Rise Time ↔ Bandwidth Calculator

Work out a soil's available water capacity and the readily available water between irrigations from field capacity, wilting point and root-zone depth.

Estimate daily reference evapotranspiration ET₀ (mm/day) from just max/min temperature, latitude and month using the FAO-56 Hargreaves–Samani equation — the basis for irrigation scheduling.

From one nozzle's flow, ground speed and nozzle spacing, get a boom sprayer's application rate (L/ha or gal/acre) and the area a full tank covers.

Enter any two of voltage, current, resistance, or power — the calculator solves for the other two using V = IR and P = VI.

Enter any two of distance, time and speed to get the third — with km/h, mph, m/s, km, miles, hours and minutes supported.

Compute density from mass and volume (ρ = m / V), or solve for the missing variable. Built-in reference table for 19 common substances.

Enter launch speed, angle and height to compute projectile range, peak height and flight time (no air resistance). Pick from Earth, Moon, Mars and more.

Compute the wind chill (feels-like temperature) from air temperature and wind speed using the 2001 Environment Canada / US NWS formula, with frostbite risk levels.

Compute dew point from air temperature and relative humidity using the Magnus formula — handy for HVAC, photography and weather analysis.

Compute kinetic energy KE = ½ m v² with mixed units (kg / g / lb and m/s / km/h / mph) and see the result in joules, kilojoules, food calories, foot-pounds and watt-hours.

Enter any three of initial amount, remaining amount, elapsed time and half-life to solve for the fourth — useful for radioactive decay, drug pharmacokinetics and radiometric dating.

Pick the colour bands and instantly read the resistance and tolerance — 4-band and 5-band notations supported, with Ω / kΩ / MΩ formatting and a closest E12 / E24 preferred-value check.

Enter two latitude/longitude pairs to compute the great-circle distance using the haversine formula (km, miles, nautical miles), with bearing and midpoint.

Solve any one of C₁, V₁, C₂, V₂ from the dilution equation C₁V₁ = C₂V₂ — a daily lab essential for chemistry, biology and pharmacy work.

Two 80 dB sound sources do not equal 160 dB. Enter multiple dB values to compute the combined SPL, and subtract background noise to recover the signal alone.

Enter up to 8 resistor values to see the series total (R₁ + R₂ + …) and the parallel total (1 / Σ(1/Rᵢ)) at the same time.

Convert between electromagnetic wavelength and frequency via c = λf, with the matching spectrum band (radio / microwave / visible / X-ray / γ) and photon energy.

Compute the capacity of vertical or horizontal cylindrical, rectangular and spherical tanks, including partial-fill volumes at a given liquid level.

Enter the pendulum length and local gravity to get the period, frequency and angular frequency, with Earth/Moon/Mars/Jupiter presets — and reverse-solve for the length needed to hit a target period.

Enter air temperature and relative humidity to get the apparent temperature (NOAA Rothfusz heat index) and the corresponding heat-stress risk band.

Enter speed, reaction time and road friction to estimate reaction, braking and total stopping distance.

Enter the refractive indices of two media and an angle of incidence — get the refraction angle and critical angle from Snell's law (n₁ sin θ₁ = n₂ sin θ₂).

Enter capacitance (F, mF, µF, nF, pF) and voltage to compute the stored energy (E = ½CV²) and charge (Q = CV) on a capacitor.

Enter altitude to compute the boiling point of water (°C / °F) and local air pressure using the ICAO standard atmosphere and the Antoine equation — useful for hiking, cooking and high-altitude baking.

Solve Q = m × c × ΔT for any one of heat energy, mass, specific heat capacity or temperature change — with presets for water, aluminium, iron, copper, glass, air and more.

Convert between pH, pOH, hydrogen-ion concentration [H⁺] and hydroxide concentration [OH⁻] — with acid / neutral / alkaline classification.

Pick the unknown (P, V, n or T), enter the other three and PV = nRT is solved instantly — works in Pa / kPa / atm / bar / mmHg / psi, m³ / L / mL, mol / mmol / kmol and K / °C / °F.

Enter two point charges and the separation distance to compute the electrostatic force between them via F = kₑ·q₁·q₂/r² (attractive or repulsive).

Enter two masses and the distance between them to compute the gravitational attraction via F = G·m₁·m₂/r².

Given any two of the three quantities (object distance u, image distance v, focal length f), solve for the remaining one and the lateral magnification.

Enter the observer eye height above the surface to compute the distance to the geometric and refraction-corrected horizon.

Given any two of spring constant k, displacement x or restoring force F, solve for the third and the elastic potential energy U = ½kx².

Enter fluid density, submerged volume and gravitational acceleration to compute the buoyant force F = ρ V g, plus whether the object floats, sinks or stays neutral.

Enter the input voltage and two series resistor values to find the divider output voltage, current and power dissipated in each resistor.

Enter the mass and radius of a celestial body (or pick Earth, Moon, Mars and other presets) to compute the minimum surface launch speed v = √(2GM/r) needed to escape its gravity.

Enter the source frequency, observer/source velocities and wave speed (sound or light) to compute the observed frequency shift due to the Doppler effect.

Enter initial temperature, ambient temperature, cooling constant and elapsed time to estimate an object's temperature with T(t) = T∞ + (T₀ − T∞)·e^(−kt), plus half-cooling time and time constant.

Enter mass and radius, then linear speed, angular speed or period, and instantly read off centripetal force F = m·v²/r, centripetal acceleration, tangential speed, angular speed, period and frequency.

Enter inductance L (H) and capacitance C (F) to compute the resonant frequency f = 1 / (2π√(LC)) of an LC tank circuit, plus its period and angular frequency.

Enter the air temperature and compute the speed of sound in dry air using v = 331.3 × √(1 + T/273.15) — output in m/s, km/h, mph, ft/s plus the "count-the-seconds" thunder distance rule.

Enter fluid density, velocity, characteristic length and viscosity to compute the Reynolds number and classify the flow as laminar, transitional or turbulent.

Enter the spring constant k and displacement x to instantly compute the elastic potential energy stored in a Hookean spring via U = ½·k·x², with conversions to kJ, kcal, ft·lbf and Wh for easy comparison.

Use A = ε·c·ℓ to compute absorbance, transmittance and concentration — any three of the four inputs determines the fourth.

Compute orbital period from semi-major axis and central body mass via T² = 4π²·a³/GM, or invert to recover the semi-major axis — with presets for the Sun, Earth, Moon and Jupiter.

Enter fluid density, velocity, drag coefficient Cd and frontal area and compute the drag force from Fd = ½·ρ·v²·Cd·A — handy for cycling, automotive, ballistic and skydiving scenarios.

Compute the hydrostatic pressure P = ρ·g·h from fluid density, depth and gravity, and convert it to kPa, bar, psi, atm, mmHg and metres of water — useful for diving, aquariums and piping.

Enter pKa together with the conjugate-base [A⁻] and weak-acid [HA] concentrations and apply the Henderson–Hasselbalch equation to compute buffer pH, the buffer ratio and the effective buffering range.

Enter mass, incline angle and friction coefficients (or pick a preset like steel-on-steel or rubber-on-concrete) to compute normal force N, maximum static friction fs,max = μs·N, kinetic friction fk = μk·N and the critical angle θc — and instantly see whether the block stays at rest or slides down the slope.

Enter driving and driven gear tooth counts plus input RPM and torque to compute gear ratio, output RPM, output torque and mechanical advantage.

Enter solute mass, molecular weight and solution volume to compute molarity (mol/L); solve for any one unknown.

Convert moment magnitude (Mw) to released energy (joules and TNT equivalent) via the Gutenberg–Richter relation log₁₀E = 1.5M + 4.8, with a comparison to landmark historical earthquakes.

Enter the actual mass of product recovered and the theoretical yield to compute % yield = actual ÷ theoretical × 100%, with an indicative classification (poor / moderate / good / excellent).

Pick a common shape (solid sphere, hollow sphere, solid cylinder, thin-walled cylinder, thin disk, slender rod about centre or end, rectangular plate) and enter mass and dimensions to get the rotational moment of inertia I = k·m·r² / k·m·L².

Enter air temperature, barometric pressure and relative humidity to compute air density (kg/m³) via the CIPM partial-pressure form and the Magnus-Tetens saturation vapour pressure — useful for cyclists, runners, pilots and aerodynamics estimates.

Enter a fall height (and optional mass / gravity) to compute fall time, impact velocity and kinetic energy via h = ½·g·t² and v = √(2·g·h). Preset gravities for Earth, Moon, Mars and more.

Enter a chemical formula (e.g. H2SO4, Ca(OH)2, CuSO4·5H2O) and the tool sums the IUPAC standard atomic weights element by element to return the molar mass in g/mol — useful for chemistry homework and lab weighing.

Enter any two of force, mass and acceleration and the tool solves for the third — handy for high-school physics, machine design, vehicle braking and elevator safety factors.

Enter mass and molar mass (g/mol), or moles, and the tool converts via n = m / M, also reporting the particle count N = n · Nₐ — the everyday workhorse of chemistry lab weighing and stoichiometry problems.

Enter a velocity v (as a fraction of c or in m/s) and compute the special-relativity Lorentz factor γ = 1/√(1 − v²/c²), together with the time-dilation factor, the length-contraction ratio and the relativistic kinetic energy.

Enter surface area A, temperature T (kelvin) and emissivity ε, then compute the radiant power P = ε σ A T⁴ under the Stefan–Boltzmann law — the standard formula for thermal radiation, the solar constant and stellar luminosity estimates.

Enter the moles (or mass + molar mass) of each component in a mixture and compute the mole fraction xᵢ = nᵢ / Σ nⱼ for every component, with the total checked to equal 1 — used in Raoult's law, gas mixtures and thermodynamic analysis.

Enter mass, frontal area, drag coefficient and air density and the tool returns the terminal velocity v_t = √(2mg / (ρ·A·Cd)) — the speed at which drag balances gravity, with presets for skydivers, raindrops, baseballs and feathers.

Enter applied force, cross-sectional area, original length and the change in length. The tool returns the engineering stress σ, strain ε, Young's modulus E = σ/ε and the elongation, alongside a reference table of common materials (steel, copper, aluminium, concrete, rubber).

Enter a mass (or energy) and the tool uses Einstein's E = mc² to compute the other side, expressing the result in joules, kilowatt-hours, megatonnes of TNT and electron-volts — and showing just how staggering even 1 g of fully-converted matter is.

Enter elevation, velocity and pressure at two points and the tool applies P + ½ρv² + ρgh = constant from Bernoulli's equation to solve for the missing variable — the foundation of pipe flow, Venturi meters, Pitot tubes and aerofoil lift.

Enter the spring constant k and mass m; the tool returns the simple-harmonic period T = 2π√(m/k), frequency f and angular frequency ω, and can also solve for k — the foundation of physics labs, mechanical vibration and MEMS sensor design.

Enter resistance R and capacitance C; the tool returns the RC circuit time constant τ = R × C, the 63.2% / 99.3% charging times (1τ / 5τ) and the voltage percentage at any time t during charging or discharging — the workhorse for electronics design, RC filters, oscillator timing and Arduino debounce.

Enter hot-reservoir temperature T_h and cold-reservoir T_c (units auto-handled in °C / K / °F); the tool returns the Carnot maximum efficiency η = 1 − T_c / T_h, the thermodynamic upper bound for any real heat engine — IC engines, thermal power stations and refrigerators.

Enter any one of linear speed, radius, period, frequency or RPM; the tool interconverts angular velocity ω (rad/s), frequency f (Hz), period T, revolutions per minute (RPM) and rim speed v — for circular motion problems, motor specs, orbital mechanics, centrifuges and any spinning object.

Enter two particle masses m₁ and m₂; the tool applies the reduced-mass formula μ = m₁·m₂ / (m₁ + m₂) to give the effective inertial mass for two-body problems — the building block that reduces orbital mechanics, Keplerian motion, vibrational spectroscopy and collision analysis to single-body equations.

Enter a wavelength or frequency; the tool applies the Planck relation E = hf = hc/λ to return the energy of a single photon in joules and electron-volts (eV), plus the visible-light colour band where applicable — the foundation of physics, chemistry and spectroscopy problems.

Enter a blackbody temperature in kelvin; the tool applies Wien's displacement law λ_max·T = b to return the peak wavelength and corresponding frequency, and can invert to estimate temperature from a measured wavelength — the foundation of solar spectra, thermal imaging and CMB analysis.

Enter initial length L₀, temperature change ΔT and the linear expansion coefficient α (or pick a preset material); the tool returns ΔL = α·L₀·ΔT and the new length — used for bridge expansion joints, rail track gaps, pipework thermal allowances and more.

Enter initial and final angular velocity plus the time interval (or tangential acceleration and radius); the tool returns the angular acceleration α and converts between rad/s² and rpm/s — a basic quantity in rotational dynamics, rotor engineering and motor analysis.

Enter any three of plane spacing d, diffraction angle θ, order n and wavelength λ; the tool solves Bragg's law nλ = 2d sinθ for the missing one — the cornerstone of X-ray crystallography and materials characterisation.

Enter initial velocity u, final velocity v and elapsed time t; the tool returns the average acceleration a = (v − u) / t along with the displacement s = ½(u + v)·t — the entry-point of Newtonian kinematics.

Enter a particle's mass and speed; the tool applies the de Broglie relation λ = h / (mv) to return the matter-wave wavelength, momentum and kinetic energy — the cornerstone of intro quantum mechanics, electron-microscope resolution and cold-neutron experiments.

Enter a mass in kilograms, solar masses or Earth masses; the tool applies r_s = 2GM/c² to return the Schwarzschild radius (event horizon) and equivalent density — the headline number for any black-hole conversation in astrophysics, GR intro courses and pop-science Q&A.

Enter battery capacity (in Wh or Ah + voltage), load power, depth of discharge and inverter efficiency; the tool returns the realistic usable runtime in hours. Use it to size UPS units, solar-storage banks, portable power stations and 12 V car fridges.

Enter the total current and two parallel resistor values; the tool applies the current-divider rule I₁ = I·R₂/(R₁+R₂) to compute each branch current, the equivalent resistance, the shared voltage and the power dissipated in each resistor — a staple for circuit analysis, PCB design and EE coursework.

Enter inductance L and frequency f; the tool returns the inductive reactance X_L = 2πfL and the angular frequency ω — used in filter design, impedance matching and AC circuit analysis.

