Physics formulas — 233 solvers

Acceleration Down a Frictionless Incline

a = g \sin\theta

MechanicsPhysicsAcceleration of an object sliding freely down a frictionless incline, set only by gravity and the slope angle.

Acceleration Down an Incline with Friction

a = g\left(\sin\theta - \mu_k \cos\theta\right)

MechanicsPhysicsAcceleration of an object sliding down an incline once kinetic friction opposes the motion, from the slope angle and μₖ.

Angle of Repose (μ = tan θ)

\mu_s = \tan\theta

MechanicsPhysicsSteepest angle a surface can be tilted before an object slides, where the coefficient of static friction equals tan θ.

Angle of Twist (φ = TL/JG)

\varphi = \frac{T L}{J G}

Strength of MaterialsMechanicsPhysicsAngle of twist of a round shaft under torque, φ = TL/JG, the stiffness check that governs long drive and torque shafts.

Angular Acceleration

\alpha = \frac{\omega - \omega_0}{t}

MechanicsPhysicsAverage angular acceleration: the change in angular velocity divided by the time taken.

Angular Diameter (Small-Angle)

\theta = \frac{2R}{d}

Astronomy & GravitationPhysicsHow large an object of radius R looks from distance d, in radians — the small-angle rule behind eclipses and telescope limits.

Angular Displacement (θ = ω₀t + ½αt²)

\theta = \omega_0 t + \tfrac{1}{2} \alpha t^{2}

MechanicsPhysicsAngle turned under constant angular acceleration, the rotational twin of x = v₀t + ½at².

Angular Momentum (L = Iω)

L = I \omega

MechanicsPhysicsAngular momentum of a rotating body: moment of inertia times angular velocity.

Angular Velocity (ω = θ/t)

\omega = \frac{\theta}{t}

MechanicsPhysicsAverage angular velocity: the angle swept divided by the time taken.

Angular Velocity from Period

\omega = \frac{2\pi}{T}

MechanicsWaves & OscillationsPhysicsOne full revolution is 2π radians, so angular velocity is 2π divided by the period.

Apparent Brightness (Inverse-Square Law)

b = \frac{L}{4\pi d^{2}}

Astronomy & GravitationPhysicsReceived flux from a source of luminosity L spread over a sphere of radius d — brightness falls with the square of distance.

Apparent Depth

d' = \frac{d}{n}

OpticsPhysicsViewed from straight above, an object under water appears at depth d/n.

Atwood Machine Acceleration

a = \frac{\left(m_1 - m_2\right) g}{m_1 + m_2}

MechanicsPhysicsAcceleration of an Atwood machine — two masses joined by a rope over a frictionless pulley, driven by their difference.

Average Shear Stress (τ = V/A)

\tau = \frac{V}{A}

Strength of MaterialsMechanicsPhysicsAverage shear stress on a bolt, pin or weld throat: the transverse force divided by the area resisting it, in Pa or psi.

Axial Deformation (δ = PL/AE)

\delta = \frac{P L}{A E}

Strength of MaterialsMechanicsPhysicsElongation of an axially loaded bar from load, length, area and Young's modulus — the workhorse δ = PL/AE of hanger design.

Banked Curve Angle

\theta = \arctan\!\left(\frac{v^{2}}{r g}\right)

MechanicsPhysicsBank angle that lets a vehicle round a curve of a given radius at a given speed with no reliance on sideways friction.

Beam Deflection — Simply Supported, Centre Load

\delta = \frac{P L^{3}}{48 E I}

Strength of MaterialsMechanicsPhysicsMaximum midspan deflection of a simply supported beam with a central point load, δ = PL³/48EI, with I entered in m⁴.

Beam Deflection — Simply Supported, Uniform Load

\delta = \frac{5 w L^{4}}{384 E I}

Strength of MaterialsMechanicsPhysicsMaximum midspan deflection of a simply supported beam under a uniform load, δ = 5wL⁴/384EI, with I entered in m⁴.

Beat Frequency

f_{\text{beat}} = f_1 - f_2

Waves & OscillationsPhysicsTwo nearby tones interfere to produce a loudness pulse at their difference frequency, with f₁ the higher of the pair.

Bending Stress (σ = Mc/I)

\sigma = \frac{M c}{I}

Strength of MaterialsMechanicsPhysicsBending stress at a distance c from the neutral axis of a beam, with the area moment of inertia I entered in m⁴.

Bending Stress from Section Modulus (σ = M/S)

\sigma = \frac{M}{S}

Strength of MaterialsMechanicsPhysicsBending stress straight from the moment and a tabulated section modulus S in m³, the everyday form used with steel tables.

Bernoulli's Equation (Two Points)

P_1 + \tfrac{1}{2}\rho v_1^{2} + \rho g z_1 = P_2 + \tfrac{1}{2}\rho v_2^{2} + \rho g z_2

Fluid MechanicsPhysicsConservation of mechanical energy along a streamline: pressure, velocity and elevation trade against one another for an incompressible fluid with no friction and no pump between the two points.

Bohr Model Energy Levels

E_n = -\frac{13.606\ \mathrm{eV}}{n^{2}}

Modern PhysicsChemistryPhysicsEnergy of the hydrogen atom's nth level: −13.606 eV divided by n².

Bolt Preload from Torque (T = KDF)

T = K D F

Strength of MaterialsMechanicsPhysicsBolt preload from tightening torque using the nut factor K, T = KDF, the field method behind every published torque spec.

Bounce Height from Coefficient of Restitution

h_2 = e^{2} h_1

MechanicsPhysicsHeight a dropped ball rebounds to, from the drop height and the coefficient of restitution of the bounce.

Boyle's Law

P_1 V_1 = P_2 V_2

ThermodynamicsChemistryPhysicsAt constant temperature, pressure times volume stays constant for a fixed amount of gas.

Brayton Cycle Efficiency (Pressure Ratio)

\eta = 1 - \frac{1}{r_p^{\left(\gamma - 1\right)/\gamma}}

ThermodynamicsPhysicsAir-standard efficiency of a gas turbine from compressor pressure ratio alone. Turbine inlet temperature does not appear, which is why the equation says nothing about the specific work — the thing that actually limits real machines.

