DZI Physics and Astronomy cheat sheet

Newton's second law: F_net = dp/dt = ma for constant mass; in components, each axis is independent.

Uniform circular motion: a_c = v²/r directed toward the center; centripetal force is a net force, not a new interaction.

Work–energy theorem: W_net = ΔK; for conservative forces, define U with F = −dU/dx and conserve E = K + U.

Kepler's third law for orbits about mass M: T² = 4π²a³/(GM), with a the semimajor axis; valid for ellipses and circles.

Orbital speed in a circular orbit: v = √(GM/r); escape speed from radius r: v_esc = √(2GM/r).

Vis-viva relation: v² = GM(2/r − 1/a), giving speed anywhere on an orbit from r and a.

Angular momentum L = Iω is conserved when net external torque is zero; reduced mass μ = m₁m₂/(m₁+m₂) governs two-body relative motion.

Moment of inertia depends on axis: I = Σmᵢrᵢ²; parallel-axis theorem I = I_cm + Md².

First law: ΔU = Q − W, with W the work done by the system; for a quasistatic process W = ∫P dV.

Ideal gas: PV = nRT = Nk_BT; internal energy of a monatomic ideal gas U = (3/2)nRT.

Entropy change for a reversible path: ΔS = ∫dQ_rev/T; entropy of an isolated system never decreases.

Carnot efficiency between reservoirs: η = 1 − T_c/T_h, the upper bound for any heat engine operating between those temperatures.

Maxwell–Boltzmann distribution: mean kinetic energy per molecule = (3/2)k_BT, independent of molecular mass.

Partition function Z = Σ exp(−Eᵢ/k_BT) links microscopic states to free energy F = −k_BT ln Z.

Adiabatic quasistatic process for an ideal gas: PV^γ = constant, with γ = C_P/C_V.

Second-law statement: no cyclic device converts heat entirely into work with a single reservoir; Kelvin–Planck and Clausius forms are equivalent.

Wave speed relation: v = fλ; for a stretched string v = √(T/μ) with μ the linear mass density.

Sound in air: v ≈ 331 + 0.6T(°C) m/s; intensity level in decibels β = 10 log₁₀(I/I₀), I₀ = 10⁻¹² W/m².

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

Thin-lens and mirror equation: 1/f = 1/d_o + 1/d_i; magnification m = −d_i/d_o for lenses.

Snell's law: n₁ sin θ₁ = n₂ sin θ₂; total internal reflection occurs when sin θ_c = n₂/n₁ for n₁ > n₂.

Single-slit minima: a sin θ = mλ (m ≠ 0); double-slit maxima: d sin θ = mλ.

Diffraction grating resolves wavelengths by R = λ/Δλ = mN, where N is the number of illuminated slits.

Standing waves on a string fixed at both ends: f_n = nv/(2L); open pipe f_n = nv/(2L); closed pipe f_n = (2n−1)v/(4L).

Coulomb's law: F = k q₁q₂/r², k = 1/(4πε₀) ≈ 8.99×10⁹ N·m²/C²; field of a point charge E = kq/r².

Gauss's law: ∮E·dA = Q_enclosed/ε₀; useful for spheres, cylinders, and infinite planes.

Electric potential from a point charge: V = kq/r; potential energy of a charge pair U = kq₁q₂/r.

Capacitance: C = Q/V; parallel plates C = ε₀A/d; energy stored U = (1/2)CV².

Ohm's law and power: V = IR, P = IV = I²R = V²/R; series resistors add, parallel conductances add.

Kirchhoff's rules: junction rule conserves charge; loop rule states the sum of potential changes around a closed loop is zero.

Magnetic force on a moving charge: F = qv×B; on a current-carrying wire: F = IL×B.

Faraday's law: emf = −dΦ_B/dt; a changing magnetic flux induces an emf opposing the change (Lenz's law).

de Broglie relation: λ = h/p; photon energy E = hf = hc/λ.

Heisenberg uncertainty: ΔxΔp ≥ ħ/2 and ΔEΔt ≥ ħ/2, where ħ = h/2π.

Hydrogen energy levels: E_n = −13.6 eV/n²; transitions obey ΔE = hf with selection rule Δℓ = ±1 for electric dipole.

Photoelectric effect: maximum kinetic energy K_max = hf − φ, with φ the work function; no emission below threshold frequency.

Infinite square well energies: E_n = n²h²/(8mL²); confinement raises the ground-state energy above zero.

Schrödinger equation: Ĥψ = Eψ; |ψ|² gives probability density, normalized so ∫|ψ|²dV = 1.

Bohr angular momentum quantization: L = nħ; atomic spectra line series follow from combining Rydberg terms.

