Formulas

Physics

Mechanics

Velocity
v = Δs / Δt
v Velocity
Δs Change in Displacement or Position
Δt Change in Time

Acceleration
a = Δv / Δt
a Acceleration
Δv Change in Velocity
Δt Change in Time

Equations of Motion
v = v₀ + at
v Final Velocity
v₀ Initial Velocity
a Acceleration
t Time

Equations of Motion
s = s₀ + v₀t + ½at²
s Final Position
s₀ Initial Position
v₀ Initial Velocity
a Acceleration
t Time

Equations of Motion
v² = v₀² + 2a(s − s₀)
v Final Velocity
v₀ Initial Velocity
a Acceleration
s Final Position
s₀ Initial Position

Newton’s Second Law
∑F = ma
∑F Net Force the Vector Sum of All Forces
m Mass
a Acceleration

Weight
W = mg
W Weight Force Due to Gravity
m Mass
g Acceleration Due to Gravity

Dry Static Friction
f_s ≤ μ_s N
f_s Force of Static Friction
μ_s Coefficient of Static Friction
N Normal Force the Perpendicular Force Exerted by a Surface

Dry Kinetic Friction
f_k = μ_k N
f_k Force of Kinetic Friction
μ_k Coefficient of Kinetic Friction
N Normal Force

Centripetal Acceleration
a_c = v² / r
a_c Centripetal Acceleration
v Tangential Speed Magnitude of Velocity
r Radius of the Circular Path

Impulse
J = FΔt
J Impulse
F Average Force
Δt Time Interval Over Which the Force Acts

Impulse-Momentum Theorem
FΔt = mΔv
F Average Force
Δt Time Interval
m Mass
Δv Change in Velocity

Work
W = FΔs cos θ
W Work
F Magnitude of the Force
Δs Magnitude of the Displacement
θ Angle Between the Force and Displacement Vectors

Work-Energy Theorem
W = ΔK
W Net Work Done on an Object
ΔK Change in Kinetic Energy K Final K Initial

Kinetic Energy
K = ½mv²
K Kinetic Energy
m Mass
v Speed

Gravitational Potential Energy
U_g = mgh
U_g Gravitational Potential Energy
m Mass
g Acceleration Due to Gravity
h Height Above a Reference Point

General Potential Energy Change
ΔU = -∫ F ⋅ ds
ΔU Change in Potential Energy
F Conservative Force
ds Infinitesimal Displacement Vector
∫ … ⋅ ds Line Integral Work Done by the Force Along a Path

Power
P = ΔW / Δt
P Power
ΔW Change in Work Amount of Work Done
Δt Change in Time Time Interval

Power-Velocity Relation
P = Fv cos θ
P Instantaneous Power
F Magnitude of the Force
v Magnitude of the Velocity
θ Angle Between the Force and Velocity Vectors

Angular Velocity
ω = Δθ / Δt
ω Angular Velocity
Δθ Change in Angular Displacement
Δt Change in Time

Linear to Angular Velocity
v = ωr
v Tangential Linear Speed
ω Angular Velocity
r Radius

Angular Acceleration
α = Δω / Δt
α Angular Acceleration
Δω Change in Angular Velocity
Δt Change in Time

Equations of Rotational Motion
ω = ω₀ + αt
ω Final Angular Velocity
ω₀ Initial Angular Velocity
α Angular Acceleration
t Time

Equations of Rotational Motion
θ = θ₀ + ω₀t + ½αt²
θ Final Angular Position
θ₀ Initial Angular Position
ω₀ Initial Angular Velocity
α Angular Acceleration
t Time

Torque
τ = rF sin θ
τ Torque
r Distance from the Pivot Point to the Point Where Force is Applied Lever Arm
F Magnitude of the Force
θ Angle Between the Force Vector and the Lever Arm Vector

Newton’s Second Law for Rotation
∑τ = Iα
∑τ Net Torque
I Moment of Inertia
α Angular Acceleration

Moment of Inertia Point Mass
I = ∑mr²
I Moment of Inertia
m Mass of a Point Particle
r Perpendicular Distance from the Particle to the Axis of Rotation
∑ Sum Over All Particles

