252 formulas

Mechanics 58 formulas

Kinematics

Velocityv = Δx/Δt
Accelerationa = Δv/Δt
Kinematic Equation 1v = v₀ + at
Kinematic Equation 2x = x₀ + v₀t + (1/2)at²
Kinematic Equation 3v² = v₀² + 2a(x − x₀)
Kinematic Equation 4x = x₀ + (1/2)(v₀ + v)t
Free Fally = y₀ + v₀t − (1/2)gt²
Projectile RangeR = (v₀² sin(2θ))/g
Projectile Maximum HeightH = (v₀² sin²(θ))/(2g)
Projectile Time of FlightT = (2v₀ sin(θ))/g
Average Velocityv_avg = (x_f − x_i)/(t_f − t_i)

Dynamics

Newton's Second LawF = ma
WeightW = mg
Friction Forcef = μN
Hooke's LawF = −kx
Drag ForceF_D = (1/2)C_D ρAv²
Terminal Velocityv_t = √(2mg/(C_D ρA))
Center of Mass (Two Objects)x_cm = (m₁x₁ + m₂x₂)/(m₁ + m₂)
PressureP = F/A
Bulk ModulusB = −ΔP/(ΔV/V)

Work & Energy

WorkW = Fd cos θ
Kinetic EnergyKE = (1/2)mv²
Gravitational Potential EnergyPE = mgh
Spring Potential EnergyPE = (1/2)kx²
PowerP = W/t = Fv
Work-Energy TheoremW_net = ΔKE = (1/2)mv_f² − (1/2)mv_i²
Conservation of Mechanical EnergyKE₁ + PE₁ = KE₂ + PE₂

Momentum

Momentump = mv
ImpulseJ = FΔt = Δp
Elastic Collision (1D)v₁′ = ((m₁ − m₂)/(m₁ + m₂))v₁ + (2m₂/(m₁ + m₂))v₂
Perfectly Inelastic Collisionv′ = (m₁v₁ + m₂v₂)/(m₁ + m₂)
Coefficient of Restitutione = (v₂′ − v₁′)/(v₁ − v₂)

Rotational Motion

Angular Velocityω = Δθ/Δt
Angular Accelerationα = Δω/Δt
Linear-Angular Velocity Relationv = rω
Centripetal Accelerationa_c = v²/r = ω²r
Centripetal ForceF_c = mv²/r = mω²r
Torqueτ = rF sin θ
Moment of Inertia (Point Mass)I = mr²
Newton's Second Law (Rotation)τ = Iα
Rotational Kinetic EnergyKE_rot = (1/2)Iω²
Angular MomentumL = Iω
Rotational Kinematic 1ω = ω₀ + αt
Rotational Kinematic 2θ = θ₀ + ω₀t + (1/2)αt²
Rotational Kinematic 3ω² = ω₀² + 2α(θ − θ₀)
Moment of Inertia (Disk/Cylinder)I = (1/2)MR²
Moment of Inertia (Solid Sphere)I = (2/5)MR²
Moment of Inertia (Rod, Center)I = (1/12)ML²
Parallel Axis TheoremI = I_cm + Md²
Rolling Kinetic EnergyKE = (1/2)mv² + (1/2)Iω²

Gravitation

Universal GravitationF = G(m₁m₂/r²)
Gravitational Fieldg = GM/r²
Gravitational Potential Energy (General)U = −Gm₁m₂/r
Orbital Velocityv = √(GM/r)
Orbital PeriodT = 2π√(r³/(GM))
Escape Velocityv_e = √(2GM/r)
Surface Gravityg = GM/R²
Geosynchronous Orbit Radiusr = ∛(GMT²/(4π²))

Thermodynamics 34 formulas

Kinetic Theory

Ideal Gas LawPV = nRT
Ideal Gas Law (Molecular)PV = Nk_B T
Average Kinetic Energy (Gas)KE_avg = (3/2)k_B T
Root Mean Square Speedv_rms = √((3k_B T)/m) = √(3RT/M)
Mean Free Pathλ = (k_B T)/(√(2)πd²P)
Pressure (Kinetic Theory)P = (1/3)ρ⟨v²⟩ = (2/3)(N/V)KE_avg
Most Probable Speedv_p = √((2k_B T)/m) = √(2RT/M)
Mean Speedv_mean = √((8k_B T)/(πm)) = √(8RT/(πM))