Pauli exclusion principle: no two electrons in an atom share all four quantum numbers; fills shells n, ℓ, m_ℓ, m_s.

Lorentz factor: γ = 1/√(1 − v²/c²); time dilation Δt = γΔτ for a moving clock as seen from the lab.

Length contraction: L = L₀/γ along the direction of motion; transverse lengths are unchanged.

Relativistic energy: E = γmc², rest energy E₀ = mc², kinetic energy K = (γ − 1)mc².

Energy–momentum relation: E² = (pc)² + (mc²)²; for photons E = pc.

Mass–energy and binding: nuclear mass defect converts to binding energy via ΔE = Δm c².

Radioactive decay: N = N₀e^(−λt); half-life T₁/₂ = ln2/λ; activity A = λN.

Lorentz transformations preserve the spacetime interval s² = (cΔt)² − Δx².

Relativistic velocity addition: u = (u' + v)/(1 + u'v/c²).

Luminosity–flux relation: F = L/(4πd²); distance modulus m − M = 5 log₁₀(d/10 pc).

Hydrostatic equilibrium: dP/dr = −GM(r)ρ(r)/r²; pressure gradient balances gravity in a stable star.

Stefan–Boltzmann law: L = 4πR²σT_eff⁴ relates luminosity, radius, and effective temperature.

Wien's law: λ_max = b/T with b ≈ 2.898×10⁻³ m·K; hotter stars peak at shorter wavelengths.

Main-sequence lifetime scales as t ≈ 10 Gyr × (M/M_☉)^(−2.5), reflecting higher mass and faster fuel consumption.

Hertzsprung–Russell diagram: main sequence runs from hot, luminous to cool, dim; giants and white dwarfs occupy separate regions.

Chandrasekhar limit ≈ 1.4 M_☉ is the maximum mass of a white dwarf supported by electron degeneracy pressure.

Stellar spectral classes in temperature order: O, B, A, F, G, K, M, from hottest to coolest.

Hubble–Lemaître law: v = H₀d; recession velocity is proportional to distance for nearby galaxies.

Redshift–velocity relation: z = (λ_obs − λ_emit)/λ_emit; for low z, v ≈ cz.

Friedmann equation: H² = (8πG/3)ρ − kc²/a² + Λc²/3 governs cosmic expansion.

Critical density: ρ_c = 3H₀²/(8πG); Ω = ρ/ρ_c determines whether the universe is open, flat, or closed.

Cosmological redshift scales the scale factor: 1 + z = a_obs/a_emit.

Galaxy rotation curves imply large dark-matter halos; orbital speeds stay roughly flat beyond the luminous disk.

Cosmic microwave background temperature ≈ 2.7 K; its small anisotropies map density fluctuations in the early universe.

Hubble time t_H = 1/H₀ sets the approximate age scale of the universe; for H₀ ≈ 70 km/s/Mpc, t_H ≈ 14 Gyr.

Mean of repeated measurements: x̄ = (1/N)Σxᵢ; standard error of the mean is σ/√N.

Standard deviation of a sample: s = √[Σ(xᵢ − x̄)²/(N − 1)]; use N − 1 for the unbiased estimate.

Error propagation for independent variables: if z = f(x, y), then σ_z² = (∂f/∂x)²σ_x² + (∂f/∂y)²σ_y².

Relative uncertainty multiplies for products and divides for quotients; absolute uncertainties add in quadrature for sums.

Least-squares linear fit: slope b = Σ(xᵢ − x̄)(yᵢ − ȳ)/Σ(xᵢ − x̄)²; intercept a = ȳ − bx̄.

Chi-square goodness of fit: χ² = Σ[(Oᵢ − Eᵢ)²/Eᵢ]; reduced χ² near 1 indicates a consistent model.

Significant figures reflect measurement precision; the result of multiplication has the same number of significant figures as the least precise input.

Systematic errors shift all measurements in one direction; random errors scatter and are reduced by averaging.

Cross-domain link: stellar luminosity uses the Stefan–Boltzmann law (domain 7) with blackbody radiation (domain 2).

Cross-domain link: cosmological redshift (domain 8) follows from the relativistic Doppler effect (domain 6).

Cross-domain link: atomic energy levels (domain 5) explain spectral lines used to classify stars (domain 7).

Cross-domain link: photon energy E = hc/λ (domain 5) connects optics (domain 3) to quantum transitions (domain 5).

Cross-domain link: gravitational orbits (domain 1) set the rotation curves used to infer dark matter (domain 8).

Cross-domain link: measurement uncertainty (domain 9) bounds every quoted astronomical distance and luminosity.

Cross-domain link: thermodynamic equilibrium (domain 2) underlies hydrostatic and thermal balance in stellar interiors (domain 7).