Rotational Work
W = τΔθ
W Work Done by a Torque
τ Torque
Δθ Angular Displacement

Rotational Power
P = τω
P Power Delivered by a Torque
τ Torque
ω Angular Velocity

Angular Momentum Point Mass
L = mrv sin θ
L Angular Momentum
m Mass
r Distance from the Point Mass to the Pivot Point
v Tangential Velocity
θ Angle Between the Position and Velocity Vectors

Angular Momentum Rigid Body
L = Iω
L Angular Momentum
I Moment of Inertia
ω Angular Velocity

Angular Impulse
H = τΔt
H Angular Impulse
τ Average Torque
Δt Time Interval

Angular Impulse-Momentum Theorem
τΔt = ΔL
τ Average Torque
Δt Time Interval
ΔL Change in Angular Momentum

Newton’s Law of Universal Gravitation
F_g = G m₁ m₂ / r²
F_g Force of Gravitational Attraction
G Gravitational Constant
m₁ m₂ Masses of the Two Objects
r Distance Between the Centers of the Two Masses

Gravitational Potential Energy General
U_g = -G m₁ m₂ / r
U_g Gravitational Potential Energy Defined as Zero at Infinite Separation
G Gravitational Constant
m₁ m₂ Masses of the Two Objects
r Distance Between Their Centers

Escape Speed
v_esc = √(2Gm / r)
v_esc Escape Speed Minimum Speed to Escape a Gravitational Field
G Gravitational Constant
m Mass of the Celestial Body e g Planet Star
r Radius of the Celestial Body

Hooke’s Law
F = -kΔx
F Restoring Force Exerted by the Spring
k Spring Constant Stiffness
Δx Displacement from the Spring’s Equilibrium Position

Spring Potential Energy
U_s = ½k(Δx)²
U_s Elastic Potential Energy Stored in the Spring
k Spring Constant
Δx Displacement from Equilibrium

Period of Simple Harmonic Oscillator
T = 2π√(m / k)
T Period Time for One Complete Cycle
m Mass of the Oscillating Object
k Spring Constant

Density
ρ = m / V
ρ Density
m Mass
V Volume

Buoyancy Force
B = ρ_fluid g V_displaced
B Buoyant Force
ρ_fluid Density of the Fluid
g Acceleration Due to Gravity
V_displaced Volume of Fluid Displaced by the Object

Mass Flow Rate
q_m = Δm / Δt
q_m Mass Flow Rate
Δm Mass
Δt Time Interval

Volume Flow Rate
q_V = ΔV / Δt
q_V Volume Flow Rate
ΔV Volume
Δt Time Interval

Equation of Continuity
ρ₁A₁v₁ = ρ₂A₂v₂
ρ Density of the Fluid
A Cross Sectional Area
v Fluid Velocity
Subscripts 1 2 Refer to Two Different Points in the Flow

Bernoulli’s Equation
P₁ + ρgy₁ + ½ρv₁² = P₂ + ρgy₂ + ½ρv₂²
P Pressure
ρ Density of the Fluid
g Acceleration Due to Gravity
y Height Above a Reference Level
v Fluid Speed
Subscripts 1 2 Refer to Two Different Points Along a Streamline

Drag Force
F_d = ½ ρ C_d A v²
F_d Drag Force
ρ Density of the Fluid
C_d Drag Coefficient Depends on the Object’s Shape
A Cross Sectional Area Area Perpendicular to Flow
v Speed of the Object Relative to the Fluid

Thermal Physics

Linear Thermal Expansion
ΔL = α L₀ ΔT
ΔL Change in Length
α Coefficient of Linear Expansion
L₀ Original Length
ΔT Change in Temperature

Volumetric Thermal Expansion
ΔV = 3α V₀ ΔT
ΔV Change in Volume
α Coefficient of Linear Expansion
V₀ Original Volume
ΔT Change in Temperature

Average Molecular Kinetic Energy
⟨K⟩ = (3/2)kT
⟨K⟩ Average Translational Kinetic Energy of a Molecule
k Boltzmann Constant
T Absolute Temperature in Kelvin

Root Mean Square Speed
v_rms = √(3kT / m)
v_rms Root Mean Square Speed of Molecules
k Boltzmann Constant
T Absolute Temperature
m Mass of a Single Molecule