Entropy

Entropy Change (Ideal Gas)ΔS = nC_V ln(T_f/T_i) + nR ln(V_f/V_i)
Boltzmann EntropyS = k_B ln W
Gibbs Free EnergyG = H − TS
Helmholtz Free EnergyF = U − TS

Waves 37 formulas

Simple Harmonic Motion

SHM Displacementx = A cos(ωt + φ)
SHM Velocityv = −Aω sin(ωt + φ)
SHM Accelerationa = −ω²x
SHM Maximum Velocityv_max = Aω
Spring Oscillation PeriodT = 2π√(m/k)
Pendulum PeriodT = 2π√(L/g)
SHM Total EnergyE = (1/2)kA² = (1/2)mω²A²
Physical Pendulum PeriodT = 2π√(I/(mgd))
Torsional Oscillation PeriodT = 2π√(I/κ)
Damped Oscillationx = Ae^(−bt/(2m)) cos(ω′t)
Quality Factor (Q Factor)Q = mω₀/b = ω₀/Δω
Resonance AmplitudeA_max at ω = ω₀

Wave Properties

Wave Equationv = fλ
Period and FrequencyT = 1/f
Angular Frequencyω = 2πf = 2π/T
Wave Numberk = 2π/λ
Wave IntensityI = P/A
Intensity-Amplitude RelationI ∝ A²
Inverse Square Law (Intensity)I = P/(4πr²)
Wave Speed on Stringv = √(T/μ)
Reflection CoefficientR = ((v₂ − v₁)/(v₂ + v₁))²
Transmission CoefficientT = 1 − R = 4v₁v₂/(v₁ + v₂)²
Group Velocityv_g = dω/dk
Phase Velocityv_p = ω/k

Sound Waves

Speed of Sound in Gasv = √(γRT/M)
Sound Level (Decibels)β = 10 log₁₀(I/I₀)
Beat Frequencyf_beat = |f₁ − f₂|
Mach NumberM = v/v_sound
Shock Wave Angle (Mach Cone)sin(θ) = v_sound/v = 1/M
Sound Intensity Level Additionβ_total = 10 log₁₀(Σ10^(β_i/10))

Doppler Effect

Doppler Effect (General)f′ = f((v ± v_o)/(v ∓ v_s))
Doppler Effect (Approaching)f′ = f((v + v_o)/(v − v_s))
Doppler Effect (Receding)f′ = f((v − v_o)/(v + v_s))
Doppler Shift (Light)Δλ/λ = v/c

Superposition

Standing Wave on Stringf_n = (n/(2L))√(T/μ)
Standing Wave (Open Pipe)f_n = nv/(2L)
Standing Wave (Closed Pipe)f_n = nv/(4L)

Electricity & Magnetism 48 formulas

Electrostatics

Coulomb's LawF = k(q₁q₂/r²)
Electric FieldE = F/q = k(Q/r²)
Electric PotentialV = k(Q/r)
Electric Potential EnergyU = k(q₁q₂/r) = qV
Uniform Electric FieldE = V/d
Gauss's Law∮E⃗ · dA⃗ = Q_enc/ε₀
Electric Dipole Momentp = qd
Electric Dipole Field (Axial)E = 2kp/r³
Electric FluxΦ_E = EA cos θ

Electric Circuits

Ohm's LawV = IR
Electric PowerP = IV = I²R = V²/R
Electric CurrentI = dQ/dt
ResistanceR = ρL/A
Resistors in SeriesR_eq = R₁ + R₂ + R₃ + ...
Resistors in Parallel1/R_eq = 1/R₁ + 1/R₂ + ...
Kirchhoff's Voltage LawΣV = 0
Kirchhoff's Current LawΣI_in = ΣI_out