Brewster's Angle

\tan\theta_B = \frac{n_2}{n_1}

OpticsPhysicsThe incidence angle at which reflected light is completely polarized.

Bulk Modulus (K = ΔP·V₀/ΔV)

K = \frac{\Delta P \, V_0}{\Delta V}

Strength of MaterialsMechanicsPhysicsBulk modulus from the pressure rise and the volume change it produces; water sits near 2.2 GPa and hydraulic oil near 1.5 GPa.

Buoyant Force (Archimedes' Principle)

F_b = \rho V g

Fluid MechanicsPhysicsThe upward force on a submerged body equals the weight of the fluid it displaces, with g = 9.80665 m/s².

Cantilever Deflection — End Load

\delta = \frac{P L^{3}}{3 E I}

Strength of MaterialsMechanicsPhysicsTip deflection of a cantilever carrying a point load at its free end, δ = PL³/3EI, with I entered in m⁴ as a plain number.

Capacitance (C = Q/V)

C = \frac{Q}{V}

Electricity & MagnetismPhysicsDefines capacitance as the charge stored per volt applied across a capacitor.

Capacitive Reactance (X_C = 1/2πfC)

X_C = \frac{1}{2\pi f C}

Electricity & MagnetismPhysicsA capacitor's opposition to AC current, falling as frequency rises.

Carnot Efficiency

\eta = 1 - \frac{T_c}{T_h}

ThermodynamicsPhysicsThe maximum possible efficiency of any heat engine operating between two absolute temperatures.

Centripetal Acceleration (a = v²/r)

a_c = \frac{v^2}{r}

MechanicsPhysicsInward acceleration of an object moving in a circle at constant speed.

Centripetal Acceleration (a = ω²r)

a_c = \omega^{2} r

MechanicsPhysicsCentripetal acceleration written in terms of angular velocity rather than linear speed.

Centripetal Force (F = mv²/r)

F_c = \frac{m v^2}{r}

MechanicsPhysicsNet inward force required to keep a mass moving in a circle at constant speed.

Charles's Law

\frac{V_1}{T_1} = \frac{V_2}{T_2}

ThermodynamicsChemistryPhysicsAt constant pressure, gas volume is directly proportional to absolute temperature.

Coefficient of Restitution

e = \frac{v_2 - v_1}{u_1 - u_2}

MechanicsPhysicsRatio of separation speed to approach speed in a collision, measuring how much of the relative motion survives impact.

Colebrook–White Friction Factor

\frac{1}{\sqrt{f}} = -2 \log_{10}\!\left(\frac{\varepsilon}{3.7 D} + \frac{2.51}{Re \sqrt{f}}\right)

Fluid MechanicsHVAC & HydronicsPhysicsThe reference equation for turbulent friction in a rough pipe, and the curve every Moody diagram is drawn from. It has f on both sides, so this page solves it by iteration rather than by any closed formula.

Combined Gas Law

\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

ThermodynamicsChemistryPhysicsFor a fixed amount of gas, pressure times volume over absolute temperature stays constant between any two states.

Conservation of Momentum (Two Bodies)

m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2

MechanicsPhysicsConservation of linear momentum in a two-body collision, solving any one mass or velocity from the other five.

Continuity Equation (A₁v₁ = A₂v₂)

A_1 v_1 = A_2 v_2

Fluid MechanicsWater TreatmentPhysicsFor incompressible flow, the same volume per second passes every cross-section of the pipe.

Coulomb's Law

F = \frac{k_e \, q_{1} q_{2}}{r^{2}}

Electricity & MagnetismPhysicsElectrostatic force between two point charges, with kₑ = 8.9875517923×10⁹ N·m²/C².

Critical Angle for Total Internal Reflection

\sin\theta_c = \frac{n_2}{n_1}

OpticsPhysicsBeyond this angle of incidence, light in the denser medium reflects totally instead of refracting.

Cross Product Magnitude

|\vec{a}\times\vec{b}| = |\vec{a}|\,|\vec{b}|\sin\theta

Vectors & MatricesTrigonometryPhysicsGives the length of the cross product of two vectors from their magnitudes and the angle between them, equal to the area they span.

Darcy–Weisbach Head Loss

h_f = f \, \frac{L}{D} \, \frac{v^{2}}{2g}

HVAC & HydronicsFluid MechanicsPhysicsThe rigorous pipe friction equation: head loss from friction factor, length-to-diameter ratio and velocity head, with g = 9.80665 m/s².

De Broglie Wavelength

\lambda = \frac{h}{m v}

Modern PhysicsPhysicsEvery moving particle has a wavelength: Planck's constant over its momentum mv.

Decibel Sound Level

\beta = 10 \log_{10}\!\left(\frac{I}{I_0}\right)

Waves & OscillationsPhysicsSound level in decibels compares an intensity to the threshold of hearing, I₀ = 10⁻¹² W/m².

Density

\rho = \tfrac{m}{V}

PhysicsChemistryWater TreatmentMass per unit volume.

Diffraction Grating Equation

m \lambda = d \sin\theta

OpticsWaves & OscillationsPhysicsBright beams leave a grating at angles where the path difference d sin θ is a whole number of wavelengths.

Displacement (Uniform Acceleration)

d = v_0 t + \tfrac{1}{2} a t^2

MechanicsPhysicsDistance travelled under constant acceleration, starting from an initial velocity, over a time t.

Displacement from Average Velocity

d = \frac{v_0 + v}{2} \, t

MechanicsPhysicsDisplacement as the average of initial and final velocities multiplied by the elapsed time, valid for uniform acceleration.

Displacement from Final Velocity (d = vt − ½at²)

d = v t - \tfrac{1}{2} a t^2

MechanicsPhysicsThe fifth kinematic equation: displacement from the FINAL velocity and the time, for when the starting speed is the unknown.

Doppler Effect (Approaching Observer)

f' = \frac{f (v + v_o)}{v}

Waves & OscillationsPhysicsA listener moving toward a stationary source meets wavefronts more often and hears a higher frequency.

