Pancyprian Physics cheat sheet

Kinematics: for constant acceleration, v = u + at, s = ut + ½at², v² = u² + 2as; average velocity = (u+v)/2 only when acceleration is uniform.

Projectile motion separates into independent axes: horizontal velocity constant, vertical acceleration g downward; time of flight from vertical motion alone.

Newton's second law: ΣF = ma in an inertial frame; weight W = mg, friction f ≤ μN with equality at the point of slipping.

Momentum p = mv; impulse J = FΔt = Δp; in any collision total momentum is conserved, kinetic energy only in elastic ones.

Collisions: perfectly elastic gives relative speed of separation equal to relative speed of approach; perfectly inelastic bodies move together.

Work-energy: W = Fs cosθ, kinetic energy ½mv², gravitational potential energy mgh near Earth; work by non-conservative forces equals change in mechanical energy.

Circular motion: a = v²/r = ω²r toward the centre; centripetal force is the net inward resultant, never an extra force to add.

Moment of inertia and angular momentum: τ = Iα, L = Iω conserved when net external torque is zero.

SHM: a = −ω²x with ω = 2π/T; x = A sin(ωt + φ), v = ωA cos(ωt + φ), maximum speed ωA at equilibrium.

SHM energy: total E = ½mω²A² is constant; kinetic and potential energy exchange, each averaging half the total.

Simple pendulum T = 2π√(L/g) for small angles; mass-spring T = 2π√(m/k), independent of amplitude.

Wave equation v = fλ; for a stretched string v = √(T/μ) where μ is mass per unit length.

Superposition: in phase gives constructive interference (path difference nλ), antiphase gives destructive ((n+½)λ).

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

Sound: intensity level in decibels β = 10 log(I/I₀) with I₀ = 10⁻¹² W m⁻²; Doppler shift f' = f(v ± v_o)/(v ∓ v_s).

Resonance occurs when driving frequency equals natural frequency; damping lowers amplitude and slightly reduces frequency.

Coulomb's law F = kQ₁Q₂/r² with k ≈ 8.99×10⁹ N m² C⁻²; field E = F/q, and for a point charge E = kQ/r².

Uniform field between parallel plates E = V/d; potential energy change of a charge q moving through p.d. V is qV.

Ohm's law V = IR; resistors in series add, in parallel 1/R = Σ1/Rᵢ; power P = VI = I²R = V²/R.

Kirchhoff's rules: sum of currents at a junction is zero; sum of emfs equals sum of p.d.s around any closed loop.

Magnetic force on a moving charge F = qvB sinθ, on a current-carrying wire F = BIL sinθ; direction from Fleming's left-hand rule.

Field of a long straight wire B = μ₀I/(2πr); solenoid B = μ₀nI; parallel currents attract, antiparallel repel.

Faraday's law: induced emf = −dΦ/dt with Φ = BA cosθ; Lenz's law fixes direction so as to oppose the change producing it.

Transformer: V_s/V_p = N_s/N_p, and for an ideal transformer I_pV_p = I_sV_s.

Temperature scales: T(K) = θ(°C) + 273.15; heat Q = mcΔθ, and latent heat Q = mL at constant temperature.

Ideal gas law pV = nRT with R ≈ 8.31 J mol⁻¹ K⁻¹; also pV = NkT with k ≈ 1.38×10⁻²³ J K⁻¹.

Kinetic theory: average translational kinetic energy per molecule = (3/2)kT; mean square speed relates to temperature, not pressure alone.

First law: ΔU = Q − W, where W is work done by the gas; W = pΔV only at constant pressure.

Isothermal: pV constant; adiabatic: no heat exchange, Q = 0 so ΔU = −W; isochoric: W = 0 so ΔU = Q.

Efficiency of a heat engine η = W/Q_H = 1 − Q_C/Q_H; Carnot limit η = 1 − T_C/T_H with absolute temperatures.

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

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

Reflection: angle of incidence equals angle of reflection; plane mirror image is virtual, same size, laterally inverted.

Refraction: n₁ sinθ₁ = n₂ sinθ₂; n = c/v; total internal reflection only when going from denser to rarer and θ exceeds sin⁻¹(n₂/n₁).

Thin lens and mirror formula 1/f = 1/u + 1/v with sign conventions; magnification m = v/u = image height/object height.

Lens power P = 1/f in dioptres with f in metres; converging lens f > 0, diverging lens f < 0.

Young's double slit: fringe spacing w = λD/d; bright fringes at path difference nλ, dark at (n+½)λ.

Diffraction grating: d sinθ = nλ gives principal maxima; smaller slit spacing gives wider spread of orders.

Single-slit diffraction: first minimum at a sinθ = λ; narrower slit produces broader central maximum.

Dispersion arises because refractive index varies with wavelength; violet bends more than red in glass.

Photoelectric effect: photons of energy E = hf = hc/λ; maximum kinetic energy of emitted electrons K_max = hf − Φ.

Threshold frequency f₀ = Φ/h; no emission occurs below it regardless of intensity, and stopping potential V_s satisfies eV_s = K_max.

de Broglie wavelength λ = h/p; photon momentum p = E/c = h/λ.

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

Radioactive decay N = N₀e^{−λt}; half-life T½ = ln2/λ; activity A = λN measured in becquerels.

Nuclear notation ᴬZX: A nucleons, Z protons; alpha decay reduces A by 4 and Z by 2, beta-minus raises Z by 1.

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

Relativity basics: time dilation t = t₀/√(1 − v²/c²), length contraction, and rest energy E₀ = mc².

Pressure p = F/A; in a liquid p = ρgh at depth h; total pressure at depth includes atmospheric pressure.

Pascal's principle: pressure applied to an enclosed fluid is transmitted undiminished; hydraulic press F₁/A₁ = F₂/A₂.

Archimedes: upthrust equals weight of displaced fluid, U = ρ_fluid V_submerged g; floating bodies displace their own weight.

Continuity equation A₁v₁ = A₂v₂ for incompressible flow; Bernoulli p + ½ρv² + ρgh = constant along a streamline.

Viscosity: Stokes' law F = 6πηrv for a small sphere; terminal velocity when viscous drag plus upthrust equals weight.

Poiseuille flow rate through a tube proportional to r⁴ and to pressure difference, inversely proportional to length and viscosity.

Hooke's law σ = Eε, i.e. stress = Young's modulus × strain, valid in the linear elastic region below the proportional limit.

Elastic potential energy per unit volume = ½ × stress × strain; ductile materials show plastic deformation, brittle ones fracture without it.

Read analogue instruments to half the smallest division; record the reading with the same number of decimal places as the resolution.

Uncertainty of a single reading is typically half the smallest scale division; of repeated readings use the standard deviation of the mean.

Combining uncertainties: absolute uncertainties add for sums and differences; percentage uncertainties add for products and quotients.

A quantity raised to power n has percentage uncertainty n times its own percentage uncertainty; constants have zero uncertainty.

Plot the manipulated variable on the x-axis and the measured response on the y-axis; a straight line confirms the hypothesised relationship.

Gradient uncertainty from the steepest and shallowest lines of best fit through the error bars, not from individual points.

Percentage difference = |experimental − accepted|/accepted × 100%; percentage error compares a measured value with its true value.

Systematic errors shift all readings in one direction and are not reduced by averaging; random errors scatter and decrease with repetition.

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