Enter the object speed and the local air temperature (the tool derives a from temperature); returns the Mach number M = v ⁄ a and classifies the regime as subsonic, transonic, supersonic or hypersonic — a standard tool for aviation and fluid dynamics.

Pick a pulley configuration (fixed, single movable, block-and-tackle or differential), enter the load; the tool returns the input force required, ideal & actual mechanical advantage and the rope length pulled — staple tool for physics class and rigging.

Enter the enthalpy change ΔH, entropy change ΔS and absolute temperature T; the tool applies ΔG = ΔH − TΔS to give the free-energy change and flags whether the reaction is spontaneous (ΔG < 0), at equilibrium (ΔG = 0) or non-spontaneous (ΔG > 0) — a staple of chemical thermodynamics, biochemistry and materials science.

Enter specific impulse Isp, initial mass m₀ and final (dry) mass m_f; the tool applies Δv = Isp · g₀ · ln(m₀ ⁄ m_f) to give the ideal velocity change a rocket can deliver — fundamental for mission planning, orbital mechanics and Kerbal Space Program players alike.

Enter the wavelength or frequency of incident light plus the metal's work function φ; the tool applies Einstein's photoelectric equation KE = hf − φ to give the maximum kinetic energy of ejected electrons, the stopping voltage V₀ and whether the threshold frequency is met.

Enter the material thermal conductivity k, area A, thickness L and temperature difference ΔT; the tool applies Fourier's law Q = k·A·ΔT / L to give the steady-state heat flow (W) and derives the R-value (thermal resistance) — fundamental for wall insulation, glazing, CPU heatsink and energy-audit calculations.

Enter the parallel fraction p of a program and the number of processors N; the tool applies Amdahl's law S = 1 / ((1 − p) + p ⁄ N) to give the theoretical maximum speedup — essential for parallel computing, GPU programming, HPC capacity planning and data-engineering decisions.

Enter the signal and noise power (or voltage); the tool applies SNR(dB) = 10·log₁₀(P_s ⁄ P_n) or 20·log₁₀(V_s ⁄ V_n) and classifies the result for communications, audio or instrumentation use — fundamental for telecom, recording and measurement.

Enter the activation energy Eₐ, pre-exponential factor A and temperature T; the tool returns the reaction rate constant k via Arrhenius k = A·exp(−Eₐ ⁄ RT) and the rate ratio between two temperatures — the standard tool for chemical kinetics, food shelf-life and drug stability.

Enter the intensity at one distance; the tool applies I₂ = I₁ × (d₁ ⁄ d₂)² to return intensity at any other distance, along with the equivalent dB change (ΔL = 20·log₁₀(d₁ ⁄ d₂)) — a workhorse for photography, stage lighting, audio engineering, radiation safety and radio link planning.

Enter current, one-way wire length, conductor size (AWG) and system type (single-phase / three-phase); the tool returns voltage drop and % drop via V_drop = 2·I·R·L (1-φ) or √3·I·R·L (3-φ) and warns when it exceeds the NEC 3% guideline — essential for home wiring, solar runs and long-distance equipment feeds.

Enter capacitance C and frequency f; the tool returns the capacitive reactance X_C = 1/(2πfC) and the angular frequency ω — used for filter design, AC circuit analysis and impedance matching, complementing the existing inductor reactance tool.

Enter mass, incline angle and coefficient of friction; the tool applies Newton's second law to return net force along the slope, acceleration, normal force and the push needed to hold or move the object — a staple for high-school / first-year physics and basic mechanical design.

Enter the two masses m₁ and m₂ hung from a fixed pulley; the tool returns the system acceleration via a = (m₁ − m₂)g ⁄ (m₁ + m₂), the rope tension T and the heavier-mass direction — the standard quick-reference for a classic mechanics textbook problem.

Enter the drop height h and the rebound height h′; the tool applies e = √(h′ ⁄ h) to compute the coefficient of restitution (COR), the energy-loss percentage and the post-impact/pre-impact speed ratio — the standard tool for physics labs and sports engineering (tennis, basketball, golf).

Enter two masses m₁, m₂ and pre-collision velocities v₁, v₂; the tool applies momentum conservation m₁v₁ + m₂v₂ = (m₁ + m₂)v′ to find the common post-collision velocity, plus energy lost (to heat/sound/deformation) and pre/post kinetic energies — the standard tool for physics class and accident-reconstruction.

Enter two temperatures plus one saturation vapor pressure (and the molar enthalpy of vaporisation ΔHvap); the tool applies ln(P₂ ⁄ P₁) = −ΔHvap ⁄ R × (1 ⁄ T₂ − 1 ⁄ T₁) to return the vapor pressure at the other temperature, or the boiling point at altitude. A staple of chemistry, HVAC, brewing, cooking and meteorology.

Enter the solute mass, molar mass, solvent mass, the solvent freezing-point depression constant Kf and the van't Hoff factor i; the tool applies ΔTf = i · Kf · m to return the temperature drop and the new freezing point. A staple of general chemistry, food science, road de-icing and antifreeze formulation.

Enter the redshift z and the Hubble constant H₀ (default Planck 2018 value 67.4 km/s/Mpc); the tool returns recession velocity v = cz, comoving distance d = v ⁄ H₀, and the relativistic-Doppler form v_rel = c · ((1+z)² − 1) ⁄ ((1+z)² + 1) for high-z work. The standard back-of-the-envelope tool for estimating cosmological distances.

Enter two or more independent sound levels (dB); the tool combines them using L_total = 10 log₁₀(Σ 10^(Lᵢ ⁄ 10)). Explains why two equally loud machines together produce only a 3 dB increase rather than double — essential for engineering acoustics and noise assessments.

Enter real power P (kW) plus either apparent power S (kVA) or the power factor cos φ; the tool computes the missing two using S² = P² + Q² and PF = cos φ, and returns the phase angle. Essential for industrial loads, motor sizing and UPS planning.

Enter the incident photon wavelength and scattering angle; the tool returns the Compton shift Δλ = (h ⁄ m_e c)(1 − cos θ), the scattered-photon wavelength and the photon energy loss. Cornerstone result of quantum mechanics, X-ray scattering and radiation physics.

Enter primary / secondary turns (or voltages); the tool returns the turns ratio V₁ ⁄ V₂ = N₁ ⁄ N₂ = I₂ ⁄ I₁, the output voltage, the output current and the impedance reflection (N₁ ⁄ N₂)². Cornerstone of electrical engineering, audio output transformer design and power-distribution planning.

Enter a distance (km, AU, light-years or parsecs); the tool uses c = 299 792 458 m/s to return the one-way light-travel time, with quick references for Earth–Moon, Earth–Sun and Earth–Mars distances. Useful for astronomy, deep-space communications and intro relativity classes.

Enter one-way distance (km or mi), vehicle class (small / medium / large petrol, diesel, hybrid, PHEV, BEV, motorbike), passenger count and whether the trip is return-only or round-trip; the tool applies DEFRA 2024 well-to-wheel emission factors to return total vehicle CO₂, per-passenger share, annualised (× 250 commuting-day) emissions and a tree-month offset.

Enter panel area, panel efficiency (%), average peak sun hours and the system performance ratio; the tool applies E = A × η × H × PR to return daily, monthly and annual kWh — the first sizing tool for any rooftop PV project.

Enter the moles available for two reactants plus their stoichiometric coefficients from the balanced equation; the tool identifies the limiting reagent, returns the theoretical yield of product (mol) and the excess reagent remaining. The canonical homework calculation in A-level, AP and first-year university chemistry.

Enter rotor diameter, wind speed, air density and the power coefficient Cp (Betz limit 0.593); the tool applies P = 0.5 × ρ × A × v³ × Cp to return instantaneous electrical power (kW), an annual-energy estimate and an equivalent number of households powered. The first-cut sizing tool for small wind turbines and off-grid systems.

Enter altitude (and optionally sea-level pressure/temperature); the tool applies the ISA barometric formula p(h) = p₀ × (1 − Lh ⁄ T₀)^(gM ⁄ RL) to return ambient pressure in hPa, atm and psi. Standard reference for hiking, drones, mountaineering, aviation and cooking corrections at altitude.

Enter any three of initial pressure/volume and final pressure/volume of an isothermal gas; the tool returns the fourth via P₁V₁ = P₂V₂ — the foundational formula for scuba tank compression, syringes, engine pistons and air springs.

Enter solute concentration, temperature and the van 't Hoff factor i; the tool returns osmotic pressure via π = i·M·R·T — the foundational equation for cell osmosis, reverse-osmosis desalination membranes and isotonic IV fluid tonicity.

Enter the focal length, f-number, and sensor circle-of-confusion; the tool returns the hyperfocal distance via H = f² ⁄ (N × c) + f — the defining number for landscape, street and astro photography focus-point selection.

Enter up to 8 capacitances and pick series or parallel; the tool returns the equivalent capacitance via C_parallel = ΣCᵢ or 1⁄C_series = Σ(1⁄Cᵢ) — the basic tool for circuit design and filter calculations.

Enter flow rate Q, total head H, fluid density and pump efficiency; the tool returns the required electrical power via P_shaft = ρ·g·Q·H ⁄ η in kW and HP — fundamental sizing equation for water supply, irrigation, fire-fighting and HVAC system design.

Heat pump COP = heat output Q ÷ electrical work input W. Enter the heat delivered and electricity used; the tool returns the COP and contrasts it with the Carnot upper bound COP_max = T_hot ⁄ (T_hot − T_cold) — useful for sizing AC, heat pumps, underfloor heating and water heaters.

The Sabine formula RT60 = 0.161·V / A (metric) estimates the time it takes for sound to decay 60 dB inside a room. Enter the room volume and the total effective absorption (Σ Sᵢ·αᵢ); the tool returns RT60 and compares it against recommended ranges for studios, classrooms and concert halls.

Enter the standard cell potential E°, number of electrons transferred n, reaction quotient Q and temperature; the tool returns the actual cell potential via the Nernst equation E = E° − (RT/nF) ln Q — used in electrochemistry, battery design, corrosion analysis and nerve-membrane potentials.

Enter solute mass / volume and total solution mass / volume; the tool returns mass percent (% w/w), mass-volume percent (% w/v) and volume percent (% v/v) — standard in chemistry class, food labelling and pharmaceutical formulation.

Enter initial pressure, volume, temperature and the heat-capacity ratio γ; the tool applies the isentropic relations PVγ = const and TVγ−1 = const to give the new P, V and T — standard for internal-combustion engines, compressors and rocket nozzles.

Enter air temperature and relative humidity (and optionally leaf temperature); the tool applies the Tetens saturation-vapor-pressure formula to return the vapor pressure deficit in kPa — the single most-used environmental metric for indoor hydroponics, commercial greenhouses, cigar humidors and fruit ripening rooms.

Enter the diameters of the two pulleys and the center-to-center distance; the tool returns the V-belt length via the standard closed-form L = 2C + π(D₁ + D₂)/2 + (D₁ − D₂)² / (4C) and shows the wrap angles on each pulley. Useful for industrial fans, lathes and any open belt drive.

Enter field elevation, QNH altimeter setting and OAT; the tool applies the standard FAA Pilot's Handbook formula and reports pressure altitude (PA), the ISA standard temperature, the deviation from ISA and the density altitude (DA) — essential for takeoff performance, high-altitude training and drone lift planning.

Enter focal length, aperture, focus distance and sensor format; the tool returns the hyperfocal distance, near-limit, far-limit and total depth of field — the core tool landscape, product and portrait photographers use to plan how much of a scene will be acceptably sharp.

Enter the installation latitude; the tool applies the standard NREL Landau / Heliene rules to give the year-round optimal fixed tilt, the summer tilt and the winter tilt — the headline number every rooftop PV, off-grid system and solar-thermal installer sizes by.

Enter any two of frequency f, wavelength λ and wave speed v; the tool applies v = f × λ to return the third together with the period T, angular frequency ω and angular wavenumber k. Built-in presets cover sound in air, fresh water, sea water and steel, plus EM in vacuum, water and glass — the most basic wave relation used across acoustics, radio, optics and ocean waves.

Enter hot-side inlet and outlet temperatures, cold-side inlet and outlet temperatures, and pick counter-flow or parallel-flow; the tool applies LMTD = (ΔT₁ − ΔT₂) / ln(ΔT₁ / ΔT₂) to give the log mean temperature difference — the fundamental quantity in shell-and-tube, plate, AC-condenser and chemical-process heat-exchanger sizing.

Enter the seconds between seeing the flash and hearing the thunder; the tool uses the speed of sound (≈ 343 m/s, light arrives essentially instantly) to compute distance in km and miles, with NOAA "30/30 rule" guidance on when to seek shelter.

Enter the balloon diameter and the payload weight; the tool applies lift ≈ (ρ_air − ρ_He) × V to give the net lift per balloon (≈ 1 g per litre at sea level), how many balloons you need, and notes on envelope weight and altitude density effects.

Enter antenna gain in dBi (or as a linear ratio); the tool applies G = 10^(dBi/10) to convert between dBi and ratio, with reference ranges for common antenna types (whip, Yagi, parabolic, patch, WiFi router) — essential for picking WiFi, two-way radio or satellite antennas.

Enter skid mark length and road surface friction coefficient μ; the tool applies v = √(2μgd) to estimate the minimum entry speed when braking began (km/h and mph) — used in crash reconstruction, insurance assessment and driving safety analysis.

Enter aperture f-number, shutter time and ISO; the tool returns EV = log₂(N²/t) − log₂(ISO/100), maps it to a typical scene (sunny day, indoor, moonlight) and lists five equivalent-exposure pairs (same EV, different aperture / shutter combinations) — handy for Sunny-16, night and moonlight photography.

Enter sample rate (Hz), bit depth (8 / 16 / 24 / 32 bit), channel count (mono / stereo / 5.1 / 7.1 / Atmos) and duration; the tool instantly returns the Nyquist limit, raw PCM bitrate, bytes-per-second and the uncompressed WAV file size.

Enter latitude (° N or S); the tool applies v = (2π · R · cos φ) ÷ T (where R is Earth's equatorial radius and T = 86 164 s, one sidereal day) to compute the linear speed of the ground at that latitude in m/s, km/h and mph — explains why equatorial launch sites (Cape Canaveral, Kourou) get a "free kick" for rocket launches.

Enter initial volume V₁, initial temperature T₁ and final temperature T₂ (in °C or K); the tool applies Charles's Law V₁/T₁ = V₂/T₂ (at constant pressure) to find the final volume V₂. Any one variable can be left blank to solve for it.

