Formula solver index — formula.expert

Speed, Distance & Time

v = \tfrac{d}{t}

PhysicsAverage speed from distance travelled and elapsed time.

Newton's Second Law

F = m a

PhysicsForce equals mass times acceleration.

Density

\rho = \tfrac{m}{V}

PhysicsChemistryWater TreatmentMass per unit volume.

Kinetic Energy

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

PhysicsEnergy of a mass in motion.

Final Velocity (Uniform Acceleration)

v = v_0 + a t

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

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.

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.

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.

Linear Momentum (p = mv)

p = m v

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

Impulse (J = FΔt)

J = F \, \Delta t

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

Centripetal Acceleration (a = v²/r)

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

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

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.

Work (W = Fd cos θ)

W = F d \cos\theta

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

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.

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).

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).

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.

Pressure (P = F/A)

P = \frac{F}{A}

MechanicsPhysicsPressure as perpendicular force spread over an area.

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.

Hooke's Law

F = k x

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

Elastic Potential Energy

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

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

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.

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.

Linear Speed from Rotation (v = ωr)

v = \omega r

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

Centripetal Acceleration (a = ω²r)

a_c = \omega^{2} r

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

Angular Acceleration

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

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

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.

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.

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.

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.

Angular Momentum (L = Iω)

L = I \omega

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

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.

Rotational Power (P = τω)

P = \tau \omega

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

Volumetric Flow Rate (Q = Av)

Q = A v

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

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.

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.

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.

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.

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.

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.

Reynolds Number

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

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

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.

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.

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.

Gauge and Absolute Pressure

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

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

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.

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.

Charles's Law

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

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

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.

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.

Thermal Efficiency

\eta = \frac{W}{Q_h}

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

Carnot Efficiency

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

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

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).

Latent Heat

Q = m L

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

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.

Heat Conduction Rate

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

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

Wave Speed (v = fλ)

v = f \lambda

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

Period-Frequency Relation

T = \frac{1}{f}

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

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).

Fundamental Frequency of a String

f = \frac{v}{2L}

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

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.

Sound Intensity (I = P/A)

I = \frac{P}{A}

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

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.

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.

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.

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.

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.

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.

Harmonic Frequencies

f_n = n f_1

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

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.

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.

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.

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.

Magnification from Heights (m = h_i/h_o)

m = \frac{h_i}{h_o}

OpticsPhysicsMagnification as the ratio of image height to object height.

Focal Length of a Spherical Mirror

f = \frac{R}{2}

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

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.

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.

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.

Brewster's Angle

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

OpticsPhysicsThe incidence angle at which reflected light is completely polarized.

Lens Power in Diopters

P = \frac{1}{f}

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

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.

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.

Apparent Depth

d' = \frac{d}{n}

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

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.

Ohm's Law

V = I R

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

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.

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.

Two Resistors in Series

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

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

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.

Electric Charge (Q = It)

Q = I t

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

RC Time Constant

\tau = R C

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

RC Capacitor Discharge

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

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

Electrical Energy (E = Pt)

E = P t

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

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.

Capacitance (C = Q/V)

C = \frac{Q}{V}

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

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.

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.

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.

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.

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 Capacitors in Parallel

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

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

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.

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).

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.

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.

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.

Inductive Reactance (X_L = 2πfL)

X_L = 2\pi f L

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

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.

RL Time Constant (τ = L/R)

\tau = \frac{L}{R}

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

LC Resonant Frequency

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

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

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.

Resistance of a Wire (R = ρL/A)

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

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

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.

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.

De Broglie Wavelength

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

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

Mass-Energy Equivalence (E = mc²)

E = m c^{2}

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

Photoelectric Effect

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

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

Lorentz Factor

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

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

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.

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.

Radioactive Activity (A = λN)

A = \lambda N

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

Wien's Displacement Law

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

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

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.

Orbital Velocity

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

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

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.

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.

Gravitational Field Strength

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

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

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.

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.

Schwarzschild Radius

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

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

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.

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.

Ideal Gas Law

P V = n R T

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

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 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 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 Velocity

v_{\max} = A \omega

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

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.

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.

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.

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.

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.

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).

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.

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 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 — 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 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 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.

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.

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.

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.

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.