Doppler Effect (Approaching Source)

f' = \frac{f v}{v - v_s}

Waves & OscillationsPhysicsObserved frequency rises when a sound source approaches, as each wavefront is emitted closer to the listener.

Dot Product from Magnitudes and Included Angle

\vec{a}\cdot\vec{b} = |\vec{a}|\,|\vec{b}|\cos\theta

Vectors & MatricesTrigonometryPhysicsGives the scalar product of two vectors from their lengths and the angle between them, the geometric face of the dot product.

Double-Slit Fringe Spacing

\Delta y = \frac{\lambda L}{d}

OpticsWaves & OscillationsPhysicsSpacing between adjacent bright fringes on a screen a distance L behind two slits d apart.

Drag Force (F = ½CdρAv²)

F_D = \tfrac{1}{2} C_d \rho A v^{2}

MechanicsPhysicsAerodynamic drag on a body moving through a fluid, growing with the square of speed and with frontal area and density.

Drop Height of a Horizontally Launched Projectile

y = \tfrac{1}{2} g t^{2}

MechanicsPhysicsDistance a horizontally launched projectile falls in a given time, independent of how fast it was thrown sideways.

Dynamic Pressure (q = ½ρv²)

q = \tfrac{1}{2} \rho v^{2}

Fluid MechanicsPhysicsThe kinetic energy per unit volume of a moving fluid — the pressure of motion itself.

Elastic Collision — Final Velocity of Body 1

v_1 = \frac{\left(m_1 - m_2\right) u_1 + 2 m_2 u_2}{m_1 + m_2}

MechanicsPhysicsFinal velocity of the first body in a one-dimensional elastic collision, where both momentum and kinetic energy survive.

Elastic Potential Energy

U = \tfrac{1}{2} k x^{2}

MechanicsPhysicsEnergy stored in an ideal spring displaced x from its rest length.

Electric Charge (Q = It)

Q = I t

Electricity & MagnetismPhysicsTotal charge transferred by a steady current flowing for a given time.

Electrical Energy (E = Pt)

E = P t

Electricity & MagnetismPhysicsEnergy consumed by a device drawing constant power over a period of time.

Electrical Power (P = I²R)

P = I^{2} R

Electricity & MagnetismPhysicsPower dissipated as heat in a resistance carrying a current (Joule heating).

Electrical Power (P = V²/R)

P = \frac{V^{2}}{R}

Electricity & MagnetismPhysicsPower dissipated in a resistance held at a fixed voltage.

Electrical Power (P = VI)

P = V I

Electricity & MagnetismPhysicsPower delivered to a component as the product of the voltage across it and the current through it.

Energy Stored in a Capacitor

E = \tfrac{1}{2} C V^{2}

Electricity & MagnetismPhysicsEnergy banked in a capacitor's electric field from its capacitance and voltage.

Energy Stored in an Inductor

E = \tfrac{1}{2} L I^{2}

Electricity & MagnetismPhysicsEnergy held in an inductor's magnetic field: half the inductance times current squared.

Escape Velocity

v = \sqrt{\frac{2GM}{r}}

Astronomy & GravitationMechanicsPhysicsMinimum launch speed needed to escape the gravity of a mass M starting from distance r, with no further propulsion.

Euler Critical Buckling Load

P_{cr} = \frac{\pi^{2} E I}{(K L)^{2}}

Strength of MaterialsMechanicsPhysicsEuler's critical buckling load for a slender column, using the end-condition factor K and the area moment of inertia in m⁴.

Factor of Safety

FS = \frac{\sigma_{u}}{\sigma_{allow}}

Strength of MaterialsMechanicsPhysicsFactor of safety as ultimate or yield strength divided by the allowable working stress, the engineer's declared margin of ignorance.

Fan Affinity Law — Airflow vs Speed

\frac{Q_2}{Q_1} = \frac{N_2}{N_1}

HVAC & HydronicsFluid MechanicsPhysicsFan airflow in CFM changes in direct proportion to wheel speed, the first law used when re-sheaving a belt-driven air handler.

Fan Affinity Law — Power vs Speed

\frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^{3}

HVAC & HydronicsFluid MechanicsPhysicsFan brake power varies with the cube of wheel speed — the law behind variable-air-volume energy savings and behind burnt-out re-sheaved motors.

Fan Affinity Law — Static Pressure vs Speed

\frac{SP_2}{SP_1} = \left(\frac{N_2}{N_1}\right)^{2}

HVAC & HydronicsFluid MechanicsPhysicsFan static pressure rises with the square of wheel speed, the reason a modest re-sheave can overpressurise ductwork and blow out flex connections.

Fan Brake Horsepower

BHP = \frac{Q \cdot SP}{6356 \, \eta}

HVAC & HydronicsFluid MechanicsPhysicsShaft power a fan absorbs; the 6356 divisor assumes cubic feet per minute, inches of water gauge and horsepower at the given efficiency.

Faraday's Law of Induction

\varepsilon = N \frac{\Delta\Phi}{\Delta t}

Electricity & MagnetismPhysicsThe EMF induced in a coil of N turns by flux changing at ΔΦ/Δt (magnitude form).

Fick's First Law of Diffusion

J = D\,\frac{C_1 - C_2}{L}

ChemistryPhysicsSteady-state diffusive flux down a concentration gradient. Written here for a slab of thickness L with a fixed concentration on each face, which is the form used for membranes, coatings and stagnant films. The flux J is in mol per square metre per second.

Final Velocity (Uniform Acceleration)

v = v_0 + a t

MechanicsPhysicsFinal velocity after accelerating uniformly from an initial velocity for a given time.

Focal Length of a Spherical Mirror

f = \frac{R}{2}

OpticsPhysicsA spherical mirror focuses parallel rays at half its radius of curvature.

Force Between Parallel Wires

F = \frac{\mu_0 I_1 I_2 \ell}{2\pi d}

Electricity & MagnetismPhysicsMagnetic force between two parallel current-carrying wires separated by distance d.

Fundamental Frequency of a String

f = \frac{v}{2L}

Waves & OscillationsPhysicsLowest standing-wave frequency of a string fixed at both ends.