Enter snow depth (cm or inches) and snow density (kg/m³, or pick a preset for fresh / powder / wet / packed snow); the tool returns the equivalent water-column height after melting (mm or inches) and the mass per square metre (kg) — used for avalanche risk, roof load and irrigation water supply.

Two modes: (1) compute μ = F/N from friction force F and normal force N; (2) compute static coefficient μ_s = tan θ from the incline angle θ at which an object just starts to slip. A staple of physics labs and engineering classes.

Enter Young's modulus E, cross-section (solid round / rectangle / pipe), length L and an effective-length factor K (0.5 fixed-fixed, 0.7 fixed-pinned, 1.0 pinned-pinned, 2.0 fixed-free); the tool computes the minimum moment of inertia, radius of gyration, slenderness ratio and the critical buckling load P_cr — the foundation tool for slender-column stability design.

Enter the cutter diameter and recommended surface speed (SFM or m/min); the tool applies RPM = (SFM × 12) ⁄ (π × D) to give the spindle speed — the standard machinist conversion used for drills, end mills and lathes.

Enter filament diameter (1.75 mm or 2.85 mm), length in metres and material (PLA / PETG / ABS / TPU / Nylon / PC — density auto-fills); the tool applies m = π·(d/2)²·L·ρ to give the mass in grams, and inverts to answer "how many metres on a 1 kg spool".

Enter roof snow depth (cm or in) and snow type (fresh 50 kg/m³, settled 200, wet 400, ice 917); the tool computes the surface load in kPa, psf and kg/m² so you can compare to local building codes (IBC, NBC, Eurocode 1).

Enter daily high and low temperatures and a crop base temperature (corn 10 °C, wheat 4 °C, cotton 15 °C, etc.); the tool applies the standard ((Tmax + Tmin) ⁄ 2) − Tbase formula to give the day's GDD and a running cumulative total — used by growers, gardeners and IPM scouts to predict bloom, harvest and pest emergence windows.

Enter your Earth age; the tool uses each planet's orbital period around the Sun (Mercury 87.97 d, Venus 224.7 d, Mars 686.97 d, Jupiter 11.86 yr…) to compute your equivalent age on each of the other seven planets plus Pluto.

Enter water volume (litres), starting temperature and the kettle wattage; the tool applies Q = m·c·ΔT to give the energy (kJ / Wh) and time needed to reach 100 °C (or a custom target) and also reports the latent heat needed to fully evaporate.

Enter tank volume (litres / gallons), incoming and target water temperature, heater rated power (W or BTU/h) and efficiency; the tool applies Q = m·c·ΔT to give the time needed to bring a full tank up to temperature.

Enter the initial and final principal quantum numbers n₁ and n₂; the tool applies E_n = −13.6/n² eV and the Rydberg formula 1/λ = R(1/n₁² − 1/n₂²) to give the energy levels, transition energy, photon wavelength and frequency (Lyman / Balmer / Paschen series).

Enter the transmit frequency (MHz / GHz) and the distance (m / km / mi); the tool applies the Friis free-space path loss formula FSPL(dB) = 20·log₁₀(d) + 20·log₁₀(f) + K, useful for Wi-Fi, LoRa, 4G/5G and satellite link-budget estimates.

Enter the battery system voltage (12 / 24 / 48 V), chemistry (LiFePO₄, Li-Ion NMC, lead-acid / AGM) and measured open-circuit voltage; the tool linear-interpolates the standard rest-voltage SoC table to give the percent state of charge, remaining headroom to low-voltage cut-off, and warns when you are at the floor.

Enter the surface air temperature and dew point; the tool applies the standard aviation approximation 125 m per °C of dew-point spread to estimate the cumulus cloud base (lifting condensation level, LCL).

Enter the lens focal length (mm) and sensor size (full-frame, APS-C, M4/3, 1-inch, etc.); the tool returns the horizontal, vertical and diagonal angle of view in degrees — useful for lens choice, framing estimates and landscape / portrait planning.

Enter the supply voltage, the LED forward voltage (per colour, per series count) and the target current (mA); the tool applies Ohm's law to give the limiting resistor value (Ω), the minimum wattage and the closest standard E12 value.

Enter the native focal length and the sensor format (full-frame 1.0×, APS-C 1.5/1.6×, M4/3 2.0×, 1-inch 2.7×, etc.); the tool returns the 35 mm-equivalent focal length and the depth-of-field-equivalent aperture, making cross-system lens comparisons straightforward.

Enter the span L, Young's modulus E, second moment of area I and the load type (centre point P or uniformly distributed w) for a simply supported beam; the tool applies δ = PL³/(48EI) or δ = 5wL⁴/(384EI) to give the mid-span deflection and maximum bending moment — a quick sanity check for woodworking, civil and mechanical work.

Enter the current (A), allowable temperature rise ΔT (°C), copper weight (oz/ft², typically 0.5 / 1 / 2) and whether the trace is on an external or internal layer; the tool applies the IPC-2221 formula I = k·ΔT^0.44·A^0.725 to give the required trace width (mm / mil) — a quick PCB layout sanity check.

Enter a note name (C, C♯, D … ) and octave (e.g. A4, C5); the tool applies the 12-tone equal-temperament formula f = 440 × 2^((n − 69)/12) to give the frequency (Hz). Reverse mode: enter a Hz value and the tool reports the nearest note and cents deviation — useful for tuning, synth design and acoustics work.

Enter the VSWR (voltage standing-wave ratio) or the return loss in dB; the tool applies Γ = (VSWR − 1)/(VSWR + 1) and RL = −20·log₁₀|Γ| in both directions, and reports the reflected/transmitted power percentages — a quick sanity check for antenna, RF and transmission-line work.

Enter the telescope aperture (mm), main focal length, eyepiece focal length and Barlow factor; the tool returns magnification M = f_tel × Barlow ÷ f_eye, exit-pupil diameter (D ÷ M), focal ratio, maximum useful magnification (~ 2 × aperture mm), minimum useful power (~ aperture ÷ 7 mm dark-adapted) and the Dawes / Rayleigh diffraction limits — the numbers that decide whether an eyepiece (or planned purchase) is the right match.

Enter the string length (mm), tension (N) and linear mass density (g/m); the tool applies f₁ = (1/2L)·√(T/μ) to give the fundamental and the first 8 harmonics — useful for guitar, violin or piano string tuning and for classroom demonstrations of standing waves.

Enter the cylinder bore (mm), stroke (mm) and cylinder count; the tool applies V = π · (bore/2)² · stroke · cylinders to give engine displacement in cc, L and cubic inches — useful for motorcycles, car tuning and mechanical-engineering teaching.

Enter the driver pulley diameter, driven pulley diameter and input RPM; the tool applies RPM_out = RPM_in × (D_drive / D_driven) to give output RPM, reduction ratio and belt linear speed (m/s) — useful for mechanical, agricultural and CNC retrofits.

Enter the runway heading (°T), wind direction and wind speed (kt or m/s); the tool applies crosswind = V·|sin α| and headwind = V·cos α to give orthogonal components — used by pilots, gliders and UAV operators to assess take-off and landing risk.

Enter vehicle power (hp) and total weight (lb or kg); the tool applies Hale's empirical formulas ET = 5.825 × (W/HP)^(1/3) and trap-speed = 234 × (HP/W)^(1/3) to estimate the 1/4-mile elapsed time and terminal speed — a drag-strip rule-of-thumb.

Enter the waterline length (LWL, ft or m); the tool applies the Froude-derived hull_speed = 1.34 × √LWL (kn) to estimate the theoretical top speed of a displacement hull — a quick benchmark for sailboats, fishing boats and small yachts; exceeding it requires a semi-planing or planing hull form.

Enter the thread's major diameter (mm or in) and pitch or TPI; the tool applies tap_drill = D − (%/100) × 1.0825 × pitch to give the required drill — the everyday tap-drill chart for tapping, machining and DIY woodworking, with 50–100 % engagement adjustment.

Enter the bolt diameter (mm or in), target clamp load (preload, kN or lbf) and nut friction factor K (dry 0.20, oiled 0.15, anti-seize 0.10); the tool applies T = K·F·d to give the recommended tightening torque — essential for mechanical assembly and maintenance.

Enter aircraft (or UAV) mass and wing area; the tool returns W/S (N/m² or lb/ft²) — a fundamental performance parameter: high W/S = stable cruise but long takeoff and high stall speed; low W/S = short takeoff and agile, but gust-sensitive.

Enter two apparent magnitudes; the tool applies Pogson's law m₁ − m₂ = −2.5 × log₁₀(F₁/F₂) to give the brightness (flux) ratio between any two stars, planets or deep-sky objects, and can convert apparent to absolute magnitude given distance.

Enter resistance (Ω) and either voltage (V) or current (A); the tool applies P = V²/R = I²R to give power dissipation (W) and recommends the next-up standard 1/8 W, 1/4 W, 1/2 W, 1 W or 2 W package (with a 50 % safety margin).

Enter the operating frequency (MHz); the tool applies L_metres = 143 / f_MHz (or L_feet = 468 / f_MHz) to give the total length and each leg of a half-wave dipole antenna — essential for amateur radio (HAM), CB, ADS-B and WiFi engineering.

Enter the feedback resistor Rf and input resistor Rin, choose inverting or non-inverting topology; the tool returns the closed-loop gain (V/V and dB) using G = −Rf / Rin (inverting) or G = 1 + Rf / Rin (non-inverting) — essential for analog circuit design.

Enter the resistance R and capacitance C; the tool applies f_c = 1 / (2π × R × C) to give the −3 dB cutoff frequency, the time constant τ = RC and the angular cutoff ω_c — essential for audio circuits, sensor interfaces and power-supply ripple filtering.

Enter the line-to-line voltage V_LL (or phase voltage V_LN), the line current I_L and the power factor cosφ; the tool applies P = √3 × V_LL × I_L × cosφ to give the real power (W), apparent power (VA) and reactive power (VAR) of a three-phase AC system — essential for industrial electrical engineering and switchboard / breaker sizing.

Enter any one measure of a sinusoidal AC voltage (or current) — RMS, peak, peak-to-peak or rectified average — the tool applies V_peak = √2 × V_rms, V_pp = 2 × V_peak and V_avg = (2/π) × V_peak to derive the other three — essential for audio, power-supply design and oscilloscope readings.

Enter the power in W / mW / µW / nW or in dBm; the tool applies P(mW) = 10^(dBm/10) to convert either way, plus the equivalent V_rms / V_peak / V_pp on a 50 Ω (or custom) impedance — essential for RF, Wi-Fi, Bluetooth, LTE and antenna-link budgets.

Enter the inductance L and resistance R; the tool applies τ = L / R to give the RL time constant, the 5τ settling time and the current at 1τ–5τ. Supplying a voltage also returns the steady-state current and stored inductor energy.

Enter the number of turns N, the solenoid length L and the current I; the tool applies B = µ₀ × (N/L) × I to give the interior magnetic flux density (Tesla and Gauss), plus the flux Φ = B × A — essential for electromagnetism, relay and loudspeaker design.

Enter mass and choose a substance (water, ethanol, copper, etc.); the tool applies Q = m × L to give the latent heat needed to melt or vaporize it at constant temperature — used in thermodynamics, HVAC, refrigeration and steam engineering.

Pick a material (steel, stainless, aluminium, copper, brass, titanium…) and a profile (round, square, flat, hexagonal), enter the dimensions and length — the tool returns the theoretical mass via ρ × V in both kg and lb — essential for mechanical design, fabrication and freight estimation.

Enter old RPM, new RPM and the baseline flow / head / power; the tool applies the affinity laws (Q ∝ N, H ∝ N², P ∝ N³) to predict the new operating point — essential for HVAC tuning, circulating pumps and variable-frequency-drive (VFD) energy savings.

Pick a lever class (1st — fulcrum centred; 2nd — load centred; 3rd — effort centred), enter the effort arm, load arm and the known effort or load; the tool applies F_eff × d_eff = F_load × d_load to give the mechanical advantage, the unknown force and worked examples (scissors, wheelbarrow, tweezers etc.) — an essential simple-machines tool for physics teaching.

Enter mass, speed, latitude and Earth (or custom) angular velocity; the tool applies F_c = 2 m × v × Ω × sin φ to give the Coriolis force magnitude and direction — rightward in the Northern Hemisphere, leftward in the Southern, the foundational equation behind cyclones, ocean gyres and long-range ballistics.

Enter the basic reproduction number R₀, the susceptible fraction S and/or vaccination coverage v with vaccine effectiveness VE; the tool returns the effective reproduction number Re = R₀ × S × (1 − v·VE) and flags whether the outbreak grows (Re > 1), plateaus (Re ≈ 1) or shrinks (Re < 1) — the core metric of public-health response.

Enter your Earth weight (kg or lb); the tool applies the surface gravity of each Solar-System body (Moon, Mercury, Venus, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto, plus the Sun) to give your apparent weight there — a popular tool for astronomy teaching, science outreach and classroom activities.

Enter linear speed (or angular speed) and radius to get centripetal acceleration a = v²/r = ω²r, the equivalent g-force, the orbital period and — if you supply a mass — the centripetal force F = m·a. The standard tool for circular-motion problems, rotating-machinery design and cornering analysis.

Enter the initial amount N₀, half-life T½ and elapsed time t; the tool applies the exponential law N = N₀ · e^(−λt) = N₀ · (1/2)^(t/T½) to return the remaining amount, fraction decayed, decay constant λ and mean lifetime τ — applicable to radioactive isotopes, drug pharmacokinetics, first-order chemical reactions and C-14 archaeological dating.

Enter R1, R2 and C; the tool applies f = 1.44 / ((R1 + 2R2)·C), T_H = 0.693·(R1+R2)·C and T_L = 0.693·R2·C to get the astable multivibrator frequency, period, high/low times and duty cycle — essential for hobby electronics, LED blinkers and PWM design.

Enter the masses mᵢ and positions xᵢ of several point particles; the tool applies x_cm = Σ(mᵢ·xᵢ) / Σmᵢ to give the centre of mass and total mass — a foundational tool for mechanics, balance problems, statics and rocket-thrust-line analysis.

Enter the radius r, bank angle θ and static tyre-road friction coefficient μ; the tool applies v² = g·r·(tanθ + μ)/(1 − μ·tanθ) to give the maximum safe speed, the frictionless "design speed" √(g·r·tanθ) and the centripetal acceleration in g — essential for highway design, motorsport and roller-coaster engineering.