Capacitance

CapacitanceC = Q/V
Parallel Plate CapacitorC = ε₀(A/d)
Capacitor EnergyU = (1/2)CV² = Q²/(2C)
Capacitors in Series1/C_eq = 1/C₁ + 1/C₂ + ...
Capacitors in ParallelC_eq = C₁ + C₂ + ...
RC Time Constantτ = RC
Capacitor with DielectricC = κε₀(A/d)
Energy Density (Electric Field)u = (1/2)ε₀E²

Magnetism

Magnetic Force on ChargeF = qvB sin θ
Magnetic Force on WireF = BIL sin θ
Biot-Savart LawdB = (μ₀I/(4π))((dl sin θ)/r²)
Magnetic Field (Long Wire)B = μ₀I/(2πr)
Magnetic Field (Solenoid)B = μ₀nI
Energy Density (Magnetic Field)u = B²/(2μ₀)
Cyclotron Frequencyf = qB/(2πm)
Cyclotron Radiusr = mv/(qB)
Hall VoltageV_H = BId/(nqA)
Ampere's Law∮B⃗ · dl⃗ = μ₀I_enc

Electromagnetic Induction

Faraday's Lawℰ = −dΦ_B/dt
Magnetic FluxΦ_B = BA cos θ
Motional EMFℰ = BLv
Self-Inductanceℰ = −L(dI/dt)
Inductor EnergyU = (1/2)LI²
RL Time Constantτ = L/R
Mutual Inductance EMFℰ = −M(dI/dt)
Transformer EquationV_s/V_p = N_s/N_p

AC Circuits

RMS Voltage/CurrentV_rms = V_peak/√(2)
Capacitive ReactanceX_C = 1/(ωC)
Inductive ReactanceX_L = ωL
Impedance (RLC Series)Z = √(R² + (X_L − X_C)²)
Resonant Frequency (LC)f = 1/(2π√(LC))

Optics 34 formulas

Reflection

Law of Reflectionθ_i = θ_r
Mirror Equation1/f = 1/d_o + 1/d_i
Mirror MagnificationM = −d_i/d_o = h_i/h_o
Focal Length and Radiusf = R/2
Fresnel Reflection (Normal Incidence)r = (n₁ − n₂)/(n₁ + n₂), R = r²
Spherical Mirror Focal Length1/f = 2/R

Refraction

Snell's Lawn₁ sin θ₁ = n₂ sin θ₂
Refractive Indexn = c/v
Critical Anglesin θ_c = n₂/n₁
Apparent Depthd_apparent = d_real/n

Lenses

Thin Lens Equation1/f = 1/d_o + 1/d_i
Lensmaker's Equation1/f = (n − 1)(1/R₁ − 1/R₂)
Lens MagnificationM = −d_i/d_o
Lens PowerP = 1/f
Combined Lens PowerP_total = P₁ + P₂

Wave Optics

Double-Slit Interferenced sin θ = mλ
Fringe SpacingΔy = λL/d
Single-Slit Diffractiona sin θ = mλ
Diffraction Gratingd sin θ = mλ
Resolving Power of GratingR = λ/Δλ = mN
Thin Film Interference2nt = (m + 1/2)λ
Optical Path LengthOPL = n · d
Constructive Interference ConditionΔ = mλ

Polarization

Brewster's Angletan θ_B = n₂/n₁
Malus's LawI = I₀ cos² θ

Resolution

Rayleigh Criterionθ_min = 1.22(λ/D)

Optical Instruments

Magnifying Glass MagnificationM = 1 + D/f
Microscope MagnificationM = −L/f_o · D/f_e
Telescope MagnificationM = −f_o/f_e
Numerical ApertureNA = n sin θ
F-Number (f-stop)f/# = f/D
Depth of FieldDOF ≈ 2Ncs²/f²

Prisms

Dispersion (Angular)dθ/dλ = dθ/dn · dn/dλ
Prism Minimum Deviationn = sin((A + δ_m)/2)/sin(A/2)