Fundamental of a Closed Pipe

f = \frac{v}{4L}

Waves & OscillationsPhysicsA pipe closed at one end resonates with a quarter wavelength inside, giving a fundamental of v/4L.

Gas Density from Molar Mass

\rho = \frac{PM}{RT}

ChemistryThermodynamicsPhysicsGives an ideal gas's density from its molar mass, pressure, and absolute temperature using R = 8.314462618 J/(mol·K).

Gauge and Absolute Pressure

P_{abs} = P_{gauge} + P_{atm}

Fluid MechanicsThermodynamicsPhysicsAbsolute pressure is the gauge reading plus the surrounding atmospheric pressure.

Gay-Lussac's Law

\frac{P_1}{T_1} = \frac{P_2}{T_2}

ThermodynamicsChemistryPhysicsAt constant volume, gas pressure is directly proportional to absolute temperature.

Gear Ratio

GR = \frac{N_{out}}{N_{in}}

MechanicsPhysicsGear ratio of a meshing pair as the driven gear's tooth count divided by the driver's, setting the torque and speed trade.

Graham's Law of Effusion

\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}

ChemistryPhysicsRelates the effusion rates of two gases to the inverse square root of their molar masses.

Gravitational Field Strength

g = \frac{GM}{r^{2}}

Astronomy & GravitationMechanicsPhysicsGravitational acceleration produced by a mass M at distance r from its center.

Gravitational Potential Energy (Orbital)

U = -\frac{GMm}{r}

Astronomy & GravitationMechanicsPhysicsGravitational potential energy of a mass m at distance r from a central mass M, taking zero at infinite separation.

Gravitational Potential Energy (U = mgh)

U = m g h

MechanicsPhysicsEnergy stored by raising a mass to height h near Earth's surface, with g = 9.80665 m/s².

Half-Life and Decay Constant

t_{1/2} = \frac{\ln 2}{\lambda}

ChemistryPhysicsConverts between a half-life and the exponential decay constant via the factor ln 2 = 0.6931471806.

Half-Life Decay

N = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}

ChemistryPhysicsGives the quantity remaining after repeated halvings over an elapsed time measured in half-lives.

Harmonic Frequencies

f_n = n f_1

Waves & OscillationsPhysicsThe nth harmonic of a vibrating system is n times its fundamental frequency.

Heat Conduction Rate

P = \tfrac{k A \Delta T}{d}

ThermodynamicsPhysicsSteady-state heat flow through a slab by Fourier's law of conduction.

Hooke's Law

F = k x

MechanicsPhysicsRestoring force of an ideal spring, proportional to its displacement from rest.

Hoop Stress in a Thin-Walled Cylinder

\sigma_{h} = \frac{p d}{2 t}

Strength of MaterialsMechanicsPhysicsHoop (circumferential) stress in a thin-walled pipe or pressure vessel, σ = pd/2t — exactly twice the longitudinal stress.

Horizontal Velocity Component

v_x = v \cos\theta

MechanicsPhysicsHorizontal component of a projectile's launch velocity — the part of the speed that carries it downrange at a constant rate.

Hubble's Law

v = H_0 d

Astronomy & GravitationPhysicsRecession velocity of a galaxy: its distance times the Hubble constant — the expansion of the universe read as a straight line.

Hydraulic Power (P = ρgQh)

P = \rho g Q h

Fluid MechanicsWater TreatmentPhysicsPower needed to lift a flow Q through a head h, with g = 9.80665 m/s².

Hydrostatic Pressure (P = ρgh)

P = \rho g h

MechanicsPhysicsWater TreatmentGauge pressure at depth h in a fluid of density ρ, using g = 9.80665 m/s².

Ideal Gas Law

P V = n R T

ChemistryPhysicsRelates pressure, volume, amount, and temperature of a gas. R = 8.314 J/(mol·K).

Impulse (J = FΔt)

J = F \, \Delta t

MechanicsPhysicsImpulse delivered by an average force acting over a contact time, equal to the change in momentum.

Index of Refraction (n = c/v)

n = \frac{c}{v}

OpticsPhysicsHow much a medium slows light: the ratio of c to the speed of light in the medium.

Inductive Reactance (X_L = 2πfL)

X_L = 2\pi f L

Electricity & MagnetismPhysicsAn inductor's opposition to AC current, rising in proportion to frequency.

Inverse-Square Law for Sound

I = \frac{P}{4\pi r^{2}}

Waves & OscillationsPhysicsA point source's intensity falls with the square of distance as its power spreads over an expanding sphere.

Kepler's Third Law (Ratio Form)

\frac{T_1^{2}}{T_2^{2}} = \frac{a_1^{3}}{a_2^{3}}

Astronomy & GravitationMechanicsPhysicsFor two bodies orbiting the same central mass, the squares of their periods are in the same ratio as the cubes of their orbital sizes.

Kinetic Energy

E_k = \tfrac{1}{2} m v^{2}

PhysicsEnergy of a mass in motion.

Kinetic Friction Force (f = μₖN)

f_k = \mu_k N

MechanicsPhysicsFriction force resisting a sliding surface, equal to the coefficient of kinetic friction times the normal force.

Laminar Friction Factor (f = 64/Re)

f = \frac{64}{Re}

HVAC & HydronicsFluid MechanicsPhysicsIn laminar pipe flow the Darcy friction factor depends only on Reynolds number — roughness plays no part below about Re = 2300.

Latent Heat

Q = m L

ThermodynamicsPhysicsHeat absorbed or released when a mass changes phase at constant temperature.

LC Resonant Frequency

f = \frac{1}{2\pi\sqrt{LC}}

Electricity & MagnetismWaves & OscillationsPhysicsThe natural oscillation frequency of an inductor-capacitor pair.

Length Contraction

L = L_0 \sqrt{1 - \beta^{2}}

Modern PhysicsPhysicsA moving object's rest length L₀ contracts to L along its direction of motion; β = v/c.

Lens Magnification (m = −d_i/d_o)

m = -\frac{d_i}{d_o}

OpticsPhysicsImage magnification from the ratio of image to object distance; the minus sign tracks inversion.