Enter a radiopharmaceutical's physical half-life T_phys and biological half-life T_bio; the tool applies 1/T_eff = 1/T_phys + 1/T_bio to give the effective half-life, cumulated activity and patient-release timing — the standard formula for nuclear-medicine dosimetry, radioactive-iodine therapy and PET/SPECT planning.

Enter the refractive indices n₁ (denser) and n₂ (rarer) and the tool applies sin θ_c = n₂/n₁ to give the critical angle for total internal reflection — essential for fibre-optic design, prisms, scuba masks and diamond-cut analysis.

Enter the lens index n, surface radii R₁ and R₂, and optional centre thickness d; the tool applies the lensmaker equation 1/f = (n − 1)·[1/R₁ − 1/R₂ + (n − 1)·d/(n·R₁·R₂)] to return the focal length f, dioptric power P = 1/f and lens-shape classification — useful for lens design, photography and ophthalmic prescriptions.

Enter substance density ρ and reference fluid density (water at 4 °C = 1000 kg/m³ by default); the tool returns specific gravity SG = ρ_sub / ρ_ref plus the related °Brix, °API and °Plato scales used in brewing, petroleum and food industries — chemistry, brewing, gemmology, geology and fluids.

Enter inductance L (H, mH, µH) and frequency f (Hz, kHz, MHz); the tool applies X_L = 2πfL to give the inductive reactance in ohms — the companion to the capacitive-reactance tool for AC circuits, filters, transformers and power-supply design.

Enter an element's atomic number Z and the characteristic X-ray series (Kα, Kβ, Lα); the tool applies Moseley's law √ν = a · (Z − σ) to return the characteristic frequency, wavelength and photon energy — central to element identification, XRF analysis and atomic-physics teaching.

Enter the energy gap ΔE, temperature T and (optionally) degeneracies g₁ and g₂; the tool applies the Boltzmann distribution n₂/n₁ = (g₂/g₁) · e^(−ΔE/kT) to return the relative population of the upper vs the lower state — essential for laser physics, semiconductors, chemical kinetics and astrophysical spectral analysis.

Enter thrust F and propellant mass flow rate ṁ; the tool returns the rocket engine specific impulse Isp (seconds) and effective exhaust velocity v_e = Isp · g₀ — the core parameter for propulsion design and the Tsiolkovsky rocket equation.

Enter air temperature (°C) and compute the saturation vapor pressure e_s (hPa, kPa, mmHg) of water using the Magnus / Tetens empirical formula — essential for meteorology, HVAC, brewing, and agriculture.

Enter dry-bulb temperature (°C) and relative humidity (%) to estimate the wet-bulb temperature T_w via the Stull (2011) empirical formula — the key indicator for human heat-stress limits, HVAC cooling-tower design, and heatwave physiology.

Pick a built-in substance (water, ethanol, methanol, acetone, benzene, toluene, n-hexane, ammonia) or enter custom Antoine constants A, B, C; the tool reports the saturation vapor pressure of the pure substance at the chosen temperature — the bread-and-butter empirical formula for chemistry, distillation and storage safety.

Enter the four arm resistors R₁, R₂, R₃, R_x, source voltage V_s and an optional galvanometer R_g; the tool solves R_x = R₂·R₃/R₁ for the unknown resistor (null method) and reports the out-of-balance voltage V_AB and detector current (deflection method) — the standard circuit behind electronics lab work, strain gauges, RTDs, thermistors and load cells.

Enter the cavity volume, neck radius, neck length, air temperature and end-correction model (flanged / unflanged); the tool applies the Helmholtz formula f = (c/2π)·√(A/(V·L_eff)) for the resonant frequency and pins it to the nearest musical note — the standard tool for bottle whistles, bass-reflex (ported) speaker design, exhaust silencers and HVAC noise control.

Enter the initial temperature T₀, ambient temperature T_env, cooling constant k and elapsed time t — Newton's law of cooling T(t) = T_env + (T₀ − T_env)·e^(−kt) returns the object's temperature at any time, or inverted to give the time to reach a target. The standard tool for forensic time-of-death estimation, food cooling, engine heat-loss and how-long-will-my-coffee-stay-hot.

Enter pipe radius r, length L, pressure drop ΔP and dynamic viscosity µ (or pick a preset: water, blood plasma, SAE 30 oil, glycerin, air); the tool returns laminar volumetric flow Q, mean velocity v̄, mass flow, wall shear stress and the Reynolds number Re — and tells you when the flow has left the laminar regime.

Enter engine stroke (mm) and RPM; the tool applies v̄ = 2 · S · N ÷ 60 to give mean piston speed in m/s and ft/min and compares against typical limits (street, high-performance, race, F1) — the classic engine-design rule of thumb for reliability and piston wear.

Enter bore (mm), stroke (mm) and combustion-chamber (clearance) volume in cc; the tool computes per-cylinder displacement V_d = π·(B/2)²·S, the static compression ratio CR = (V_d + V_c)/V_c, the recommended octane (87 / 91 / 95 / 100) and the typical naturally-aspirated vs forced-induction limits — essential for engine builds, head shaving and piston swaps.

Enter the incident polarized intensity I₀ and the angle θ between two polarizer transmission axes; the tool returns the transmitted intensity I = I₀·cos²θ and the percentage — the basic formula behind LCDs, 3D glasses, photographic CPL filters and optical experiments.

Enter the refractive indices n₁ and n₂ of the two media; the tool returns Brewster's angle θ_B = arctan(n₂/n₁), the unique incidence angle at which reflected light is fully s-polarized — the design angle for CPL filters, anti-glare coatings and laser-cavity windows.

Enter the sphere radius, particle density, fluid density and dynamic viscosity; the tool returns the terminal settling velocity of a small sphere in a viscous fluid via Stokes' law — used for sedimentation tests, blood-cell separation, flocculation modelling, fine-particle pollutant settling and microfluidics.

Enter a black-body absolute temperature T (K) to get the peak emission wavelength via Wien's displacement law λ_max · T = 2.8977719 × 10⁻³ m·K; the reverse direction also recovers T from a measured λ_max — used in stellar colour-temperature work, incandescent lighting, infrared thermography and solar-spectrum analysis.

Enter the plate area A, separation d and relative permittivity εᵣ; the tool returns capacitance C = ε₀ εᵣ A / d, and (if a voltage V is supplied) also the stored energy E = ½CV², charge Q = CV, electric field E = V/d and surface charge density σ — a staple of electrical engineering, introductory physics and PCB design.

Enter the number of turns N, cross-section area A (cm²), length ℓ (cm) and core relative permeability µᵣ of a solenoid; the tool applies L = µ₀ µᵣ N² A / ℓ for the inductance in H, mH or µH — useful for air-core, ferrite-core and steel-core coil design, transformer first-cut sizing and LC resonant-circuit component picks.

Enter the dynamic viscosity μ, specific heat c_p and thermal conductivity k of a fluid; the tool applies Pr = μ·c_p / k for the dimensionless Prandtl number and compares against common-fluid benchmarks (air ~0.71, water ~7, oils > 100, liquid metals ~0.01) — the key parameter for convective heat transfer, boundary-layer analysis and heat-exchanger design.

Enter substrate concentration [S], maximum velocity V_max and Michaelis constant K_M; the tool applies the Michaelis-Menten equation v = V_max·[S]/(K_M + [S]) to give the reaction rate and the saturation fraction v/V_max. Supplying the total enzyme concentration [E]₀ adds the turnover number k_cat = V_max/[E]₀ and the catalytic efficiency k_cat/K_M — the standard model in enzymology, drug metabolism, fermentation engineering and biochemistry.

Enter pipe length L, flow rate Q, inner diameter D and the Hazen-Williams roughness coefficient C; the tool applies h_f = 10.67 · L · Q^1.852 / (C^1.852 · D^4.87) (SI units) to return frictional head loss (m of water) and unit head loss (m / 100 m) — the engineering yardstick for municipal water mains, fire-pump sizing, irrigation and industrial cooling-water systems.

Enter the nameplate rating P_rated (MW), the period's actual energy E (MWh) and the period length (h); the tool returns the capacity factor CF = E / (P_rated · t) — the standard yardstick for benchmarking wind farms, solar plants, nuclear and thermal plants — alongside typical reference bands (onshore wind 25–45 %, solar PV 15–25 %, nuclear 90 %+).

Enter the initial state (P₁, V₁, T₁), the polytropic index n and one final-state quantity; the tool applies PV^n = constant together with the ideal-gas law to solve simultaneously for P₂, V₂, T₂, the work W and the heat Q — covering the isothermal (n = 1), adiabatic (n = γ), isobaric (n = 0) and isochoric (n = ∞) special cases, a core tool for thermodynamics, internal-combustion cycles and compressor design.

Enter the waterline length (LWL); the tool applies the classical naval-architecture rule v(knots) ≈ 1.34 × √(LWL_ft) to estimate the theoretical top speed of a displacement hull (the wave-resistance limit reached at Froude ≈ 0.4) — the starting-point top-speed estimate for sailboats, monohull yachts and trawlers.

Enter object height and the sun's altitude angle above the horizon; the tool uses length = height / tan(angle) to return the ground shadow length — a basic tool for golden-hour photography planning, solar-panel shading analysis and architectural sunlight studies.

Enter electricity use (kWh per month or year) and your grid emission factor (g CO₂e / kWh — the tool preloads 30+ major regions from IEA 2023 data); it returns equivalent CO₂ emissions in kg / metric tonnes plus intuitive comparisons against driving km, transatlantic flights and trees needed for one year of offset.

Enter chimney height, indoor and outdoor temperatures, and atmospheric pressure; the tool applies the stack-effect formula ΔP = 0.0342·P·h·(1/T_out − 1/T_in) to return the natural draft pressure in pascals and inches of water — the core physics behind wood stoves, fireplaces, gas water heaters and natural-ventilation design.

Enter the slope length L and tip width W of a wedge (axe, chisel, knife, doorstop), the tool returns the ideal mechanical advantage IMA = L / W and multiplies by friction efficiency to give the real force-multiplication ratio — the classic simple-machine problem in physics teaching, blade design and mechanical engineering.

Enter a DNA primer sequence (A/T/C/G); the tool auto-selects the Wallace rule (≤ 13 nt, Tm = 2·(A+T) + 4·(G+C)) or the salt-corrected Marmur–Doty / Howley GC% formula (> 13 nt) and returns the melting temperature, suggested annealing temperature, GC %, length and base composition — the daily bench tool for molecular biology, PCR, qPCR and sequencing-primer design.

Enter engine thrust (N / kgf / lbf), total mass and gravity (default Earth g = 9.80665 m/s²; Moon and Mars presets included). The tool returns thrust-to-weight ratio (TWR), net acceleration above gravity, maximum static climb angle and the thrust / weight in Newtons — the entry-level core metric for rockets, fighter jets, drones, model rockets and electric VTOL design.

Enter two masses (m₁, m₂) and their pre-collision velocities (u₁, u₂); the tool applies momentum and kinetic-energy conservation to return post-collision velocities (v₁, v₂), the relative-velocity ratio, total momentum and total kinetic energy — the foundational formula for A-level physics, billiards, Newton's cradle, bowling-pin physics and mechanical impact analysis.

Enter two frequencies in Hz; the tool returns the beat frequency fᵦ = |f₁ − f₂|, the beat period 1/fᵦ and the average frequency (f₁+f₂)/2 — the standard tool for tuning instruments (violin double-stops, guitar harmonics, piano), radar mixing, ultrasonic testing and optical interference physics.

Enter the grating density (lines / mm), wavelength and diffraction order m; the tool applies d sin θ = m λ to return the diffraction angle θ — the standard formula for grating spectrometers, laser-diffraction experiments and spectrograph design.

Enter the seven Drake parameters (star formation rate R*, fraction with planets fp, habitable planets per star ne, fraction that develop life fl, that develop intelligence fi, that develop detectable comms fc, civilisation lifetime L); the tool returns N = R* · fp · ne · fl · fi · fc · L — the interactive form of the canonical SETI / astrobiology thought experiment.

Enter magnetic flux density B, current I, wire length L and the angle θ between current and field; the tool applies F = B I L sin θ to return the magnetic force — the workhorse formula for motor design, electromagnet sizing and Hall-effect experiments.

Enter inductance L and capacitance C (with optional resistance R); the tool computes the resonant frequency f₀ = 1/(2π·√(LC)) plus angular frequency ω₀, quality factor Q = (1/R)·√(L/C) and −3 dB bandwidth Δf — the foundational tool for oscillator, filter and radio-tuning design.

Enter observer eye-height (ship bridge, lighthouse, aircraft cruise altitude); the tool applies d ≈ √(2 · R · h) with a standard atmospheric-refraction correction (k ≈ 1.13) and returns the line-of-sight horizon distance in km / nautical miles / miles — fundamental for sailors, pilots, radio operators and landscape photographers.

Enter any number of resistor values; the tool returns the series total (R_total = Σ Rᵢ) and the parallel total (1 / R_total = Σ 1/Rᵢ) side by side, showing the reciprocal-sum intermediate — fundamental for circuit design, PCB layout, voltage dividers and current-limiting resistors.

Enter inductance L and current I; the tool applies E = ½ L I² to return the energy stored in the inductor's magnetic field in joules, millijoules and kilojoules — the core equation for switched-mode power supplies (SMPS), boost / buck converters, induction heating, transformer design and energy-recovery circuits.

Enter a bearing in degrees (0–360°) and the tool returns the standard 4-, 8-, 16- and 32-wind compass names (N, NE, ENE, etc.) plus the angular range each point covers — used in sailing, hiking navigation, radio communications and antenna pointing.

Enter daily mean temperatures (one or a series) and a base temperature (default 18 °C / 65 °F); the tool computes HDD = max(0, base − mean) and CDD = max(0, mean − base) per day and totals them — the international metric for heating/cooling energy estimation, HVAC sizing and agricultural growing-season analysis.

Enter your dive time, average depth, pressure consumed and tank volume; the tool returns the surface-equivalent SAC (L / min) and the bar-per-minute rate on that tank, then estimates how long the next dive can last at a chosen planning depth, start pressure and reserve — formulas from PADI, BSAC, DAN and the US Navy Diving Manual.

