Physics formula sheets
Start with the model and its assumptions. These relationships use SI units unless you convert every quantity consistently.
Vectors & forces
| Relationship | Formula | When it applies |
|---|---|---|
| Components | Aₓ = A cos θ; Aᵧ = A sin θ | θ is measured counterclockwise from +x. |
| Magnitude & direction | |A| = √(Aₓ² + Aᵧ²); θ = atan2(Aᵧ, Aₓ) | Direction is undefined when |A| = 0. |
| Unit vector | Â = A / |A| | Dimensionless; requires A ≠ 0. |
| Vector products | A · B = |A||B| cos θ; |A × B| = |A||B| sin θ | Dot product is scalar; cross product follows the right-hand rule. |
| Newton’s second law | ΣF = ma | In an inertial frame, for constant mass. |
| Weight & friction | W = mg; fₛ ≤ μₛN; fₖ = μₖN | Static friction adjusts to prevent relative slipping, up to its maximum. |
| Spring force | Fₓ = −kx | x is displacement from the relaxed position; linear elastic regime. |
Motion, momentum & energy
| Relationship | Formula | When it applies |
|---|---|---|
| Average motion | vavg = Δr/Δt; average speed = distance/Δt | Distance is path length; displacement is change in position. |
| Constant acceleration | v = v₀ + at; Δx = v₀t + ½at²; v² = v₀² + 2aΔx | Apply per component only when that acceleration is constant. |
| Projectile motion | x = v₀ cos θ · t; y = y₀ + v₀ sin θ · t − ½gt² | Uniform downward gravity, negligible air resistance. |
| Gravitation | F = GMm/r²; U = −GMm/r | Outside a spherically symmetric source or for point masses; r is centre-to-centre separation and U = 0 at infinity. |
| Momentum & impulse | p = mv; J = ∫F dt = Δp | Total momentum is conserved when net external impulse is zero. |
| Work & kinetic energy | W = ∫F · dr; K = ½mv²; Wnet = ΔK | Classical nonrelativistic motion. |
| Potential energy & power | Ug = mgh; Us = ½kx²; P = dW/dt = F · v | mgh assumes nearly uniform gravity; x is spring deformation. |
Rotation
| Relationship | Formula | When it applies |
|---|---|---|
| Circular motion | v = rω; aᶜ = v²/r = rω² | v denotes speed and ω angular-speed magnitude; centripetal acceleration points toward the centre. |
| Torque & inertia | τ = r × F; I = Σmᵢrᵢ²; Στ = Iα | rᵢ is perpendicular distance from the axis; use torque components about a stationary pivot or the centre of mass, with constant I. |
| Angular momentum & energy | L = Iω; Krot = ½Iω² | L = Iω applies about a principal or fixed symmetry axis. |
| Rolling | vCM = Rω; K = ½mvCM² + ½Iω² | Rolling without slipping. |
Thermal physics & fluids
| Relationship | Formula | When it applies |
|---|---|---|
| Temperature & heat | T/K = T/°C + 273.15; Q = mcΔT; Qphase = mL | A temperature difference has the same numerical value in K and °C. |
| Heat transfer | Pcond = kAΔT/ℓ; Prad,net = εσA(T⁴ − Tenv⁴) | Radiation temperatures must be in kelvin. |
| Expansion | Δℓ = αℓ₀ΔT; ΔV ≈ 3αV₀ΔT | Small changes in isotropic solids; other substances use their volume expansion coefficient β. |
| Ideal gas | PV = nRT | Absolute pressure, temperature in kelvin; ideal dilute-gas approximation. |
| Hydrostatics & buoyancy | P = P₀ + ρgh; FB = ρfluid gVdisplaced | Use the fluid density, not the object density, for buoyancy. |
| Continuity & Bernoulli | A₁v₁ = A₂v₂; P + ½ρv² + ρgy = constant | Steady incompressible flow; Bernoulli along a streamline with negligible losses. |
Waves & harmonics
| Relationship | Formula | When it applies |
|---|---|---|
| Wave relationships | v = fλ = ω/k; ω = 2πf; k = 2π/λ | Phase speed for a monochromatic wave. |
| Sound level | β = 10 log₁₀(I/I₀); I₀ = 10⁻¹² W/m² | Add incoherent sound intensities before converting to decibels. |
| Refraction & photons | n₁ sin θ₁ = n₂ sin θ₂; E = hf = hc/λ; p = h/λ | Angles are measured from the normal. Photon equations use vacuum wavelength; energy is per photon. |
| Mass–spring motion | ω₀ = √(k/m); T = 2π√(m/k) | Ideal linear spring, no damping. |
| Pendulums | Tsimple ≈ 2π√(ℓ/g); Tphysical ≈ 2π√(I/(mgd)) | Small angular amplitude; d is pivot-to-centre-of-mass distance. |
| Standing waves | λₙ = 2ℓ/n; fₙ = nv/(2ℓ) | String fixed at both ends; n loops and n + 1 nodes including endpoints. |
| Damping | x = A e⁻ᵞᵗ cos(ωdt + φ); ωd² = k/m − γ² | Underdamped viscous damping; γ = b/(2m). Exponential decay does not stop at a finite time. |
Find definitions and the original reference tables in the glossary →