BCS Written Physics cheat sheet

Newton's second law: F_net = dp/dt = ma for constant mass; work-energy theorem W_net = ΔK.

Kinematics with constant acceleration: v = u + at, s = ut + ½at², v² = u² + 2as.

Projectile motion: horizontal velocity constant, vertical acceleration g; range R = u²sin2θ/g on level ground.

Momentum conservation holds when net external force is zero; collisions conserve momentum, only elastic ones conserve kinetic energy.

Circular motion: centripetal acceleration a = v²/r = ω²r, directed to the centre; force F = mv²/r.

Rotational analogues: τ = Iα, L = Iω, K_rot = ½Iω²; parallel-axis theorem I = I_cm + Md².

Simple harmonic motion: a = −ω²x, T = 2π√(m/k) for a spring, T = 2π√(l/g) for a simple pendulum.

Impulse J = ∫F dt = Δp; friction f ≤ μN with limiting value f_max = μ_sN.

Newton's law of gravitation: F = Gm₁m₂/r², with G ≈ 6.67 × 10⁻¹¹ N·m²/kg².

Orbital speed v = √(GM/r); escape velocity v_e = √(2GM/R) = √(2gR).

Kepler's third law: T² ∝ a³, or T² = 4π²a³/(GM).

Gravitational potential energy U = −GMm/r; potential V = −GM/r; field g = GM/r².

Continuity equation for incompressible flow: A₁v₁ = A₂v₂.

Bernoulli's equation: P + ½ρv² + ρgh = constant along a streamline for ideal flow.

Archimedes' principle: buoyant force equals weight of displaced fluid; floatation when average density is below fluid density.

Poiseuille flow rate Q = πr⁴ΔP/(8ηL); Reynolds number Re = ρvD/η, laminar below about 2000.

Stress = force/area, strain = fractional deformation; Young's modulus Y = stress/strain in the linear region.

Hooke's law: extension proportional to load up to the elastic limit; energy stored U = ½Fx = ½kx².

Shear modulus G = shear stress/shear strain; bulk modulus K = −ΔP/(ΔV/V).

Poisson's ratio σ = lateral strain / longitudinal strain, typically 0.2–0.4 for metals.

Surface tension γ = force/length; excess pressure in a bubble ΔP = 4γ/r, in a droplet ΔP = 2γ/r.

Capillary rise h = 2γcosθ/(ρgr).

Viscosity: viscous force F = ηA(dv/dy); terminal velocity v_t = 2r²(ρ−σ)g/(9η).

Thermal expansion: ΔL = LαΔT, ΔV = VγΔT with γ ≈ 3α for isotropic solids.

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

Ideal gas law: PV = nRT = Nk_BT; R ≈ 8.314 J/(mol·K), k_B ≈ 1.38 × 10⁻²³ J/K.

C_p − C_v = R for an ideal gas; γ = C_p/C_v equals 5/3 monatomic, 7/5 diatomic.

Adiabatic relation: PV^γ = constant; isothermal work W = nRT ln(V₂/V₁).

Carnot efficiency η = 1 − T_c/T_h with temperatures in kelvin.

Second law: entropy of an isolated system never decreases; ΔS = Q_rev/T.

Heat conduction: dQ/dt = kAΔT/L; Stefan–Boltzmann law P = εσAT⁴.

Kinetic theory: mean kinetic energy per molecule = (3/2)k_BT; rms speed v_rms = √(3RT/M).

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

Progressive wave: y = A sin(ωt − kx), with k = 2π/λ and ω = 2πf.

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

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

Doppler effect: f' = f(v ± v_o)/(v ∓ v_s), with signs chosen for approach or recession.

Energy of a wave is proportional to amplitude squared; intensity falls as 1/r² for a point source in 3D.

Polarisation is a property of transverse waves only; Malus's law I = I₀cos²θ.

Group velocity v_g = dω/dk; phase velocity v_p = ω/k; equal for non-dispersive media.