Heat Flow Rate
P = ΔQ / Δt
P Power Rate of Heat Transfer
ΔQ Heat Transferred
Δt Time Interval

Stefan-Boltzmann Law
P = εσA(T⁴ – T₀⁴)
P Net Radiated Power
ε Emissivity of the Object’s Surface
σ Stefan Boltzmann Constant
A Surface Area
T Temperature of the Object
T₀ Temperature of the Surroundings

Wien’s Displacement Law
λ_max = b / T
λ_max Wavelength of Peak Emission Blackbody Radiation
b Wien’s Displacement Constant
T Absolute Temperature

Change in Internal Energy Ideal Monatomic Gas
ΔU = (3/2)nRΔT
ΔU Change in Internal Energy
n Number of Moles
R Ideal Gas Constant
ΔT Change in Temperature

Entropy Change
ΔS = ΔQ_rev / T
ΔS Change in Entropy
ΔQ_rev Heat Transferred in a Reversible Process
T Absolute Temperature

Efficiency of a Heat Engine
η = 1 – Q_c / Q_h
η Efficiency
Q_c Heat Exhausted to the Cold Reservoir
Q_h Heat Absorbed from the Hot Reservoir

Maximum Carnot Efficiency
η_carnot = 1 – T_c / T_h
η_carnot Maximum Possible Carnot Efficiency
T_c Absolute Temperature of the Cold Reservoir
T_h Absolute Temperature of the Hot Reservoir

Coefficient of Performance Refrigerator
COP_real = Q_c / (Q_h – Q_c)
COP Coefficient of Performance
Q_c Heat Removed from the Cold Reservoir Inside
Q_h Heat Delivered to the Hot Reservoir Outside

Ideal Gas Law
PV = nRT
P Pressure
V Volume
n Number of Moles of Gas
R Ideal Gas Constant
T Absolute Temperature

First Law of Thermodynamics
ΔU = Q – W
ΔU Change in Internal Energy of the System
Q Heat Added to the System
W Work Done by the System on its Surroundings

Work Done in Isothermal Expansion
W = nRT ln(V_f / V_i)
W Work Done by the Gas
n Number of Moles
R Ideal Gas Constant
T Constant Absolute Temperature
V_f Final Volume
V_i Initial Volume

Heat Transfer at Constant Pressure
Q = n C_p ΔT
Q Heat Transferred
n Number of Moles
C_p Molar Specific Heat at Constant Pressure
ΔT Change in Temperature

Waves & Optics

Wave Velocity
v = fλ
v Wave Velocity Speed
f Frequency
λ Wavelength

Wave Function Sinusoidal
y(x,t) = A sin(2π(x/λ – ft) + φ)
y Wave Displacement at Position x and Time t
x Position
t Time
A Amplitude Maximum Displacement
λ Wavelength
f Frequency
φ Phase Constant Initial Phase

Beat Frequency
f_beat = |f₁ – f₂|
f_beat Beat Frequency the Frequency of the Amplitude Variation
f₁ f₂ Frequencies of the Two Interfering Waves

Intensity of a Wave
I = P / A
I Intensity Power per Unit Area
P Power
A Area

Sound Intensity Level
β = 10 log(I / I₀)
β Sound Intensity Level in Decibels dB
I Sound Intensity
I₀ Reference Intensity Threshold of Hearing Typically 10⁻¹² W m²

Doppler Effect Approaching Source
f_o = f_s [v / (v – v_s)]
f_o Observed Frequency
f_s Source Frequency
v Speed of Sound in the Medium
v_s Speed of the Source Toward the Observer

Snell’s Law of Refraction
n₁ sinθ₁ = n₂ sinθ₂
n Index of Refraction of a Medium
θ Angle Measured from the Normal Perpendicular
Subscripts 1 2 Refer to the First and Second Media

Critical Angle for Total Internal Reflection
sin θ_c = n₂ / n₁
θ_c Critical Angle
n₁ Index of Refraction of the Initial Denser Medium
n₂ Index of Refraction of the Second Less Dense Medium

Lens Mirror Equation
1/f = 1/d_o + 1/d_i
f Focal Length
d_o Object Distance
d_i Image Distance