Lens Power in Diopters

P = \frac{1}{f}

OpticsPhysicsThe optician's unit: lens power in diopters is the reciprocal of focal length in meters.

Linear Momentum (p = mv)

p = m v

MechanicsPhysicsMomentum as the product of an object's mass and velocity.

Linear Speed from Rotation (v = ωr)

v = \omega r

MechanicsPhysicsA point at radius r on a rotating body moves with linear speed ωr.

Longitudinal Stress in a Thin-Walled Cylinder

\sigma_{l} = \frac{p d}{4 t}

Strength of MaterialsMechanicsPhysicsLongitudinal (axial) stress in a thin-walled cylinder under internal pressure, σ = pd/4t — exactly half the hoop stress.

Lorentz Factor

\gamma = \frac{1}{\sqrt{1 - \beta^{2}}}

Modern PhysicsPhysicsThe relativistic stretch factor, written in terms of β = v/c.

Mach Number

Ma = \frac{v}{\sqrt{\gamma R T / M}}

Fluid MechanicsPhysicsSpeed divided by the local speed of sound in the same gas. Below about Ma = 0.3 a gas can be treated as incompressible, which is the assumption behind most duct and pipe sizing.

Machine Efficiency

\eta = \frac{W_{out}}{W_{in}}

MechanicsPhysicsEfficiency of a machine as useful work out divided by work in, with the shortfall lost to friction, heat, and noise.

Magnetic Field of a Solenoid

B = \frac{\mu_0 N I}{L}

Electricity & MagnetismPhysicsThe uniform field inside a long coil of N turns and length L carrying current I.

Magnetic Flux (Φ = BA cos θ)

\Phi = B A \cos\theta

Electricity & MagnetismPhysicsThe magnetic field threading a surface: field times area times the cosine of the tilt angle.

Magnetic Force on a Current-Carrying Wire

F = B I L \sin\theta

Electricity & MagnetismPhysicsMagnitude of the force on a straight current-carrying wire in a uniform magnetic field.

Magnetic Force on a Moving Charge

F = q v B \sin\theta

Electricity & MagnetismPhysicsMagnitude of the magnetic (Lorentz) force on a charge moving through a magnetic field.

Magnification from Heights (m = h_i/h_o)

m = \frac{h_i}{h_o}

OpticsPhysicsMagnification as the ratio of image height to object height.

Mass-Energy Equivalence (E = mc²)

E = m c^{2}

Modern PhysicsPhysicsThe rest energy locked in mass: multiply by the speed of light squared.

Max Bending Moment — Centre Point Load

M = \frac{P L}{4}

Strength of MaterialsMechanicsPhysicsMaximum bending moment in a simply supported beam carrying one point load at midspan, M = PL/4, occurring under the load.

Max Bending Moment — Uniform Load

M = \frac{w L^{2}}{8}

Strength of MaterialsMechanicsPhysicsMaximum bending moment at midspan of a simply supported beam under a uniformly distributed load, the classic M = wL²/8.

Maximum Speed on a Flat Curve

v_{\max} = \sqrt{\mu_s g r}

MechanicsPhysicsFastest a vehicle can round a flat, unbanked curve before friction can no longer supply the centripetal force.

Maximum Static Friction (f = μₛN)

f_{s,\max} = \mu_s N

MechanicsPhysicsLargest static friction force available before an object breaks loose and slides, from the static coefficient and normal force.

Mechanical Advantage of a Lever

MA = \frac{d_e}{d_l}

MechanicsPhysicsMechanical advantage of a lever as the ratio of effort arm to load arm, showing how much the lever multiplies force.

Moment of Inertia: Point Mass

I = m r^{2}

MechanicsPhysicsRotational inertia of a compact mass circling at radius r from the axis.

Moment of Inertia: Solid Disk

I = \tfrac{1}{2} m r^{2}

MechanicsPhysicsRotational inertia of a uniform solid disk or cylinder about its central axis.

Moment of Inertia: Solid Sphere

I = \tfrac{2}{5} m r^{2}

MechanicsPhysicsRotational inertia of a uniform solid sphere about an axis through its center.

Motional EMF (ε = BLv)

\varepsilon = B L v

Electricity & MagnetismPhysicsVoltage generated across a conductor of length L moving at speed v through a field B.

Newton's Law of Universal Gravitation

F = \frac{G m_1 m_2}{r^2}

MechanicsPhysicsAttractive force between two masses, with G = 6.6743 × 10⁻¹¹ m³ kg⁻¹ s⁻² (CODATA 2018).

Newton's Second Law

F = m a

PhysicsForce equals mass times acceleration.

Newton's Second Law for Rotation (τ = Iα)

\tau = I \alpha

MechanicsPhysicsNet torque equals moment of inertia times angular acceleration — F = ma for spinning things.

Normal (Axial) Stress

\sigma = \frac{P}{A}

Strength of MaterialsMechanicsPhysicsAxial stress in a bar or hanger rod — the internal force divided by the cross-sectional area that carries it, in Pa or psi.

Normal Force on an Incline (N = mg cos θ)

N = m g \cos\theta

MechanicsPhysicsNormal force pressing a resting mass into an incline, equal to the component of its weight perpendicular to the slope.

Normal Strain (ε = δ/L)

\varepsilon = \frac{\delta}{L}

Strength of MaterialsMechanicsPhysicsNormal strain as the change in length divided by the original length, a dimensionless ratio usually quoted in microstrain.

Nuclear Radius (r = r₀A^⅓)

r = r_0 A^{1/3}

Modern PhysicsPhysicsRadius of a nucleus of mass number A: r₀ times the cube root of A, because nuclear matter packs at one fixed density.

Ohm's Law

V = I R

Electricity & MagnetismPhysicsRelates the voltage across a conductor to the current through it and its resistance.

Orbital Period

T = 2\pi \sqrt{\frac{r^{3}}{GM}}

Astronomy & GravitationMechanicsPhysicsTime for one circular orbit of radius r around a central mass M — Kepler's third law in Newtonian form.

Orbital Velocity

v = \sqrt{\frac{GM}{r}}

Astronomy & GravitationMechanicsPhysicsSpeed of a body in a circular orbit of radius r around a central mass M.