Enter usable battery capacity (kWh), starting SOC %, reserve SOC % and consumption (Wh/km or mi/kWh); the tool reports remaining range and projected SOC on arrival, with adjustments for highway speed, cold weather and cabin heating.

Enter the tank oxygen fraction (e.g. EAN32 = 0.32) and target PO₂ ceiling (1.4 ata recreational, 1.6 ata deco / contingency); the tool returns Maximum Operating Depth (MOD, m / ft) and Equivalent Air Depth (EAD) — formulas from PADI, TDI and the NOAA Diving Manual.

Enter air temperature (°C) and relative humidity (%); the tool applies the Magnus-Tetens equation (Alduchov-Eskridge 1996) to return how many grams of water vapour are actually in each cubic metre of air (absolute humidity, g/m³), plus dew point and vapour pressure — used for HVAC design, museums / instruments / wine cellars, mould / damp control and greenhouse / barn ventilation.

Enter two point charges (µC) and the separation (m); the tool applies F = k · q₁ · q₂ / r² and returns the electrostatic force magnitude (auto-formatted with SI prefixes — µN, mN, N, kN, MN) and the direction: like signs repel, opposite signs attract — useful for physics class, electronics, electrostatics demos and ESD estimation.

Enter a wind speed (m/s, km/h, mph or knots) and air density (default ICAO sea-level 15 °C standard atmosphere 1.225 kg/m³); the tool applies q = ½·ρ·v² to give the dynamic pressure (Pa, psf), multiplies by an exposed area to give the fully-blocked theoretical force (N, kgf, lbf) and labels the Beaufort scale — useful for signs, solar panels, sails, scaffolding and drone flight envelopes.

Choose a conductor material (copper, aluminium, silver, gold, tungsten, nichrome, stainless…), enter wire length and cross-sectional area (or AWG gauge); the tool applies R = ρ·L / A to give resistance (Ω) and adjusts the standard 20 °C resistivity for the operating temperature — useful for PCB traces, voltage drop, heating elements and transformer secondaries.

Enter the masses and temperatures (°C) of two streams of water; the tool applies conservation of heat — m₁·c·(T_f − T₁) + m₂·c·(T_f − T₂) = 0 — to give the equilibrium temperature T_f = (m₁·T₁ + m₂·T₂) / (m₁ + m₂), the combined mass and the heat (J / kJ) transferred — useful for coffee brew-water temperature, bath-tub mixing, lab and quick chemical-engineering estimates.

Enter your elevation (m or ft) and the original recipe baking temperature and time; the tool follows USDA / King Arthur Baking guidelines to recommend higher oven temperatures (+15–25 °F), shorter bake time (−5 to −15 %), less sugar (−2 to −3 Tbsp / cup), more liquid (+2 to +4 Tbsp / cup) and less leavening (−1/8 to −1/4 tsp baking powder) — and shows the boiling point of water at that elevation for reference.

Enter the supply voltage and PWM duty cycle (0–100 %); the tool returns the average voltage V_avg = D · V, the rectangular-pulse RMS voltage V_rms = V · √D, and can reverse-solve the duty needed for a target V_avg. Useful for Arduino LED dimming, motor speed control and heater PWM.

Enter latitude, longitude, date, local time and time-zone offset; the tool applies the NOAA solar-position equations to return the sun’s altitude and azimuth (0° = north, clockwise), along with solar declination, the equation of time and a sun-above-horizon flag — handy for photographers (golden hour), solar-panel siting and gardeners chasing the sun.

Enter the duct dimensions (round diameter or rectangular W × H) and the air velocity (FPM or m/s); the tool returns the volumetric flow Q = V × A in CFM, m³/h and L/s — useful for HVAC duct sizing, kitchen exhaust and fume-hood ventilation.

Enter your annual CO₂ footprint in tonnes or kg; the tool uses the US EPA / FAO estimate that one mature tree absorbs roughly 21–22 kg of CO₂ per year and returns the number of trees needed — plus the corresponding forest area for a typical planting density.

Enter battery capacity (mAh), pack voltage (cell count or V), safe-discharge limit (typically 80 %) and the average current draw in flight (A); the tool returns the theoretical flight time in minutes and notes the hover vs cruise difference — useful for FPV, aerial photography, agricultural spraying and inspection-drone planning.

Enter a lens focal length plus your sensor format (Full Frame, APS-C, Micro Four Thirds, 1-inch, phone 1/2.3-inch…) or a custom crop factor; the tool returns the 35mm-equivalent focal length and horizontal angle of view via FOV = 2·atan(w / 2f) — essential for comparing lenses across sensor sizes in photo and video.

Enter room area, ceiling height, climate zone (cold / temperate / mild) and insulation quality; the tool uses the US DOE Heating BTU rule of thumb (25–60 BTU/ft²) to return required heater output in BTU/hr and kW — useful for sizing oil, electric or heat-pump heaters and underfloor systems.

Enter water volume (L / gal), starting temperature, target temperature, heater wattage (kW) and efficiency; the tool uses Q = m × c × ΔT to compute heating time in hours, energy used (kWh) and electricity cost — useful for hot tubs, swimming pools and large aquariums.

Enter each hop addition’s weight (g), alpha-acid %, boil time (min), batch volume (L) and original gravity (OG); the tool applies the Glenn Tinseth bitterness equation (the homebrew de-facto standard) to return per-hop IBU contributions and the total IBU of the batch.

Enter the scale length, target pitch (note name) and string gauge (or unit weight); the tool applies Mersenne's law T = (2 L f)² × μ to return each string's tension in pounds-force and newtons — essential when switching string gauges, dropping tunings or designing multi-scale guitars.

Compute circular orbital speed (v = √(GM/r)) and orbital period for a satellite given the central body mass and orbital radius.

Compute the angular momentum L = Iω of a rigid body from moment of inertia I (kg·m²) and angular speed ω (rad/s or RPM); inline hints for common shape moments.

Enter power (W) and voltage (V) to read current in amps. Supports DC, single-phase AC (with power factor) and three-phase AC. Useful for sizing breakers, picking extension cords or working out the current draw of an appliance.

Enter the circuit amperage (15 / 20 / 30 A), voltage (120 / 230 V) and the total wattage of connected devices; using the NEC "continuous load ≤ 80 % of rated" rule the tool returns the remaining safe headroom in watts.

Enter LED strip W/metre, length and voltage (5 V / 12 V / 24 V); with a 20 % safety headroom the tool returns the required PSU wattage and amperage.

Enter air temperature (°C / °F) plus dew point (°C) or relative humidity (RH %); the tool returns the humidex value and its four-band comfort rating using Environment Canada's Masterton-Richardson formula.

Enter air temperature (°C / °F) and wind speed (km/h / mph); using NOAA / Environment Canada's joint 2001 wind-chill formula plus the Tikuisis 2003 frostbite-time model, returns the wind-chill (WCT) temperature, estimated time-to-frostbite for exposed skin and a 4-tier risk classification.

Enter cooling capacity (kW / BTU/h), daily runtime and sensible heat ratio (SHR); returns daily condensate generation in litres and US gallons per ASHRAE latent-heat methodology.

Enter starting and target circular orbit altitudes plus central body (Earth / Moon / Mars / Sun); using the vis-viva equation, returns the two Hohmann impulses Δv₁, Δv₂, total Δv and transfer time.

Enter load tension, rope-on-post friction coefficient and wrap angle (number of turns); using the Euler-Eytelwein capstan equation T_load / T_hold = e^(μθ), returns the holding force and mechanical advantage.

Enter 1–10 parallel resistor values (Ω / kΩ / MΩ); using 1/R_total = Σ(1/R_i) the tool instantly returns equivalent resistance, current split and power dissipation in each resistor.

Enter maximum sustained wind speed (mph / kph / m/s / knots); the tool returns the NOAA NHC Saffir-Simpson Hurricane Wind Scale category (Tropical Storm / Cat 1–5) with the official damage descriptor.

Enter the S-minus-P wave arrival-time difference (seconds) and typical crustal wave speeds to estimate the distance from your seismic station to the earthquake epicentre, in kilometres and miles.

Enter your base shutter speed and the ND filter strength (stops) to compute the long-exposure shutter time after the filter is added.

Use the GN = aperture × distance identity — lock any two of aperture, distance and guide number to solve for the third, with ISO and flash-power adjustments.

Enter the seconds between a lightning flash and the thunderclap (with air temperature) to find the storm distance in km, miles, metres and feet, with temperature-corrected speed of sound.

Enter altitude (m / ft) to find atmospheric pressure (hPa / psi / mmHg) and the % of sea-level pressure, using the ISA barometric formula.

Enter torque (lb·ft or N·m) and engine speed (RPM) to compute power via HP = torque × RPM ÷ 5252 (imperial) or kW = torque × ω (SI), with two-way HP ↔ kW conversion.

Look up Mohs hardness 1-10 with reference minerals (talc, quartz, diamond) plus household scratch references (fingernail, copper coin, glass, steel knife).

Enter estimated tornado wind speed (mph / km/h) to get the original Fujita (F0-F5) or Enhanced Fujita (EF0-EF5) category and corresponding damage description.

Enter date, longitude and time zone — use the Equation of Time to find when actual solar noon occurs relative to clock noon (12:00) on that day.

Enter liquid surface tension, density, contact angle and tube radius to get the capillary rise height (mm / m) using Jurin's law (h = 2σcosθ / ρgr).

Enter your guitar scale length and the tool returns the distance of every fret from the nut (mm and inches) using the equal-temperament Rule of 17.817 — for luthiers, repairs and DIY guitar builds.

Enter the basic reproduction number R₀ (or current Rₜ) and serial interval (days) to get daily growth rate, doubling time and 30-day case multiplier.

Enter PPFD (μmol·m⁻²·s⁻¹) and photoperiod hours — instantly compute DLI (mol·m⁻²·d⁻¹) and check it against the recommended range for low / medium / high / very-high light plants.

Enter capacitance C (F), resistance R (Ω), initial voltage V₀ and target voltage V to get the discharge time t = RC·ln(V₀/V), the time constant τ and the voltage at 1τ and 5τ.

Enter the number of disks, per-disk capacity and RAID level — get the *usable capacity*, *fault-tolerance* (how many disks can fail) and the *write-penalty multiplier* for NAS / server storage planning.

Enter the wheelbase and the maximum steer angle of the front wheels; the tool applies the Ackermann approximation r = wheelbase / sin(θ) to give the minimum turning radius and full turning-circle diameter.

Enter frequency, total path length and obstacle position to get the first Fresnel-zone radius and 60 % clearance needed — the staple link-planning check for Wi-Fi, microwave point-to-point and LoRa.

Enter latitude and use mode (year-round, summer-priority, winter-priority) to get the recommended fixed tilt angle for a solar panel plus optional seasonal-adjustment angles.

Enter electrode gap and pressure factor; the tool uses the 30 kV/cm rule of thumb plus Paschen-curve minimums to estimate dielectric breakdown voltage — handy for HV design, ESD review and lab safety sizing.

Type the 3-digit code (103 = 10 kΩ), 4-digit precision code (1002 = 10.0 kΩ) or letter-marked low-value code (R47 = 0.47 Ω) printed on an SMD chip resistor and instantly read the resistance and nearest E-series preferred value.

Enter rated DC output and efficiency (%) to see real wall-plug input wattage, heat loss, annual kWh and running cost — plus where it sits on the 80 PLUS Bronze / Gold / Platinum / Titanium ladder.

Enter any one of SEER (seasonal), EER (steady-state) or COP (coefficient of performance) — instantly get the other two plus an annual kWh estimate from your unit's cooling capacity and runtime.

Enter battery bank capacity (Wh), depth of discharge (DoD), daily load (Wh) and inverter efficiency — get the days your storage can run without any solar input.

Estimate stick (MMA), MIG and TIG welding current from metal thickness and electrode diameter using the industry rule of thumb "≈ 30–40 A per mm of thickness".

Enter true wind speed/direction plus your own (boat, bike or drone) speed and heading — get apparent wind speed and angle via vector addition.

Sound travels at 343 m/s — the farther the speaker, the larger the *delay*. Enter speaker distance and air temperature to get the ms delay for home-theatre or live-PA lip-sync setup.

Enter a mass and radial distance (or pick a preset: Earth, Sun, white dwarf, neutron star, black hole) to compute how much slower a clock runs deep in a gravity well versus far away, using the Schwarzschild metric.

Enter the starting paper thickness and number of folds n to see the 2ⁿ-fold thickness — from millimetres to the distance to the Moon — a classic exponential-growth visualisation.

Compute mass flow rate kg/s from fluid density, pipe inner diameter and average velocity — a core equation for HVAC, process piping and rocket-propulsion design.

Compute Darcy-Weisbach friction head loss h_f (metres of fluid) from pipe length, diameter, velocity and friction factor — the core formula behind pump sizing, fire-protection and HVAC pipe-network design.

Enter small piston force and area, plus the large piston area or diameter — get output force, mechanical advantage and stroke ratio from Pascal's principle.

Compute volumetric flow rate and throat velocity through a Venturi meter from inlet/throat diameters, pressure drop and fluid density (Bernoulli's equation).

Compute the inductance L = μ₀ μ_r N² A ⁄ (2π r) of an ideal toroidal coil from turns, mean radius, cross-section area and core relative permeability.

Enter telescope focal length, eyepiece focal length and eyepiece apparent FOV (AFOV) to get magnification and true FOV in degrees — compared against the Moon, Jupiter, M31 and other common targets.

Enter aperture, magnification and your eye-pupil diameter to get the exit pupil in mm and classify the result as empty magnification, planetary, deep-sky, rich-field — or "wasted light" when the exit pupil overflows your iris.

From latitude, date and local solar time, compute the sun's elevation (altitude) angle above the horizon and the corresponding zenith angle.

Convert any energy (joules, kWh, calories) to its TNT mass equivalent (grams, kg, tons, kilotons, megatons of TNT) — used to scale explosions, asteroid impacts and natural-disaster yields.

Compute the quality factor Q = f₀ / BW from resonant frequency and the −3 dB bandwidth — for RLC circuits, mechanical oscillators and acoustic systems. Also shows damping ratio ζ and ring-down time τ.

Enter a desired orbital period (minutes to days) and Kepler's third law (T² ∝ a³) returns the orbital semi-major axis and altitude above Earth's surface. Default 23 h 56 m 4 s (one sidereal day) gives the geostationary orbit (GEO) ≈ 35 786 km.

