AAPT NCPT (Level 1) cheat sheet

Newton's second law: net force equals mass times acceleration; valid only in inertial frames.

Uniform circular motion requires centripetal acceleration v^2/r directed toward the center; speed constant, velocity changing.

Work-energy theorem: net work equals change in kinetic energy, W_net = ΔK = ½mv_f² − ½mv_i².

Momentum is conserved when net external impulse is zero; in collisions p_i = p_f along each direction.

Elastic collision: kinetic energy and momentum both conserved; perfectly inelastic: objects stick together, maximum KE loss.

Torque τ = r F sinθ about a pivot; static equilibrium requires ΣF = 0 and Στ = 0.

Angular momentum L = Iω is conserved when net external torque is zero (e.g., spinning skater pulling arms in).

Simple harmonic motion: a = −ω²x, with ω = √(k/m) for a spring and ω = √(g/L) for a small-angle pendulum.

First law: ΔU = Q − W, where W is work done by the gas; sign conventions must be stated before use.

Ideal gas law: PV = nRT = NkT; average translational KE per molecule = (3/2)kT.

Root-mean-square speed v_rms = √(3kT/m) = √(3RT/M); lighter molecules move faster at the same temperature.

Isothermal process: T constant, ΔU = 0, so Q = W = nRT ln(V_f/V_i).

Adiabatic process: Q = 0, so ΔU = −W; PV^γ = constant with γ = C_p/C_v.

Second law: entropy of an isolated system never decreases; heat flows spontaneously from hot to cold.

Carnot efficiency η = 1 − T_c/T_h with temperatures in kelvin; no real engine exceeds it.

Internal energy of an ideal gas depends only on temperature; for a monatomic gas U = (3/2)nRT.

Wave speed v = fλ; for a stretched string v = √(T/μ) where μ is linear mass density.

Standing waves on a string fixed at both ends: f_n = nv/(2L), n = 1, 2, 3, ...

Doppler shift: observed frequency rises when source and observer approach, falls when they recede.

Snell's law: n₁ sinθ₁ = n₂ sinθ₂; total internal reflection occurs only going from higher to lower n beyond the critical angle.

Thin lens/mirror: 1/f = 1/d_o + 1/d_i, magnification m = −d_i/d_o; real images form on the same side as outgoing rays for mirrors.

Double-slit maxima: d sinθ = mλ; single-slit minima: a sinθ = mλ with m ≠ 0.

Thin-film interference depends on path difference plus any phase shift of λ/2 on reflection from a higher-index medium.

Diffraction grating resolves wavelengths; larger number of slits N sharpens principal maxima.

Coulomb's law: F = kq₁q₂/r², k ≈ 8.99×10⁹ N·m²/C²; force is along the line joining charges.

Electric field E = F/q; field lines point from positive to negative charge and never cross.

Gauss's law: net flux through a closed surface equals Q_enclosed/ε₀; useful for symmetric charge distributions.

Ohm's law V = IR for ohmic resistors; power dissipated P = IV = I²R = V²/R.

Series: R_eq = ΣR, current common; parallel: 1/R_eq = Σ1/R, voltage common.

Kirchhoff's rules: junction rule conserves charge; loop rule conserves energy around any closed loop.

Magnetic force on a moving charge F = qvB sinθ, perpendicular to both v and B; on a wire F = BIL sinθ.

Faraday's law: induced emf equals −dΦ_B/dt; Lenz's law gives the induced current direction opposing the flux change.

Photon energy E = hf = hc/λ; h ≈ 6.63×10⁻³⁴ J·s, hc ≈ 1240 eV·nm.

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

de Broglie wavelength λ = h/p; larger momentum means shorter wavelength.

Mass-energy equivalence E = mc²; 1 u ≈ 931.5 MeV/c².

Radioactive decay: N = N₀e^(−λt), half-life t₁/₂ = ln2/λ ≈ 0.693/λ.

Bohr model: photon energy for a transition equals the energy-level difference; hydrogen levels E_n = −13.6 eV/n².

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

Heisenberg uncertainty principle: Δx·Δp ≥ ħ/2, limiting simultaneous knowledge of position and momentum.

Precision describes repeatability of measurements; accuracy describes closeness to the true value.

For independent random errors, uncertainties add in quadrature: δz = √((∂z/∂x·δx)² + (∂z/∂y·δy)²).

For z = x·y or z = x/y, fractional uncertainties add: δz/z = √((δx/x)² + (δy/y)²).

Report a result as value ± uncertainty with matching decimal places and consistent significant figures.

Standard deviation of the mean is s/√N; averaging N measurements reduces random error by √N.

Plot data to test linearity; slope and intercept from a best-fit line carry units of y per x.

Systematic errors shift all readings the same way and are not reduced by averaging; calibrate or correct for them.

A control or blank measurement establishes baseline; repeat trials and randomize order to detect drift.