v ∓ v_r), with signs chosen so approach raises the observed frequency.

Superposition: constructive interference when path difference = nλ, destructive when path difference = (n + ½)λ.

Snell's law: n₁·sinθ₁ = n₂·sinθ₂; refractive index n = c/v, so light slows in a denser medium.

Total internal reflection occurs only going from denser to less dense medium, when θ₁ exceeds θ_c with sinθ_c = n₂/n₁.

Thin lens and spherical mirror equation: 1/f = 1/s + 1/s'; magnification m = −s'/s (lens) or m = −s'/s with sign conventions stated.

Lens maker's equation: 1/f = (n − 1)·(1/R₁ − 1/R₂); converging lens f > 0, diverging lens f < 0.

Young's double slit: fringe spacing Δy = λ·D/d; maxima at d·sinθ = nλ, minima at d·sinθ = (n + ½)λ.

Single-slit diffraction minima: a·sinθ = nλ (n = 1, 2, …); central maximum is twice as wide as the secondary ones.

Diffraction grating: d·sinθ = nλ; smaller grating spacing d gives larger angular separation of orders.

Dispersion: n decreases with increasing wavelength, so violet refracts more than red in glass.

Postulates: physical laws are identical in all inertial frames, and the speed of light in vacuum is c for every inertial observer.

Lorentz factor γ = 1/√(1 − v²/c²) ≥ 1; time dilation Δt = γ·Δτ for the proper time Δτ.

Length contraction L = L₀/γ applies only to the dimension parallel to the motion; proper length is measured in the rest frame.

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

Energy–momentum relation: E² = (p·c)² + (m·c²)²; for photons m = 0 and E = p·c = h·f.

Velocity addition: u = (u' + v)/(1 + u'·v/c²); the result never exceeds c.

Simultaneity is frame-dependent: events simultaneous in one inertial frame are generally not simultaneous in another.

For v ≤ 0.1c, γ ≈ 1 + v²/(2c²), so classical mechanics is a good approximation.

Photon energy E = h·f = h·c/λ, with h = 6.63×10⁻³⁴ J·s; photon momentum p = h/λ.

Photoelectric effect: K_max = h·f − W₀, where W₀ is the work function; threshold frequency f₀ = W₀/h.

Stopping potential relation: e·V₀ = K_max; no emission occurs below the threshold frequency regardless of intensity.

de Broglie wavelength λ = h/p = h/(m·v); matter waves become measurable for electrons and lighter particles.

Bohr model: E_n = −13.6 eV/n² for hydrogen; transition photon energy h·f = E_f − E_i.

Bohr angular momentum quantization: L = n·h/(2π); orbit radius r_n = n²·a₀ with a₀ ≈ 0.053 nm.

Heisenberg uncertainty principle: Δx·Δp ≥ ħ/2, with ħ = h/(2π); it is a fundamental limit, not a measurement flaw.

Wave–particle duality: interference and diffraction reveal wave behaviour, while the photoelectric effect reveals particle behaviour.

Nuclear notation ᴬ_ZX: Z protons, A − Z neutrons; isotopes share Z but differ in A.

Mass defect and binding energy: ΔE = Δm·c²; 1 u = 931.5 MeV/c², so 1 u of defect releases 931.5 MeV.

Alpha decay: A decreases by 4, Z by 2; beta-minus decay: n → p + e⁻ + ν̄, so Z increases by 1.

Beta-plus decay and electron capture reduce Z by 1; gamma emission changes neither A nor Z.

Radioactive decay law: N = N₀·e^(−λt); activity A = λ·N; half-life T½ = ln2/λ ≈ 0.693/λ.

After n half-lives the remaining fraction is (1/2)^n; the activity follows the same exponential law.

Conservation laws in nuclear reactions: charge Z, nucleon number A, energy–mass and momentum are conserved.

Fission of heavy nuclei and fusion of light nuclei both release energy because the products are more tightly bound (higher binding energy per nucleon).