Orifice Plate Flow

Q = \frac{C_d}{\sqrt{1 - \beta^{4}}}\cdot\frac{\pi d^{2}}{4}\sqrt{\frac{2\,\Delta P}{\rho}}

Fluid MechanicsPhysicsVolumetric flow through a square-edged orifice plate from the differential pressure across it. The 1/√(1−β⁴) term is the velocity-of-approach factor, with β the bore-to-pipe diameter ratio.

Otto Cycle Efficiency (Compression Ratio)

\eta = 1 - \frac{1}{r^{\gamma - 1}}

ThermodynamicsPhysicsAir-standard efficiency of a spark-ignition engine from its compression ratio alone. Nothing about fuel, speed or displacement enters — only how hard the charge is squeezed and the heat capacity ratio of the working gas.

Perfectly Inelastic Collision

v = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}

MechanicsPhysicsCommon velocity of two bodies that stick together after a perfectly inelastic collision, from conservation of momentum.

Period of a Spring-Mass Oscillator

T = 2\pi \sqrt{\tfrac{m}{k}}

Waves & OscillationsMechanicsPhysicsPeriod of a mass bouncing on a spring, set only by the mass and the spring stiffness.

Period-Frequency Relation

T = \frac{1}{f}

Waves & OscillationsPhysicsPeriod and frequency are reciprocals: seconds per cycle versus cycles per second.

Photoelectric Effect

KE_{max} = E_{photon} - \phi

Modern PhysicsPhysicsMaximum kinetic energy of an ejected electron: photon energy (hf) minus the work function.

Photon Energy (E = hf)

E = h f

Modern PhysicsPhysicsThe energy of a single photon: Planck's constant times the light's frequency.

Photon Energy from Wavelength (E = hc/λ)

E = \frac{h c}{\lambda}

Modern PhysicsPhysicsPhoton energy written in terms of wavelength: shorter waves, more energetic photons.

Photon Momentum (p = h/λ)

p = \frac{h}{\lambda}

Modern PhysicsPhysicsMassless but not momentum-less: a photon carries h divided by its wavelength.

Poiseuille's Law

Q = \frac{\pi \, \Delta P \, r^{4}}{8 \mu L}

Fluid MechanicsPhysicsLaminar flow rate through a round pipe — proportional to the fourth power of the radius.

Poisson's Ratio

\nu = \frac{\varepsilon_{lat}}{\varepsilon_{ax}}

Strength of MaterialsMechanicsPhysicsPoisson's ratio, the lateral contraction per unit of axial extension — close to 0.30 for steel and 0.33 for aluminium.

Power (P = W/t)

P = \frac{W}{t}

MechanicsPhysicsAverage power as work or energy delivered per unit time.

Power from Force and Velocity (P = Fv)

P = F v

MechanicsPhysicsInstantaneous power delivered by a force parallel to the velocity.

Pressure (P = F/A)

P = \frac{F}{A}

MechanicsPhysicsPressure as perpendicular force spread over an area.

Pressure Head (h = P/ρg)

h = \frac{P}{\rho g}

Fluid MechanicsWater TreatmentPhysicsConverts a pressure into the equivalent height of a fluid column, with g = 9.80665 m/s².

Projectile Maximum Height

H = \frac{v_0^{2} \sin^{2}\theta}{2g}

MechanicsPhysicsPeak height reached by a projectile launched at a given speed and angle above level ground, ignoring air resistance.

Projectile Range on Level Ground

R = \frac{v_0^{2} \sin 2\theta}{g}

MechanicsPhysicsHorizontal distance a projectile covers over level ground, from its launch speed and angle, ignoring air resistance.

Projectile Time of Flight

T = \frac{2 v_0 \sin\theta}{g}

MechanicsPhysicsTotal time a projectile stays airborne before returning to its launch height, set by launch speed and angle with g = 9.80665 m/s².

Pulley System Effort Force

F = \frac{W}{n}

MechanicsPhysicsEffort force needed to lift a load with a pulley system, divided down by the number of rope sections supporting the load.

Radioactive Activity (A = λN)

A = \lambda N

Modern PhysicsChemistryPhysicsDecays per second: the decay constant times the number of undecayed nuclei.

Rankine Cycle Thermal Efficiency

\eta = \frac{\left(h_1 - h_2\right) - \left(h_4 - h_3\right)}{h_1 - h_4}

ThermodynamicsPhysicsNet work over heat added for a steam power cycle, taken straight off the four state-point enthalpies: turbine inlet h₁, turbine exhaust h₂, condensate h₃ and feedwater leaving the pump h₄. Enter all four in the same unit; the answer is a ratio either way.

RC Capacitor Discharge

V = V_{0} \, e^{-t/\tau}

Electricity & MagnetismPhysicsExponential decay of the voltage on a capacitor discharging through a resistor.

RC Time Constant

\tau = R C

Electricity & MagnetismPhysicsCharacteristic charging/discharging time of a resistor-capacitor circuit.

Relation Between E, G and ν

E = 2G(1 + \nu)

Strength of MaterialsMechanicsPhysicsThe isotropic elastic identity E = 2G(1 + ν), linking Young's modulus, the shear modulus and Poisson's ratio in one step.

Resistance of a Wire (R = ρL/A)

R = \frac{\rho L}{A}

Electricity & MagnetismPhysicsResistance from material resistivity, length, and cross-sectional area.

Resultant of Two Vectors at an Angle

R = \sqrt{A^2 + B^2 + 2AB\cos\theta}

Vectors & MatricesTrigonometryPhysicsFinds the magnitude of the sum of two vectors from their lengths and the angle between them, the parallelogram rule in one equation.

Reynolds Number

Re = \frac{\rho v D}{\mu}

Fluid MechanicsPhysicsThe dimensionless ratio of inertial to viscous forces that decides laminar versus turbulent flow.

RL Time Constant (τ = L/R)

\tau = \frac{L}{R}

Electricity & MagnetismPhysicsHow quickly current builds or decays in an inductor-resistor circuit.