Enter significant wave height Hs and period T to compute the deep-water wave energy flux P = ρg²Hs²T / (64π) in kW per metre of wavefront — the standard metric for assessing wave-power site potential.

Enter speed and corner radius — get lateral acceleration a = v²/r expressed in g (Earth gravity), the figure used to gauge tyre grip and the side-load a driver or passenger feels.

Enter insulation R-value and inside-outside temperature difference to compute heat loss per square metre (Q = ΔT / R, W/m²) — compare before vs after upgrading insulation.

Enter a single cell’s nominal voltage, capacity (Ah) and mass, plus series count (S) and parallel count (P) — get the pack voltage, capacity, stored energy (Wh), energy density, theoretical runtime and per-cell C-rate. Works for Li-ion, LFP, LTO, NiMH and lead-acid.

Enter an indoor CO₂ reading in ppm (e.g. 800, 1500, 3000) — the tool classifies it against ASHRAE 62.1, NIOSH and EU indoor-air guidelines (fresh / acceptable / drowsy / warning / dangerous), with recommended ventilation actions and notes on cognitive impact.

Enter a sample bit depth (8 / 16 / 24 / 32 bit) and the tool returns the number of quantisation steps (2ⁿ), the theoretical dynamic range (SQNR = 6.02 n + 1.76 dB) and the corresponding signal-to-noise headroom — a core metric in recording, podcasting and mastering.

Enter density ρ (kg/m³) and sound speed c (m/s) — the tool returns the characteristic acoustic impedance Z = ρ·c. Enter two media to get the pressure and intensity reflection coefficients r and R, transmission T and transmission loss TL in dB — used in ultrasound, sonar, NDT, medical imaging and room acoustics.

Enter the span L (m), cable weight w (N/m) and horizontal tension H (N) — the tool returns the mid-span sag from the parabolic shortcut sag ≈ w·L² / (8·H) alongside the exact catenary a·(cosh(L/2a)−1). Used for overhead power lines, surveyor tape-sag correction, indoor cableways and pedestrian rope-bridge design.

Enter the octane ratings of two fuels (e.g. 87 + 93) and the volume of each (litres or gallons); the tool returns the volume-weighted blend octane and reverse-solves "how much premium do I add to reach a target octane?" — useful for tuned engines, motorsport and older vehicles requiring a specific octane.

Computes the wind force on a flat structure using F = ½·ρ·v²·C_d·A. Enter the wind speed (m/s, km/h, mph), exposed area and drag coefficient; the tool returns the dynamic pressure q, the total wind force F (newtons) and the equivalent kg-force — useful for sign-fixing, solar-panel mounting, fence design and basic structural sizing.

Compute the mean free path of an ideal gas: λ = kT / (√2 · π · d² · P), given temperature, pressure and molecular diameter. Used in vacuum science, thin-film deposition, low-pressure electronics design and semiconductor processing.

Enter arrow total weight (grain or g) and launch velocity (fps or m/s); the tool returns kinetic energy (KE = ½mv², ft·lbf and J) and momentum (p = mv, slug·ft/s and kg·m/s), then flags the suitable game class using Easton / IBO hunting energy thresholds (small game ≥ 25 ft·lbf, deer ≥ 42, elk / black bear ≥ 65, large / dangerous game ≥ 80).

Enter the genotypes of two parents (e.g. AA × Aa, Aa × Aa, Bb × bb); the tool returns the 4-cell Punnett square, offspring genotype ratios (AA : Aa : aa) and the dominant / recessive phenotype ratio — the canonical secondary-school biology tool, also useful for eye colour, hair colour and single-gene inheritance questions in cystic fibrosis, sickle-cell disease and similar conditions.

Enter two (or more) spring constants k and choose series or parallel; the tool returns the equivalent stiffness via 1/k = Σ 1/kᵢ or k = Σ kᵢ. Used in mechanics, suspension design and MEMS engineering.

Enter input voltage Vin and output voltage Vout (or work backwards from a dB value), and the tool applies G(dB) = 20·log₁₀(Vout / Vin) both directions — staple for audio, RF and instrumentation work, alongside power-ratio dB (10·log₁₀).

Enter initial state (P₁, V₁, T₁) and any two of the final-state values; the tool applies P₁V₁/T₁ = P₂V₂/T₂ to return the missing one — staple of high-school and early-university chemistry / physics.

Enter the inductance L (henries) and resistance R (ohms); the tool returns the LR time constant τ = L / R and the elapsed times to reach 63 %, 86 %, 95 % and 99 % of steady-state current — essential for electronics, relay design and transient analysis.

Enter inductance L (H) and resistance R (Ω); the tool returns the −3 dB cutoff frequency f = R/(2πL) and time constant τ = L/R for a first-order RL low- or high-pass filter — bread-and-butter electronics design.

Enter current I (amperes) and perpendicular distance r (metres); the tool returns the magnetic flux density around a long straight conductor using B = μ₀ I / (2π r) — bread-and-butter for motor, EMC and coil design.

Enter principal quantum numbers n₁ < n₂; the tool uses the Rydberg formula 1/λ = R_H (1/n₁² − 1/n₂²) to give the hydrogen spectral-line wavelength and its named series (Lyman, Balmer, Paschen…).

Enter pendulum length, bob mass, the maximum swing angle θ₀ and the current angle θ. The tool returns potential energy, kinetic energy and the total — via energy conservation E = mgL(1 − cos θ₀) and KE = E − PE.

Enter the absolute temperature T and molar mass M of a gas. The tool uses Maxwell–Boltzmann kinetic theory to compute the root-mean-square speed v_rms = √(3RT/M) — e.g. ≈ 515 m/s for room-temperature nitrogen.

Enter the position uncertainty Δx and the particle mass. The tool uses Δx·Δp ≥ ℏ/2 to return the minimum momentum uncertainty Δp_min and the corresponding velocity uncertainty Δv_min — with presets for electron, proton, neutron and more.

Enter the black-hole mass M (in solar masses or kilograms). The tool uses Hawking's 1974 result to compute the Hawking temperature T_H = ℏc³/(8πGMk_B), the Schwarzschild radius r_s and the total evaporation time t ≈ 5120πG²M³/(ℏc⁴) — a solar-mass black hole takes ~10⁶⁷ years to evaporate.

Enter the primary radius R and the primary/satellite density ratio. The tool returns the *rigid Roche limit* d ≈ 1.26·R·(ρ_M/ρ_m)^(1/3) and *fluid Roche limit* d ≈ 2.44·R·(ρ_M/ρ_m)^(1/3) — the distance at which a moon or comet is torn apart by tidal forces (Saturn's rings sit inside this distance).

Enter Secchi-disk transparency z_SD and a water-body type (ocean / lake / turbid). The tool estimates the *euphotic depth* z_eu = k · z_SD (ocean k ≈ 2.5, lake k ≈ 3.0, turbid k ≈ 2.0), the vertical attenuation coefficient K_d ≈ 1.44 / z_SD and the 10% / 0.1% light depths — the standard limnology / oceanography proxies for aquatic photosynthesis, dive visibility and aquaculture site suitability.

Enter supply voltage V_s, LED forward voltage V_LED and the desired LED current I_LED. The tool returns the required series resistor R = (V_s − V_LED)/I and its power dissipation P — essential for DIY electronics and Arduino projects.

Enter internal pressure p, mean diameter D and wall thickness t. The tool uses *thin-walled cylinder theory* (D/t > 20) to compute hoop (circumferential) stress σ_h = pD/(2t), longitudinal stress σ_l = pD/(4t) and the von Mises equivalent stress — a standard quick-check for boilers, pressure vessels and gas cylinders.

Enter any one of angular frequency ω (rad/s), frequency f (Hz) or period T (s) and the tool converts the other two via ω = 2πf and T = 1/f — the standard conversion for oscillators, AC circuits and waves.

Enter pipe internal diameter d and fluid velocity v. The tool applies the continuity equation Q = A·v to return volumetric flow rate Q in m³/s, L/min and US gpm — useful for pipes, irrigation, pumps and HVAC ducts.

Enter the object's physical diameter d and viewing distance D; the tool returns the angular diameter α = 2·arctan(d/(2D)), compares it with the small-angle approximation α ≈ d/D, and outputs degrees, arcminutes and arcseconds. Useful for astronomy (apparent size of Moon, planets), photography (subject framing) and signage design.

Enter the basic reproduction number R₀ (the average number of new infections per case in a fully susceptible population); the tool returns the herd-immunity threshold HIT = 1 − 1/R₀ — the share of the population that must be immune for an outbreak to die out — and optionally adjusts for vaccine effectiveness e via HIT/e.

Enter moles n, absolute temperature T and the initial/final volume (or pressure) ratio; the tool applies the first law to give an *isothermal* process's work W = nRT·ln(V₂/V₁), heat Q and ΔU = 0 — useful for thermodynamics, physical-chemistry and gas-compression problems.

Enter a cone half-angle θ (or its radius and distance); the tool returns the solid angle Ω = 2π(1 − cos θ) in steradians — used in photometry, radiometry, antenna patterns and field-of-view calculations.

Enter the sample's ¹⁴C/¹²C ratio as a percentage of the modern atmospheric standard; the tool returns the uncalibrated radiocarbon age t = −(t½/ln 2)·ln(N/N₀) using both the Libby (5 568 yr) and Cambridge (5 730 yr) half-life conventions.

Enter ligand concentration [L], half-maximal concentration EC50 (or IC50), Hill coefficient n and maximum response Emax; the tool returns the response E = Emax · [L]^n / (EC50^n + [L]^n), the fraction E / Emax, and the matching EC10 and EC90 — useful for pharmacological dose–response, enzyme kinetics and receptor-occupancy analysis.

Enter object distance dₒ and image distance dᵢ (or alternatively object and image heights); the tool returns the thin-lens lateral magnification m = −dᵢ ⁄ dₒ, and labels the image as upright / inverted, real / virtual, magnified / reduced — the standard tool for school optics, photography and microscopy.

Enter altitude; the tool uses the International Standard Atmosphere (ISA) model to return the matching pressure, temperature and air density — useful for altimeter settings, hiking and mountaineering, drone operation, HVAC design and weather analysis.

Enter the gas partial pressure P and Henry’s constant k_H; the tool returns the equilibrium gas solubility in liquid using both c = k_H · P (solubility form) and P = K_H · c (volatility form) — the standard model behind carbonated drinks, scuba decompression, ocean CO₂ uptake and biomedical oxygen transport.

Enter hydraulic radius, channel slope and Manning roughness n; the tool computes open-channel flow velocity V and discharge Q — the standard empirical hydraulics formula (Manning 1891) used in civil drainage, irrigation and storm-water design.

Enter the SPL L₁ at a reference distance r₁, the new distance r₂ and the source type (point or line). The tool returns L₂ = L₁ − 20·log₁₀(r₂/r₁) for point sources or L₂ = L₁ − 10·log₁₀(r₂/r₁) for line sources — the standard model for concerts, factories, traffic-noise estimates.

Enter rotor diameter, rotor speed (rpm) and wind speed; the tool returns TSR = ω·R / v and benchmarks it against the 6–8 sweet spot where 3-blade horizontal-axis wind turbines reach peak power coefficient Cp.

Enter a base temperature (commonly 18 °C / 65 °F) and the day's mean outdoor temperature; the tool returns daily HDD = max(base − T_avg, 0) and CDD = max(T_avg − base, 0) — the standard climatology metrics for sizing heating, cooling and energy use in buildings.

Enter bandwidth B (Hz) and signal-to-noise ratio S/N (dB or ratio); the tool applies the Shannon-Hartley theorem *C = B · log₂(1 + S/N)* to return the theoretical maximum data rate (bit/s) of an AWGN channel — the upper bound used in Wi-Fi, 5G, satellite and modem link-budget design.

Enter two sidereal periods T₁ and T₂; the tool returns the synodic period T_syn = 1 / |1/T₁ − 1/T₂| — the time between successive conjunctions or oppositions. Underlies observing windows, Mars-launch opportunities and lunar-solar calendar sync.

Enter the highest signal frequency f_max; the tool returns the Nyquist rate f_s ≥ 2·f_max and the corresponding folding frequency — the Shannon-Nyquist sampling theorem cornerstone, underlying every ADC, audio, wireless, radar and medical-imaging design.

Enter target distance D and target width W; the tool applies Fitts's Law T = a + b·log₂(D/W + 1) to return the movement time and the Index of Difficulty (ID) — the standard UX / HCI model for comparing button size and placement.

Enter the number of choices n plus empirical constants a and b; the tool applies Hick's Law T = a + b·log₂(n + 1) to return the user's decision time — used in UX to compare menu length, controller layout and option groupings.

Enter the initial pressure P₁, initial temperature T₁ (K) and final temperature T₂; the tool applies Gay-Lussac's Law P₁/T₁ = P₂/T₂ (at constant volume and amount of gas) to solve for the final pressure P₂ — the basic gas law behind pressure cookers, tyre pressure swings between summer and winter, and the safety of stored gas cylinders.

Enter a galactic distance d (Mpc) and the Hubble constant H₀ (km/s/Mpc, default 70); the tool applies Hubble's Law v = H₀·d to return the recession velocity v and the approximate Hubble time t_H = 1/H₀ (a rough upper bound on the age of the universe) — the most basic velocity–distance relation in observational cosmology.

Enter two latitude / longitude pairs; the tool applies Vincenty's 1975 *inverse* formula on the WGS-84 ellipsoid, iteratively solving for the geodesic distance (≈ 0.5 mm precision), initial bearing and final bearing — hundreds of times more accurate than the spherical Haversine (which errs by up to 0.5 %), and the GIS, marine and aviation industry standard.

Enter calibrated airspeed (CAS, knots), pressure altitude (feet) and OAT deviation from ISA (°C); the tool applies the ICAO Standard Atmosphere together with the *compressible impact-pressure* equation (FAA AC 91-79A) to return true airspeed (TAS), Mach number, equivalent airspeed (EAS) and live atmospheric pressure / density / speed of sound — required for pilot navigation (wind correction, leg time, fuel planning), as TAS can exceed CAS by 50 % or more at high altitude.

Enter Cp and Cv (or pick a gas type — monatomic / diatomic / polyatomic) and the tool returns the *adiabatic index* γ = Cp / Cv — a base parameter for adiabatic processes, sound speed, Carnot cycles and shock-wave calculations (ideal monatomic γ = 5/3, diatomic γ = 7/5, air ≈ 1.40).