Modern Physics 41 formulas

Special Relativity

Lorentz Factorγ = 1/√(1 − v²/c²)
Time DilationΔt = γΔt₀ = Δt₀/√(1 − v²/c²)
Length ContractionL = L₀/γ = L₀√(1 − v²/c²)
Relativistic Momentump = γmv = mv/√(1 − v²/c²)
Mass-Energy EquivalenceE = mc²
Relativistic Total EnergyE = γmc² = mc²/√(1 − v²/c²)
Relativistic Kinetic EnergyK = (γ − 1)mc²
Energy-Momentum RelationE² = (pc)² + (mc²)²
Relativistic Velocity Additionu = (u′ + v)/(1 + u′v/c²)
Relativistic Doppler Effectf′ = f√((1 − v/c)/(1 + v/c))
Proper Timedτ = dt/γ = dt√(1 − v²/c²)
Relativity of SimultaneityΔt′ = γ(Δt − vΔx/c²)

Quantum Mechanics

Photon Energy (Planck)E = hf = hc/λ
Photoelectric EffectK_max = hf − φ
Stopping PotentialeV₀ = hf − φ
de Broglie Wavelengthλ = h/p = h/(mv)
Heisenberg Uncertainty PrincipleΔx · Δp ≥ ℏ/2
Energy-Time UncertaintyΔE · Δt ≥ ℏ/2
Time-Dependent Schrödinger Equationiℏ(∂Ψ/∂t) = ĤΨ
Particle in a Box Energy LevelsE_n = n²h²/(8mL²)
Compton Scatteringλ′ − λ = (h/(m_e c))(1 − cos θ)
Photon Momentump = h/λ = E/c
Fermi EnergyE_F = (ℏ²/(2m))(3π²n)^(2/3)
Photoelectric Threshold Frequencyf_threshold = φ/h
Bragg's Law2d sin θ = nλ
Electron Wavelength (Accelerated)λ = h/√(2m_e eV)

Atomic Physics

Bohr Radiusa₀ = 4πε₀ℏ²/(m_e e²) ≈ 5.29 × 10⁻¹¹ m
Bohr Energy LevelsE_n = −(13.6 eV)/n²
Hydrogen Spectrum (Rydberg)1/λ = R_H (1/n₁² − 1/n₂²)
Hydrogen-like Atom EnergyE_n = −13.6 eV × Z²/n²
Electron Orbital Radiusr_n = n²a₀/Z
Moseley's Law√(f) = a(Z − b)

Nuclear Physics

Radioactive Decay LawN(t) = N₀e^(−λt)
Half-Lifet_½ = (ln 2)/λ ≈ 0.693/λ
ActivityA = λN = A₀e^(−λt)
Nuclear Binding EnergyE_b = Δm · c² = [Zm_p + Nm_n − M]c²
Alpha Decay Energy (Q-value)Q = (m_parent − m_daughter − m_α)c²
Beta Decayn → p + e⁻ + ν̄_e
Nuclear RadiusR = R₀A^(1/3)

Thermal Radiation

Stefan-Boltzmann Law (Blackbody Form)P = σAT⁴
Wien's Displacement Lawλ_max T = b = 2.898 × 10⁻³ m·K

Related guides

Techniques and worked methods for putting these formulas to use.

Mechanics 58 formulas

Kinematics

Dynamics

Work & Energy

Momentum

Rotational Motion

Gravitation

Thermodynamics 34 formulas

Heat & Temperature

First Law

Second Law

Kinetic Theory

Entropy

Waves 37 formulas

Simple Harmonic Motion

Wave Properties

Sound Waves

Doppler Effect

Superposition

Electricity & Magnetism 48 formulas

Electrostatics

Electric Circuits

Capacitance

Magnetism

Electromagnetic Induction

AC Circuits

Optics 34 formulas

Reflection

Refraction

Lenses

Wave Optics

Polarization

Resolution

Optical Instruments

Prisms

Modern Physics 41 formulas

Special Relativity

Quantum Mechanics

Atomic Physics

Nuclear Physics

Thermal Radiation

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