RMS and Peak Voltage

V_{rms} = \frac{V_{peak}}{\sqrt{2}}

Electricity & MagnetismPhysicsFor a sine wave, the effective (heating-equivalent) voltage is the peak divided by √2.

Rope Tension When Lifting a Mass

T = m\left(g + a\right)

MechanicsPhysicsTension in a rope lifting a mass with an upward acceleration, exceeding the static weight by the factor (g + a).

Rotational Kinetic Energy

KE_{rot} = \tfrac{1}{2} I \omega^{2}

MechanicsPhysicsKinetic energy stored in rotation: half the moment of inertia times angular velocity squared.

Rotational Power (P = τω)

P = \tau \omega

MechanicsPhysicsMechanical power delivered by a torque turning at angular velocity ω.

Scalar Projection of One Vector onto Another

\text{comp}_{\vec{b}}\vec{a} = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}

Vectors & MatricesAlgebraPhysicsMeasures how far a vector reaches along the direction of another, the length of the shadow it casts on that second vector.

Schwarzschild Radius

r_s = \frac{2GM}{c^{2}}

Astronomy & GravitationPhysicsRadius of the event horizon of a non-rotating black hole of mass M.

Sensible Heat (Q = mcΔT)

Q = m c \Delta T

ThermodynamicsPhysicsChemistryHeat needed to change a mass's temperature: mass times specific heat times the temperature change.

Shear Modulus (G = τ/γ)

G = \frac{\tau}{\gamma}

Strength of MaterialsMechanicsPhysicsShear (rigidity) modulus as shear stress divided by shear strain, roughly 0.38 of Young's modulus for common metals.

SHM Displacement at Time t

x = A \cos\left(\omega t\right)

MechanicsPhysicsDisplacement of a simple harmonic oscillator at time t, a cosine of amplitude A and angular frequency ω released from full stretch.

SHM Maximum Acceleration

a_{\max} = A \omega^{2}

MechanicsPhysicsMaximum acceleration of a simple harmonic oscillator, reached at the turning points where the restoring force is largest.

SHM Maximum Velocity

v_{\max} = A \omega

MechanicsPhysicsMaximum speed of a simple harmonic oscillator, reached at the equilibrium point, equal to amplitude times angular frequency.

Simple Pendulum Period

T = 2\pi \sqrt{\frac{L}{g}}

Waves & OscillationsMechanicsPhysicsPeriod of a simple pendulum swinging through small angles, with g = 9.80665 m/s² (standard gravity).

Slenderness Ratio (KL/r)

\lambda = \frac{K L}{r}

Strength of MaterialsMechanicsPhysicsSlenderness ratio KL/r of a compression member, the single number that decides whether a column crushes or buckles.

Snell's Law of Refraction

n_1 \sin\theta_1 = n_2 \sin\theta_2

OpticsPhysicsLight bends at an interface so that n sin θ stays the same on both sides.

Sound Intensity (I = P/A)

I = \frac{P}{A}

Waves & OscillationsPhysicsSound intensity is the acoustic power passing through each square meter of surface.

Speed in Circular Motion (v = 2πr/T)

v = \frac{2\pi r}{T}

MechanicsPhysicsSpeed of an object in uniform circular motion: one circumference (2πr, with π ≈ 3.14159265) per period.

Speed of Sound in Air

v = 331.3 + 0.606\, T_C

Waves & OscillationsPhysicsThe speed of sound in dry air grows about 0.6 m/s for every degree Celsius above freezing.

Speed, Distance & Time

v = \tfrac{d}{t}

PhysicsAverage speed from distance travelled and elapsed time.

Stefan-Boltzmann Law

P = \varepsilon \sigma A T^4

ThermodynamicsPhysicsPower radiated by a hot surface, using the Stefan–Boltzmann constant σ = 5.670374419×10⁻⁸ W/(m²·K⁴).

Stellar Luminosity

L = 4\pi R^{2} \sigma T^{4}

Astronomy & GravitationPhysicsTotal power a star radiates — the Stefan–Boltzmann law wrapped over a sphere of radius R glowing at effective temperature T.

Stellar Parallax Distance

d = \frac{1\,\text{AU}}{p}

Astronomy & GravitationPhysicsDistance to a star from its annual parallax angle p, using the small-angle form of d = AU/tan p.

Stokes' Drag (F = 6πμrv)

F = 6\pi \mu r v

Fluid MechanicsPhysicsViscous drag on a small sphere creeping through a fluid at low Reynolds number.

Swamee–Jain Friction Factor

f = \frac{0.25}{\left[\log_{10}\!\left(\frac{\varepsilon}{3.7D} + \frac{5.74}{Re^{0.9}}\right)\right]^{2}}

HVAC & HydronicsFluid MechanicsPhysicsAn explicit turbulent friction factor within about 1% of the implicit Colebrook–White equation, valid for Re from 5000 to 10⁸.

Terminal Velocity

v_t = \sqrt{\frac{2 m g}{\rho A C_d}}

MechanicsPhysicsSteady falling speed at which drag balances weight, from mass, air density, frontal area, and the drag coefficient.

Thermal Efficiency

\eta = \frac{W}{Q_h}

ThermodynamicsPhysicsFraction of heat input that a heat engine converts into useful work.

Thermal Linear Expansion

\Delta L = \alpha L_0 \Delta T

ThermodynamicsPhysicsLength change of a solid caused by a temperature change, via the linear expansion coefficient.

Thermal Stress in a Restrained Member

\sigma = E \alpha \Delta T

Strength of MaterialsMechanicsPhysicsStress raised in a fully restrained member that is heated or cooled — the cause of rail sun kinks and cracked pipe anchors.

Thin Lens Equation

\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

OpticsPhysicsRelates a thin lens's focal length to its object and image distances.

Thin-Film Constructive Interference (Bright Reflection)

2 n t = \left(m + \tfrac{1}{2}\right)\lambda

OpticsWaves & OscillationsPhysicsBright-reflection condition for a thin film with one half-wave inversion: twice the optical thickness is a half-odd number of wavelengths.

Thin-Film Destructive Interference (Dark Reflection)

2 n t = m \lambda

OpticsWaves & OscillationsPhysicsDark-reflection condition for a thin film with one half-wave inversion: twice the optical thickness is a whole number of wavelengths.