Enter the incident irradiance I₀ and the angle θ between the surface normal and the light ray; the tool applies *Lambert's cosine law* I = I₀ · cos θ — the base formula for solar-panel tilt design, photographic metering, computer graphics diffuse shading and laser-spot power-density calculations.

Enter the number of *components C* and *phases P* in a system; the tool applies *Gibbs' phase rule F = C − P + 2* to return the *degrees of freedom F* (independent intensive variables), with optional constraints for fixed-pressure (−1) and fixed-temperature (−1) systems — a cornerstone of chemical thermodynamics, alloy phase-diagram reading and geology.

Enter the stoichiometry of a slightly-soluble salt MₐXᵦ and its molar solubility s; the tool returns the *solubility product Kₛₚ = (a·s)^a · (b·s)^b*, or solves for s given Kₛₚ — the standard calculation for precipitation forecasting, ionic-equilibrium labs and qualitative-analysis cation-separation schemes.

Enter the relative speed v (or β = v/c) and the *proper time Δτ*; the tool applies the *Lorentz factor γ = 1/√(1 − β²)* to return the dilated observer-frame time *Δt = γ · Δτ*, or solves the inverse for Δτ — the workhorse equation of Einstein's 1905 special relativity and the basis of GPS clock corrections, cosmic-ray muon survival and accelerator beam-time bookkeeping.

Enter the magnetic flux density B₀ (tesla, millitesla or gauss) and a nuclide (¹H, ¹³C, ¹⁹F, ³¹P, ²³Na, electron…); the tool applies *ω = γ · B₀* and *f = γ ⁄ (2π) · B₀* to return the Larmor precession angular frequency, resonance frequency (Hz / MHz / GHz), period, and free-space RF wavelength — the workhorse calculation for MRI RF-coil tuning, NMR carrier-frequency setup and ESR X-band / Q-band experiments.

Enter a flywheel’s mass, radius, shape and speed (RPM); the tool uses E = ½ I ω² (I = k·m·r²) to compute the stored rotational kinetic energy in joules, kilojoules and watt-hours.

Converts a radiation dose (in microsieverts) into an intuitive "number of bananas", using the conventional reference of ~0.1 µSv per banana, to make everyday doses easier to picture.

Enter a speed and curve radius to find the ideal road or track banking angle that lets a vehicle round the bend without relying on friction.

Compute a material's Poisson's ratio from its lateral and axial strains (or original and changed dimensions).

Enter water volume, start and target temperature, heater power and efficiency to get the energy needed and the heating time — e.g. how long a kettle takes to boil.

Use Graham's law to find the ratio of effusion (or diffusion) rates of two gases from their molar masses — the lighter gas effuses faster.

From bar diameter (or standard size) and total length, compute the unit weight and total weight of reinforcing steel.

Count a cricket's chirps per minute and estimate the air temperature (°C and °F) using Dolbear's law.

From cruise altitude, target altitude, ground speed and descent angle, find the top-of-descent distance, required descent rate and descent time.

From a telescope's aperture, compute the Dawes' limit (smallest resolvable angular separation) to judge whether close double stars can be split.

Compute milling feed rate (mm/min or IPM) from spindle RPM, number of flutes and chip load per tooth.

Find the true average speed of a there-and-back trip when the outbound and return speeds differ — the harmonic mean, not the naive average of the two speeds.

Enter the code printed on a ceramic capacitor (e.g. 104, 223, 4R7) to get its capacitance in pF / nF / µF and tolerance.

Convert a measured NTC thermistor resistance into temperature using the Beta (B-parameter) equation.

Compute the capacitor and inductor values for a passive speaker crossover from the crossover frequency and driver impedance.

Calculate the diffraction-limited angular resolution and Airy disk size of an optical system from aperture and wavelength.

Use the NPF rule to compute the longest star-trail-free exposure from aperture, pixel pitch and focal length.

Compute the spring rate k of a helical compression spring from wire diameter, mean coil diameter, number of active coils and the material shear modulus.

From the hang angle and body weight, compute the tension on each hammock anchor and check it against the 30-degree rule of thumb.

Compute the internal volume (litres / gallons) and water mass held by a round pipe from its inside diameter and length.

When an extension bar changes the effective length of a torque wrench, find the dial setting so the fastener still receives the target torque.

Compare two driving speeds over the same distance to see how much time you actually save.

From buffer size in samples and sample rate, compute the one-way and round-trip monitoring latency of an audio interface in milliseconds.

Compute an AC motor’s synchronous speed from supply frequency and pole count, then the actual rotor speed after slip.

Estimate cruising engine RPM from road speed, tyre diameter, transmission gear ratio and final-drive (axle) ratio.

Compute the equivalent capacitance of multiple capacitors wired in series or in parallel.

Convert between relative centrifugal force (RCF / ×g) and rotor speed (RPM) for a given rotor radius.

Use the Sunny 16 rule to get the shutter speed and exposure value (EV) for a given ISO, lighting condition and chosen aperture.

From speed and curve radius, compute the ideal banking angle θ = arctan(v²/(r·g)) that needs no friction to hold the turn.

Compute open-channel discharge Q over a rectangular or V-notch weir from the head of water above the crest.

Convert cutting speed (surface speed) and tool or workpiece diameter into the spindle RPM you need on a lathe or mill, in metric or imperial.

Look up the safe current rating for a copper or aluminium conductor by AWG size and insulation temperature rating, or find the smallest gauge for a target load.

Compute an arrow’s FOC (front-of-center) percentage from its total length and balance point.

Find a hydraulic cylinder’s push (extend) and pull (retract) force from pressure, bore and rod diameter.

Calculate the torque needed to drive a load along a lead screw from load, lead, diameter and friction.

Estimate a rolling bearing’s L10 basic rating life from dynamic load rating, equivalent load and speed.

Relate valve flow coefficient Cv, pressure drop and specific gravity to liquid flow rate (or the Cv needed).

Estimate a firearm’s free recoil velocity and recoil energy (joules and ft·lbf) from bullet weight, muzzle velocity, powder charge and gun weight.

Use the Greenhill formula to estimate the barrel twist rate (1:N inches) needed to stabilise a bullet from its diameter, length and velocity.

Compute the total impedance (ohms) of multiple speakers wired in series or parallel and check it matches your amplifier.

Estimate how long a tank takes to drain through an orifice from its cross-section, hole area and water height using Torricelli’s law.

Compute conduit fill percentage from conduit inner diameter and conductor size, checking against the NEC 40% / 31% / 53% limits.

Compute an astrophotography image scale (arcsec per mm and per pixel) plus field of view from focal length and camera pixel size.

Compute a boat propeller’s theoretical speed and slip percentage from pitch, engine RPM, gear ratio and actual GPS speed.

Compute the bellows extension exposure factor, stops of compensation and magnification for macro photography from focal length and extension.

Compute a sailboat’s sail area–displacement ratio (SA/D) from sail area and displacement to gauge how powered-up it is.

Use the Joukowsky equation to estimate the water-hammer pressure rise from a change in flow velocity, fluid density and pressure-wave speed.

Estimate coaxial cable signal attenuation (dB) and remaining power from cable type, length and signal frequency.

Estimate arrow launch speed (fps) by adjusting a bow’s IBO speed rating for actual draw length, draw weight and arrow weight.

Compute the steps-per-mm firmware value from motor steps, microstepping and belt or leadscrew parameters.

Turn a measured chain length over a number of links into a wear percentage and compare it against replacement thresholds.

Compute a LiPo battery maximum continuous discharge current, power and safe charge current from its capacity and C-rating.

Estimate the faintest star magnitude a telescope can theoretically show from its aperture.

Compute the X/Y coordinates and adjacent-hole spacing for evenly spaced bolt holes on a bolt circle.

Find a spur gear's pitch diameter, outside diameter, tooth height and centre distance from its module and tooth count.

Use the mirror equation to find image distance, magnification and image type for a concave or convex mirror from its focal length (or radius) and object distance.

Estimate the peak-to-peak ripple voltage across a rectifier smoothing capacitor from load current, capacitance and rectification frequency.

Find a semiconductor junction temperature from power dissipation, ambient temperature and the thermal resistances, plus the heat-sink rating you need.

Size the series resistor and component power ratings for a Zener diode voltage regulator from the input voltage, Zener voltage and load current.

Snap any target resistance to the nearest standard E12/E24/E96 resistor value and show the percentage error.

Find the gauge block stack height needed to set a sine bar to a target angle (or the angle from a given stack height).

Use Peukert's law to find a battery's real runtime and effective delivered capacity from its rated capacity, Peukert exponent and actual discharge current.

Find the axial standing-wave (room mode) frequencies from a room's length, width and height to spot bass resonances and problem frequencies.

Compute the steady flow rate of liquid discharging through an orifice from its diameter, upstream head and discharge coefficient.

Convert a star’s annual parallax angle (arcseconds or milliarcseconds) into distance in parsecs, light-years and astronomical units.

Compute the internal volume and system resonance of a sealed subwoofer enclosure from the driver’s Thiele-Small parameters (Vas, Qts) and a target Qtc.

Find how much a solution’s boiling point rises from the molality, van’t Hoff factor and ebullioscopic constant.

Compute the Hall voltage across a conductor or semiconductor from current, magnetic flux density, carrier density and thickness.

Compute the average drift velocity of charge carriers from current, cross-sectional area and carrier density.

From feed flow, recovery and salt rejection, compute an RO system’s permeate and concentrate flow, concentration factor and permeate/concentrate TDS.

From a starting concentration, per-step dilution factor and number of steps, find the final concentration, total dilution factor and the stock/diluent volume per step.

Find the unknown molar concentration of an acid or base from the titrant concentration, the two volumes and the mole ratio.

From target distance and point-of-impact offset, compute the MOA, MIL and number of scope clicks needed to zero in.

From box volume, tuning frequency and port diameter, compute the port length needed for a ported (bass-reflex) subwoofer enclosure.

From input and output voltage, compute the ideal duty cycle of a buck or boost DC-DC converter.

From each element’s mass percentage and atomic mass, find a compound’s simplest whole-number empirical formula.

Estimate a zipline rider's arrival speed and travel time from the start–end height drop, cable length and friction loss using energy conservation.

Estimate a finished beer's SRM and EBC colour from your malt colours (°L), weights and batch volume using the Morey equation.

Estimate how far an e-bike can travel on one charge from its battery voltage, capacity and energy use per kilometre.

Work out required fuel flow and injector duty cycle (%) from engine power, injector flow rating, cylinder count and BSFC.

Find the wind correction (crab) angle a plane must hold and the resulting ground speed from true airspeed, wind and desired course.

Compute the theoretical and real output power of a micro-hydro turbine from the net head, flow rate and system efficiency.

Convert apparent power (kVA) and voltage into the line current in amps for single-phase or three-phase systems.

Estimate the pull force of a solenoid or electromagnet from turns, current, core area and air gap.

Work out a pump’s net positive suction head available (NPSHa) from surface pressure, vapour pressure, fluid density, static head and friction losses.

From dry-bulb temperature, relative humidity and pressure, compute the humidity ratio and enthalpy of moist air per kilogram of dry air.

Compute the full-load current of a single- or three-phase electric motor from its horsepower or kW, voltage, efficiency and power factor.

Compute the skin depth of an alternating current in a conductor from frequency, resistivity and relative permeability.

Find the impulse from a mass and change in velocity, or the change in momentum from a force and the time it acts.

Estimate how long a compressor takes to fill a receiver tank from its volume, the start and target pressure, and the compressor’s free-air delivery.

Convert between standard (SCFM) and actual (ACFM) compressed-air flow, correcting for pressure and temperature.

Compute the characteristic impedance, effective permittivity and propagation delay of a PCB microstrip trace from its width, dielectric height and permittivity.

Compute a coaxial cable’s characteristic impedance, velocity factor and per-length capacitance and inductance from the inner and outer conductor diameters and the dielectric constant.

Work out a fiber link’s total loss and power margin from transmit power, receiver sensitivity, fiber attenuation, connector and splice losses.

Compute the characteristic impedance and propagation delay of a buried PCB stripline trace from its width, ground-plane spacing and dielectric constant.

Use the Shannon–Hartley theorem to get a channel’s maximum error-free bit rate and spectral efficiency from its bandwidth and signal-to-noise ratio.

Compute the differential and single-ended impedance of an edge-coupled PCB differential pair from trace width, spacing, dielectric height and permittivity.

Compute the characteristic impedance and effective permittivity of a coplanar waveguide (CPWG) from its trace width, gap to the side ground planes and dielectric constant.

Compute the resistor values for Pi- and T-type resistive attenuators from the attenuation in dB and the system impedance.

Use the Friis formula to compute the total noise figure and total gain of a receiver chain from each stage’s gain and noise figure.

From an ADC’s bit depth and reference voltage, compute the quantization step (LSB), theoretical SNR, dynamic range and voltage per bit.

Find the full-power bandwidth from an op-amp slew rate and output swing, or the slew rate needed for a target frequency.

From sample rate and FFT size, compute the frequency resolution, each bin frequency, record length and Nyquist frequency.

Compute the differential characteristic impedance of a twisted pair from wire diameter, centre spacing and insulation permittivity.

Compute the cutoff frequency of each TE/TM mode of a rectangular waveguide from its inner dimensions and dielectric, highlighting the dominant TE10 mode.

Compute the RMS thermal (Johnson–Nyquist) noise voltage, noise power and noise current from resistance, temperature and bandwidth.

Find the output and input third-order intercept points (OIP3 / IIP3) from the fundamental-to-IM3 power difference in a two-tone test.

Compute the characteristic impedance of a quarter-wave matching line between two impedances, plus its physical length at a given frequency.

Compute the maximum detection range of a monostatic radar from transmit power, antenna gain, wavelength, target radar cross-section and minimum detectable signal.

Compute the far-field (Fraunhofer) boundary distance R = 2D²/λ from an antenna aperture and frequency to check whether a measurement is in the far field.

Compute an optical fiber’s numerical aperture (NA), maximum acceptance half-angle and full acceptance angle from the core and cladding refractive indices.

Compute the aliased (folded) frequency that appears after sampling a signal, given the sample rate and input frequency, and flag when the Nyquist criterion is violated.

Estimate the transmission bandwidth of an FM signal with Carson’s rule from the peak frequency deviation and the highest modulating frequency, plus the modulation index.

From the modulation order M and symbol rate (baud), compute bits per symbol, bit rate and the required Nyquist bandwidth.