Magnification
m = h_i / h_o = -d_i / d_o
m Magnification
h_i Image Height
h_o Object Height
d_i Image Distance
d_o Object Distance

Thin Lens Formula
1/f = 1/i + 1/o
f Focal Length
i Image Distance
o Object Distance

Speed of Sound in a Medium
v = √(B / ρ)
v Speed of Sound
B Bulk Modulus of the Medium
ρ Density of the Medium

Wavelength
λ = v / f
λ Wavelength
v Wave Speed
f Frequency

Acoustic Impedance
Z = ρ c
Z Acoustic Impedance
ρ Density of the Medium
c Speed of Sound in the Medium

Electricity & Magnetism

Coulomb’s Law
F = k q₁ q₂ / r²
F Electrostatic Force Between Two Point Charges
k Coulomb’s Constant 1 4πε₀
q₁ q₂ Electric Charges
r Distance Between the Charges

Electric Field
E = F / q
E Electric Field
F Force Experienced by a Test Charge
q Magnitude of the Test Charge

Electric Field and Potential
E = -ΔV / Δx
E Electric Field Component in the X Direction
ΔV Change in Electric Potential
Δx Change in Position Displacement

Electric Potential Point Charge
V = k q / r
V Electric Potential Voltage at a Distance r from the Charge
k Coulomb’s Constant
q Source Charge Creating the Potential
r Distance from the Source Charge

Capacitance Parallel Plate
C = κε₀ A / d
C Capacitance
κ Dielectric Constant
ε₀ Permittivity of Free Space
A Area of One Plate
d Distance Between the Plates

Energy Stored in a Capacitor
U = ½ Q V = ½ C V²
U Energy Stored
Q Charge on One Plate
V Voltage Potential Difference Between Plates
C Capacitance

Electric Current
I = Δq / Δt
I Electric Current
Δq Net Charge Flowing Past a Point
Δt Time Interval

Current Density
J = I / A
J Current Density
I Electric Current
A Cross Sectional Area

Ohm’s Law
V = I R
V Voltage Potential Difference
I Current
R Resistance

Resistivity
ρ = R A / L
ρ Resistivity of the Material
R Resistance of a Sample of the Material
A Cross Sectional Area of the Sample
L Length of the Sample

Electric Power
P = V I = I² R
P Power Dissipated
V Voltage Drop Across a Component
I Current Through the Component
R Resistance of the Component

Magnetic Force on a Moving Charge
F = q v B sin θ
F Magnetic Force
q Charge of the Particle
v Speed of the Particle
B Magnetic Field Strength
θ Angle Between the Velocity and Magnetic Field Vectors

Magnetic Force on a Current Carrying Wire
F = I L B sin θ
F Magnetic Force on the Wire
I Current in the Wire
L Length of the Wire Inside the Magnetic Field
B Magnetic Field Strength
θ Angle Between the Current and Magnetic Field Vectors

Biot Savart Law
dB = (μ₀ / 4π) (I ds × r̂) / r²
dB Infinitesimal Magnetic Field Produced by a Current Element
μ₀ Permeability of Free Space
I Current
ds Infinitesimal Length Vector of the Wire Direction of Current
r̂ Unit Vector Pointing from the Source Element to the Point Where the Field is Measured
r Distance from the Source Element to the Point

Magnetic Field Inside a Solenoid
B = μ₀ n I
B Magnetic Field Inside the Solenoid
μ₀ Permeability of Free Space
n Number of Turns of Wire per Unit Length n N L
I Current in the Wire

Magnetic Field of a Straight Wire
B = (μ₀ I) / (2π r)
B Magnetic Field Strength
μ₀ Permeability of Free Space
I Current in the Wire
r Perpendicular Distance from the Wire

Force Between Parallel Wires
F / L = (μ₀ I₁ I₂) / (2π d)
F / L Force per Unit Length Between the Wires
μ₀ Permeability of Free Space
I₁ I₂ Currents in the Two Wires
d Distance Separating the Wires

Magnetic Flux
Φ_B = B A cos θ
Φ_B Magnetic Flux
B Magnetic Field Strength
A Area
θ Angle Between the Magnetic Field Vector and the Area Vector Normal to the Surface