Speed of sound in air v ≈ 331 + 0.6T(°C) m/s; in a gas v = √(γRT/M).

Closed pipe harmonics: f_n = (2n − 1)v/(4L); open pipe: f_n = nv/(2L).

Sound intensity level β = 10 log₁₀(I/I₀) dB, with I₀ = 10⁻¹² W/m².

Snell's law: n₁sinθ₁ = n₂sinθ₂; refractive index n = c/v.

Critical angle sinθ_c = n₂/n₁ for n₁ > n₂; total internal reflection beyond it.

Thin lens/mirror: 1/f = 1/v + 1/u (sign conventions consistent); magnification m = v/u.

Young's double slit: fringe spacing Δy = λD/d; diffraction grating maxima d sinθ = nλ.

Rayleigh criterion: θ_min ≈ 1.22λ/D for a circular aperture.

Coulomb's law: F = kq₁q₂/r², with k = 1/(4πε₀) ≈ 9 × 10⁹ N·m²/C².

Electric field E = F/q; potential V = W/q; for a point charge E = kq/r², V = kq/r.

Ohm's law V = IR; power P = VI = I²R = V²/R.

Series: R = R₁ + R₂ + …; parallel: 1/R = 1/R₁ + 1/R₂ + …

Kirchhoff's laws: junction currents sum to zero; loop emf sum equals loop IR sum.

Capacitance C = Q/V; parallel plate C = ε₀A/d; energy stored U = ½CV².

RC circuits: charging q = Q₀(1 − e^(−t/RC)); time constant τ = RC.

Electric flux through a closed surface equals enclosed charge/ε₀ (Gauss's law).

Force on a moving charge F = qv × B; on a current element F = BIL sinθ.

Magnetic field of a long straight wire B = μ₀I/(2πr); solenoid B = μ₀nI.

Force per unit length between parallel currents F/L = μ₀I₁I₂/(2πd).

Faraday's law: emf = −dΦ/dt; Φ = BA cosθ.

Lenz's law: induced current opposes the change in flux that produces it.

Motional emf for a rod moving in a field: emf = Bvl.

Self-inductance emf = −L dI/dt; energy stored U = ½LI².

A changing electric field produces a magnetic field (Maxwell's displacement current); EM waves travel at c = 1/√(μ₀ε₀).

Bohr model: E_n = −13.6/n² eV for hydrogen; photon energy hf = E₂ − E₁.

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

Radioactive decay N = N₀e^(−λt); half-life T½ = ln2/λ ≈ 0.693/λ.

Activity A = λN, measured in becquerels (1 Bq = 1 decay/s).

Mass defect and binding energy: E = Δmc²; 1 u ≈ 931.5 MeV.

Alpha decay reduces A by 4 and Z by 2; beta-minus decay raises Z by 1 with no change in A.

Standard model quarks: u, d, c, s, t, b; a proton is uud, a neutron udd.

Conservation laws in particle reactions: charge, baryon number, lepton number, energy–momentum.

Photoelectric effect: hf = φ + K_max; stopping potential V_s satisfies eV_s = K_max.

Threshold frequency f₀ = φ/h; no emission below it regardless of intensity.

Photon energy E = hf = hc/λ; hc ≈ 1240 eV·nm.

Compton shift Δλ = (h/m_ec)(1 − cosθ).

Planck relation E = hf links photon energy to frequency; h ≈ 6.63 × 10⁻³⁴ J·s.

Blackbody radiation: peak wavelength λ_max T = 2.9 × 10⁻³ m·K (Wien's law).

Nuclear fission releases energy from binding-energy differences; fusion powers stars.

Wave–particle duality: interference confirms wave nature; photoelectric effect confirms particle nature.

Time dilation Δt = γΔt₀; length contraction L = L₀/γ; γ = 1/√(1 − v²/c²).

Relativistic energy E = γmc²; rest energy E₀ = mc²; K = (γ − 1)mc².

Relativistic momentum p = γmv; energy–momentum relation E² = p²c² + m²c⁴.