Time Dilation

\Delta t = \frac{\Delta t_0}{\sqrt{1 - \beta^{2}}}

Modern PhysicsPhysicsA moving clock's proper time Δt₀ stretches to Δt for a stationary observer; β = v/c.

Torque

\tau = r F \sin\theta

MechanicsPhysicsTurning effect of a force applied at distance r from a pivot, at angle θ to the lever arm.

Torque with a Lever Arm (τ = rF sin θ)

\tau = r F \sin\theta

MechanicsPhysicsTorque produced by a force applied at distance r from the pivot, at angle θ to the lever.

Torricelli's Law (v = √(2gh))

v = \sqrt{2 g h}

Fluid MechanicsWater TreatmentPhysicsSpeed of fluid jetting from an opening a depth h below the free surface, with g = 9.80665 m/s².

Torsional Shear Stress (τ = Tr/J)

\tau = \frac{T r}{J}

Strength of MaterialsMechanicsPhysicsTorsional shear stress at radius r in a round shaft, τ = Tr/J, peaking at the surface, with J entered in m⁴ as a plain number.

Transformer Voltage Ratio

\frac{V_{s}}{V_{p}} = \frac{N_{s}}{N_{p}}

Electricity & MagnetismPhysicsRelates the primary and secondary voltages of an ideal transformer to its turns ratio.

Two Capacitors in Parallel

C_{t} = C_{1} + C_{2}

Electricity & MagnetismPhysicsCombines two parallel capacitors by simply adding their capacitances.

Two Capacitors in Series

C_{t} = \frac{C_{1} C_{2}}{C_{1} + C_{2}}

Electricity & MagnetismPhysicsCombines two series capacitors into a total that is smaller than either one.

Two Resistors in Parallel

R_{t} = \frac{R_{1} R_{2}}{R_{1} + R_{2}}

Electricity & MagnetismPhysicsTotal resistance of two resistors connected side by side: product over sum, always less than either branch.

Two Resistors in Series

R_{t} = R_{1} + R_{2}

Electricity & MagnetismPhysicsTotal resistance of two resistors connected end to end is simply their sum.

Vector Subtraction (Component Form)

c_x = a_x - b_x

Vectors & MatricesAlgebraPhysicsSubtracts one vector from another one component at a time, the operation behind every relative-velocity and displacement-change problem.

Velocity Head (h = v²/2g)

h_v = \frac{v^{2}}{2g}

Fluid MechanicsWater TreatmentPhysicsThe kinetic energy of a flow expressed as an equivalent column height, with g = 9.80665 m/s².

Velocity-Displacement Relation (v² = v₀² + 2ad)

v^2 = v_0^2 + 2 a d

MechanicsPhysicsLinks initial and final speeds to acceleration and displacement without involving time.

Venturi Meter Flow

Q = \frac{C}{\sqrt{1 - \beta^{4}}}\cdot\frac{\pi D_2^{2}}{4}\sqrt{\frac{2\,\Delta P}{\rho}}

Fluid MechanicsPhysicsFlow through a classical Venturi tube from the inlet-to-throat differential pressure. The gently converging cone recovers most of the pressure it borrows, so the discharge coefficient sits near 0.98 rather than the orifice plate's 0.61.

Vertical Velocity Component

v_y = v \sin\theta

MechanicsPhysicsVertical component of a projectile's launch velocity — the part of the speed that fights gravity and sets the time aloft.

Volumetric Flow Rate (Q = Av)

Q = A v

Fluid MechanicsWater TreatmentPhysicsFlow through a duct or pipe: cross-sectional area times average flow velocity.

Water Hammer Surge (Joukowsky Equation)

\Delta P = \rho \, a \, \Delta v

HVAC & HydronicsFluid MechanicsPhysicsPeak pressure surge from a sudden change in flow velocity: fluid density times pressure-wave celerity times the velocity change.

Wave Speed (v = fλ)

v = f \lambda

Waves & OscillationsPhysicsThe universal wave equation: a wave's speed equals its frequency times its wavelength.

Wave Speed on a String

v = \sqrt{\frac{F}{\mu}}

Waves & OscillationsPhysicsWaves travel faster on a tighter, lighter string: speed is the square root of tension over linear density.

Weight (W = mg)

W = m g

MechanicsPhysicsWeight of a mass at Earth's surface, using standard gravity g = 9.80665 m/s² (exact by definition).

Weight Component Along an Incline (mg sin θ)

F_{\parallel} = m g \sin\theta

MechanicsPhysicsComponent of an object's weight acting down the slope of an incline — the force that drives it toward the bottom.

Wien's Displacement Law

\lambda_{max} = \frac{b}{T}

Modern PhysicsThermodynamicsPhysicsThe peak wavelength of thermal radiation, with b = 2.8978 × 10⁻³ m·K.

Work (W = Fd cos θ)

W = F d \cos\theta

MechanicsPhysicsWork done by a constant force acting at an angle to the displacement.

Work from Force and Displacement Components

W = F_x d_x + F_y d_y

Vectors & MatricesPhysicsMechanicsComputes the work done by a force from the components of the force and the displacement, without needing the angle between them.

Work–Energy Theorem

W = \tfrac{1}{2} m \left(v^{2} - v_0^{2}\right)

MechanicsPhysicsNet work done on an object equals its change in kinetic energy, linking force and distance to a change in speed.

x-Component from Magnitude and Angle

v_x = |\vec{v}| \cos\theta

Vectors & MatricesTrigonometryPhysicsResolves a vector into its horizontal part from the vector's length and the angle it makes with the positive x-axis.

y-Component from Magnitude and Angle

v_y = |\vec{v}| \sin\theta

Vectors & MatricesTrigonometryPhysicsResolves a vector into its vertical part from the vector's length and the angle it makes with the positive x-axis.

Young's Modulus (E = σ/ε)

E = \frac{\sigma}{\varepsilon}

Strength of MaterialsMechanicsPhysicsYoung's modulus as the ratio of normal stress to normal strain, the stiffness constant of a material in its elastic range.