Faraday’s Law of Induction
ℰ = -dΦ_B / dt
ℰ Induced Electromotive Force EMF or Voltage
dΦ_B / dt Rate of Change of Magnetic Flux

Self Inductance EMF
ℰ = -L dI/dt
ℰ Self Induced EMF
L Self Inductance
dI/dt Rate of Change of Current

Impedance of RLC Circuit
Z = √[R² + (X_L – X_C)²]
Z Impedance AC Analogue of Resistance
R Resistance
X_L Inductive Reactance X L ωL
X_C Capacitive Reactance X C 1 ωC

Gauss’s Law for Electricity
∮ E ⋅ dA = Q_enclosed / ε₀
∮ … ⋅ dA Closed Surface Integral Calculates Flux Through a Gaussian Surface
E Electric Field
Q_enclosed Total Charge Enclosed Inside the Surface
ε₀ Permittivity of Free Space

Gauss’s Law for Magnetism
∮ B ⋅ dA = 0
∮ … ⋅ dA Closed Surface Integral Calculates Flux Through a Gaussian Surface
B Magnetic Field
0 Signifies that There are No Magnetic Monopoles All Magnetic Field Lines are Closed Loops

Ampere’s Law with Maxwell’s Correction
∮ B ⋅ ds = μ₀ε₀ (dΦ_E / dt) + μ₀ I
∮ B ⋅ ds Line Integral of the Magnetic Field Around a Closed Loop
μ₀ Permeability of Free Space
ε₀ Permittivity of Free Space
dΦ_E / dt Rate of Change of Electric Flux Through the Loop
I Current Enclosed by the Loop

Modern Physics

Mass Energy Equivalence
E = m c²
E Rest Energy
m Rest Mass
c Speed of Light in a Vacuum

Energy of a Photon
E = h f
E Energy of a Single Photon
h Planck’s Constant
f Frequency of the Photon

Photoelectric Effect Max Kinetic Energy
K_max = h f – φ
K_max Maximum Kinetic Energy of the Ejected Photoelectron
h Planck’s Constant
f Frequency of the Incident Photon
φ Work Function Minimum Energy Needed to Eject an Electron from the Material

De Broglie Wavelength
λ = h / p
λ de Broglie Wavelength
h Planck’s Constant
p Momentum of the Particle

Heisenberg Uncertainty Principle Position Momentum
Δx Δp ≥ ħ / 2
Δx Uncertainty in Position
Δp Uncertainty in Momentum
ħ Reduced Planck’s Constant h 2π

Time Dependent Schrödinger Equation
iħ ∂Ψ/∂t = Ĥ Ψ
i Imaginary Unit √ 1
ħ Reduced Planck’s Constant
∂Ψ/∂t Partial Derivative of the Wavefunction with Respect to Time
Ĥ Hamiltonian Operator an Operator Representing the Total Energy of the System
Ψ Wavefunction of the Quantum System

Relativistic Time Dilation
Δt = Δt₀ / √(1 – v²/c²)
Δt Time Interval Measured in a Frame of Reference Where the Clock is Moving
Δt₀ Proper Time Interval Measured in the Clock’s Rest Frame
v Relative Speed Between the Two Reference Frames
c Speed of Light

Relativistic Length Contraction
L = L₀ √(1 – v²/c²)
L Length Measured in a Frame Where the Object is Moving
L₀ Proper Length Length in the Object’s Rest Frame
v Relative Speed of the Object
c Speed of Light

Relativistic Momentum
p = γ m v
p Relativistic Momentum
γ Lorentz Factor γ 1 √ 1 v² c²
m Rest Mass
v Velocity

Relativistic Energy
E = γ m c²
E Total Relativistic Energy Rest Energy Kinetic Energy
γ Lorentz Factor
m Rest Mass
c Speed of Light

Rydberg Formula
1/λ = R (1/n₁² – 1/n₂²)
λ Wavelength of the Emitted Absorbed Photon
R Rydberg Constant
n₁ n₂ Principal Quantum Numbers n₂ gt n₁

Hubble’s Law
v = H₀ d
v Recessional Velocity of a Galaxy How Fast it is Moving Away from Us
H₀ Hubble Constant Current Rate of Expansion of the Universe
d Proper Distance to the Galaxy