Heisenberg uncertainty: ΔxΔp ≥ ħ/2, ΔEΔt ≥ ħ/2.

Schrödinger equation: iħ ∂ψ/∂t = Ĥψ; |ψ|² is probability density.

Particle in a box of width L: E_n = n²h²/(8mL²).

Quantum numbers n, l, m_l, m_s; Pauli exclusion forbids identical fermions sharing all four.

Photons are massless and travel at c; no massive particle can reach c.

Crystal systems: seven types; Miller indices (hkl) label planes, spacing d_hkl.

Bragg's law: 2d sinθ = nλ for constructive X-ray diffraction.

Unit cell types: simple cubic, bcc (2 atoms), fcc (4 atoms); packing fractions 0.52, 0.68, 0.74.

Band theory: filled valence band plus empty conduction band separated by a gap gives an insulator; a small gap gives a semiconductor.

Fermi energy is the highest occupied level at absolute zero; metals have a partially filled band.

Drude model conductivity σ = ne²τ/m; resistivity rises with temperature in metals.

Hall effect sign identifies charge-carrier type; Hall voltage V_H = BI/(net).

Superconductors show zero resistance and the Meissner effect below T_c.

Intrinsic carrier concentration n_i rises steeply with temperature; n·p = n_i² in equilibrium.

n-type doping uses pentavalent donors; p-type uses trivalent acceptors.

Diode equation I = I₀(e^(eV/k_BT) − 1); forward drop ≈ 0.7 V for silicon.

A p–n junction forms a depletion region with a built-in potential barrier.

Half-wave rectifier gives one pulse per cycle; full-wave gives two; a filter capacitor smooths output.

Transistor current gain β = I_C/I_B; in active mode the emitter–base junction is forward biased, collector–base reverse biased.

Operational amplifier: ideal infinite input impedance, zero output impedance; inverting gain = −R_f/R_in.

Logic gates: AND, OR, NOT, NAND, NOR; NAND and NOR are universal gates.

Vector operators: gradient of a scalar gives a vector; divergence of a vector gives a scalar; curl gives a vector.

Gauss's theorem ∫∇·F dV = ∮F·dS; Stokes' theorem ∫(∇×F)·dS = ∮F·dl.

Laplacian ∇² appears in Laplace's and Poisson's equations and the wave equation.

Fourier transform converts between time and frequency domains; convolution becomes multiplication.

Euler's relation e^(iθ) = cosθ + i sinθ underlies phasor analysis.

Numerical integration: trapezoidal error O(h²), Simpson's rule error O(h⁴).

Newton–Raphson iteration x_{n+1} = x_n − f(x_n)/f'(x_n) converges quadratically near a simple root.

Euler method for ODEs has local error O(h²); Runge–Kutta (RK4) has local error O(h⁵).

A hypothesis must be falsifiable: it predicts an observable outcome that could prove it wrong.

A controlled experiment varies one independent variable while holding others constant.

Correlation does not establish causation; confounding variables must be excluded.

Random and systematic errors differ: random scatter reduces with averaging, systematic bias does not.

A null hypothesis states no effect; a p-value is the probability of data at least as extreme given the null.

Reproducibility requires reporting method, data, and analysis so others can repeat the work.

A theory is a well-tested explanatory framework, not a guess; a law describes a regularity.

Peer review and independent replication are the standard checks on published claims.

For repeated readings, uncertainty is the standard deviation of the mean: σ/√N.

For a single scale reading, uncertainty is typically half the smallest division.

Combined uncertainty: for sums, add absolute uncertainties; for products or quotients, add fractional uncertainties.

If y = xⁿ, the fractional uncertainty in y is n times that in x.

Vernier callipers least count = 1 main-scale division − 1 vernier division; micrometer least count usually 0.01 mm.

Significant figures: the result carries no more figures than the least precise input.

A graph slope uncertainty comes from the steepest and shallowest lines of best fit through error bars.

Zero error must be recorded and subtracted before calculating a result.