Compute a radar's maximum unambiguous range and unambiguous velocity from the pulse repetition frequency (PRF) to check for range and Doppler ambiguities.

From the RF signal and local-oscillator (LO) frequencies, compute a superheterodyne receiver’s intermediate frequency (IF) and image frequency to plan image rejection.

Compute the theoretical bit error rate (BER) of BPSK/QPSK from the energy-per-bit to noise ratio Eb/N0, plus the equivalent SNR.

Compute a radar's range resolution — the smallest separation of two targets it can distinguish — from the pulse width or signal bandwidth, ΔR = c/(2B).

Compute an antenna's effective aperture Ae = Gλ²/(4π) from its gain and frequency, relating gain to physical capture area.

Convert between noise figure (NF, dB) and equivalent noise temperature (Te, K) for LNA and receiver-system design.

Use the Friis transmission equation to compute free-space received power from transmit power, antenna gains, frequency and distance.

Use the Kraus approximation to estimate antenna directivity/gain (dBi) from the two principal-plane half-power (3 dB) beamwidths, or invert to the required beamwidth.

Compute the two-way Doppler shift from a radar’s transmit frequency and a target’s radial velocity, or invert a measured shift back to velocity.

Compute the quarter-wave line impedance, isolation resistor and branch length for an equal-split two-way Wilkinson power divider.

Compute the Q factor, series/shunt reactances and matching inductor/capacitor values for an L-network that matches a source resistance to a load.

Convert integrated single-sideband phase noise (dBc) and a carrier frequency into RMS phase jitter and RMS time jitter.

Compute the minimum Butterworth filter order required from passband/stopband attenuation and frequency specifications.

Compute the crest factor and peak-to-average power ratio (PAPR, dB) from a signal’s peak and RMS values, or peak and average power.

Compute link margin from received power and receiver sensitivity, or the extra path loss allowed for a target margin.

Compute an ADC’s effective number of bits (ENOB) from a measured SINAD, and invert to the equivalent ideal SNR.

Compute a receive system’s G/T figure of merit (dB/K) from antenna gain and system noise temperature.

Compute the processing gain of a direct-sequence spread-spectrum (DSSS) link from chip rate and data rate, and the resulting jamming margin.

Compute the polarization loss factor (PLF) and dB loss between two linearly polarized antennas from their polarization tilt-angle mismatch.

Compute the equivalent noise bandwidth (ENBW) of a Butterworth filter from its −3 dB bandwidth and order.

From a complex load impedance R + jX and reference impedance Z0, compute the complex reflection coefficient Γ (magnitude and phase), VSWR, return loss and mismatch loss.

Convert between input- and output-referred 1 dB compression points using an amplifier’s small-signal gain, and estimate the output back-off at a given operating level.

From a second-order phase-locked loop’s natural frequency and damping factor, compute the −3 dB loop bandwidth, equivalent noise bandwidth and approximate lock time.

From a varactor’s zero-bias capacitance, junction potential and grading coefficient, compute the capacitance at any reverse bias and the resulting tuning ratio.

From a pulse-Doppler radar's PRF and transmit frequency, compute the maximum unambiguous radial velocity and blind speed set by the Nyquist limit on the Doppler shift.

Compute a receiver's sensitivity and minimum detectable signal from the thermal noise floor (kTB), noise figure and required signal-to-noise ratio.

Compute the load capacitance a quartz crystal sees from the two external load capacitors and stray capacitance, and size the capacitors for a target CL.

From a receiver's input third-order intercept point and noise floor, compute the spurious-free dynamic range and the minimum detectable signal.

From a line's physical length, frequency and velocity factor, compute the electrical length in wavelengths and degrees plus the phase delay.

From the reference clock, phase-accumulator bit width and frequency tuning word, compute a direct digital synthesis (DDS) output frequency, frequency resolution and quantisation error.

From the round-trip time of a reflected pulse and the cable velocity factor, compute the distance to an open or short fault located by time-domain reflectometry (TDR).

From the input/output voltages, switching frequency and inductance, compute the peak-to-peak inductor ripple current and ripple ratio of a buck converter.

From the envelope maximum/minimum or the carrier and message amplitudes, compute the amplitude-modulation (AM) index, percentage modulation and sideband-to-total power fraction.

Convert between dBµV, dBm, µV and dBmV at a chosen system impedance for EMC work and receiver-sensitivity calculations.

Convert between Eb/N0 and the in-band signal-to-noise ratio (SNR) of a receiver using the bit rate and bandwidth (spectral efficiency).

Compute the pulse compression ratio τ·B, compressed pulse width, processing gain and range resolution from pulse width and bandwidth.

Compute the overall input third-order intercept point (IIP3) of a multi-stage RF chain from each stage gain and IIP3.

Compute a device noise figure and equivalent noise temperature from the hot/cold noise power ratio (Y-factor) and the noise-source ENR.

Compute group delay τg = −Δφ/Δω from the phase change across a frequency span, with phase-delay conversion.

Compute the image rejection ratio (IRR, dB) of a quadrature receiver from I/Q gain imbalance and phase error.

Compute the SNR improvement from integrating N radar pulses: coherent is exactly 10·log10 N, non-coherent lies between the √N and N bounds.

Compute the phase velocity, group velocity and guide wavelength of a rectangular waveguide from the operating and cutoff frequencies.

Compute the radar cross section (RCS, m² and dBsm) of canonical shapes — sphere, flat plate, cylinder and dihedral — from their dimensions and wavelength.

Compute pulse broadening and the dispersion-limited maximum bit rate from the dispersion parameter D, fiber length and source spectral width.

Compute duty cycle, average power and per-pulse energy from pulse width, pulse repetition frequency and peak power.

Convert error vector magnitude (EVM) to the equivalent signal-to-noise ratio (SNR) and modulation error ratio (MER), or back again.

Compute the occupied bandwidth and bandwidth efficiency of a raised-cosine / root-raised-cosine shaped signal from the symbol rate and roll-off factor.

Compute photodiode responsivity from wavelength and quantum efficiency, and the photocurrent from incident optical power.

Compute the divergence angle, Rayleigh range and spot size at any distance of a Gaussian laser beam from its wavelength and waist radius.

Compute the required element phase shift, grating-lobe onset and maximum grating-lobe-free scan angle of a phased array from element spacing, wavelength and steering angle.

Compute the shot-noise RMS current i = √(2qIB) from a DC current and bandwidth, with noise power and equivalent-resistance conversions.

Compute the optical link power penalty 10·log10((ER+1)/(ER−1)) dB from the extinction ratio ER = P1/P0, plus the one/zero levels relative to average power.

From coupling and directivity, compute isolation, mainline insertion loss and each port level: Isolation = Coupling + Directivity.

Use the Chu–Harrington limit to compute the minimum quality factor Q = 1/(ka)³ + 1/(ka) of an electrically small antenna and estimate fractional bandwidth ≈ 1/Q.

List the second-, third- and higher-order intermodulation products (2f1−f2, 2f2−f1, …) and harmonic frequencies from two input tones f1 and f2.

Compute a pulsed radar’s minimum detectable range R_min = c·(τ + t_rec)/2 from pulse width and receiver recovery time — the blind zone during transmission.

Compute an op-amp’s closed-loop −3 dB bandwidth f = GBW / A_cl from its gain–bandwidth product and the closed-loop gain.

Compute a parabolic dish antenna’s gain G = η·(π·D·f/c)² in dBi and its half-power beamwidth from diameter, frequency and aperture efficiency.

Compute far-field power density S = EIRP/(4π·d²) and electric field strength E = √(30·P·G)/d from transmit power, antenna gain and distance.

Compute a quartz crystal’s quality factor Q = 1/(2π·f·R₁·C₁) from its motional resistance R₁, motional capacitance C₁ and series-resonant frequency.

Compute the AC-to-DC resistance ratio R_ac/R_dc of a round conductor due to skin effect from frequency, wire diameter and material.

Compute a voltage-controlled oscillator’s tuning sensitivity Kv = Δf/ΔV (in Hz/V, MHz/V and rad/s/V) from its frequency range and tuning-voltage range.

Compute a phase-locked-loop phase-detector gain Kd — charge-pump Kd = I_cp/(2π) A/rad, plus XOR/multiplier and PFD detector types.

Compute the EMC antenna factor AF = 20·log₁₀(f) − G − 29.79 dB/m from frequency and gain, and recover field strength from the measured receiver voltage.

Compute a cascaded integrator-comb (CIC) decimation filter’s DC gain (R·M)^N, bit growth N·log₂(R·M) and required register word length.

Compute an amplifier’s Rollett stability factor K and |Δ| from two-port S-parameter magnitudes to test for unconditional stability (K>1 and |Δ|<1).

Compute a power amplifier’s power-added efficiency (PAE), drain efficiency and gain from its RF output power, input drive power and DC supply power.

Compute the jitter-limited SNR of an ADC, SNR = −20·log₁₀(2π·f·t_jitter), from the input signal frequency and RMS clock/aperture jitter.

Estimate VHF/UHF mobile radio path loss (dB) for urban, suburban and open areas with the empirical Okumura–Hata model from frequency, base/mobile antenna heights and distance.

Compute the path loss, received power and crossover distance of the two-ray ground-reflection propagation model from antenna heights and link distance.

Compute the Fresnel diffraction parameter v and single knife-edge diffraction loss (ITU-R P.526 approximation) from obstacle height, frequency and the two path-leg distances.

Compute a wireless channel's coherence bandwidth, flat/frequency-selective fading threshold and maximum ISI-free symbol rate from the multipath RMS delay spread.

Compute the DC average voltage, RMS voltage, ripple factor, form factor and peak inverse voltage (PIV) of a half-wave or full-wave rectifier from the peak input voltage.

Convert between a PT100/PT1000 platinum RTD's resistance and temperature using the Callendar–Van Dusen equation.

Compute the maximum Doppler shift, Doppler spread and channel coherence time from mobile speed and carrier frequency.

Compute percent overshoot, peak time, rise time and settling time of a second-order system from its damping ratio ζ and undamped natural frequency ωₙ.

Compute total harmonic distortion (THD-F and THD-R, in percent and dB) from the fundamental and individual harmonic amplitudes.

Compute the reflectance, transmittance and reflection loss at a normal-incidence optical interface from the refractive indices of the two media.

Compute the coherence length and coherence time of a laser from its linewidth, given in either frequency or wavelength.

Compute the free spectral range, longitudinal-mode spacing, finesse and cavity linewidth of a Fabry–Pérot / laser cavity from its length, refractive index and mirror reflectivity.

Compare the equivalent noise bandwidth, coherent gain, scalloping loss and highest side-lobe level of common FFT window functions (rectangular, Hann, Hamming, Blackman and more).

Convert an NTC thermistor resistance to temperature (and back) with the three-coefficient Steinhart–Hart equation — more accurate than the single-parameter beta model.

Compute the thermal Doppler broadening (FWHM) of a gas emission or absorption spectral line from temperature and particle mass.

Compute the relativistic Doppler shift of light at any observation angle θ — observed wavelength, frequency, redshift z and the transverse (pure time-dilation) case — from the source speed β = v/c.

Estimate the number of taps and the order of an FIR digital filter from the sample rate, transition bandwidth and stopband attenuation using the Kaiser, fred harris and Bellanger formulas.

Compute the electron plasma frequency (angular and in hertz), plasma wavelength and collisionless skin depth from electron density, and the critical density for a given frequency.

Apply the magnetic Ohm’s law to compute reluctance, permeance, magnetomotive force, flux and flux density from a magnetic circuit’s path length, area, relative permeability, turns and current.

Compute the first higher-order (TE11) mode cutoff frequency of a coaxial cable from the inner/outer conductor diameters and dielectric constant — the highest frequency at which the cable stays single-mode TEM.

Compute the Debye shielding length of a plasma and the number of particles inside a Debye sphere from the electron temperature and density.

Use Albersheim’s empirical equation to estimate the single-pulse signal-to-noise ratio required for a target detection probability Pd and false-alarm probability Pfa over N integrated pulses.

Compute the cyclotron frequency, angular frequency and period of a charged particle orbiting in a uniform magnetic field from the flux density and the particle charge/mass.

From the ultimate gain Ku and ultimate oscillation period Tu, compute the Kp, Ki and Kd gains for P / PI / PID controllers using the Ziegler–Nichols closed-loop rules.

Estimate the gain (dBi) and effective aperture of a pyramidal horn antenna from its aperture dimensions, frequency and aperture efficiency.

Use the Shockley diode equation to compute the current through a pn-junction diode at a given voltage from the saturation current, ideality factor and temperature.

Estimate the loss density of a magnetic core with the Steinmetz equation P = k·f^α·B^β from the frequency, peak flux density and material coefficients.

Estimate the EMF voltage produced by a thermocouple from the hot/cold junction temperatures and the Seebeck coefficient of the thermocouple type.

Compute a Wien bridge oscillator’s frequency of oscillation, the amplifier gain needed to sustain it, and the required Rf/Rg resistor ratio from the RC network.

Compute the minimum Chebyshev (Type I) filter order required from passband ripple, stopband attenuation and frequency specifications.

Compute the cooling power, electrical input power and coefficient of performance (COP) of a Peltier (thermoelectric) module from its Seebeck coefficient, resistance, thermal conductance, drive current and cold/hot side temperatures.

Compute a Colpitts LC oscillator’s frequency of oscillation, effective series capacitance and feedback ratio from the inductor and the two divider capacitors.

Compute a Hartley LC oscillator’s frequency of oscillation, total tank inductance and feedback ratio from the two tapped inductors and the tuning capacitor.

Use the Curie–Weiss law χ = C/(T − θ) to compute the magnetic susceptibility of a paramagnet (and its inverse) from the Curie constant, temperature and Weiss temperature.

Compute the cutoff frequency and quality factor Q of a unity-gain Sallen–Key second-order low-pass filter from its two resistors and two capacitors.

Compute the centre notch frequency of a balanced Twin-T notch filter from its R and C values.

Compute a BJT common-emitter amplifier’s transconductance and small-signal voltage gain from the collector current, load resistance and any emitter degeneration.

Estimate a power MOSFET’s switching loss and average power dissipation from the drain voltage, current, rise/fall times and switching frequency.

Size the damping resistor, capacitor and resistor power dissipation of an RC snubber for a switching node from the ringing frequency and stray capacitance.

Compute the mutual inductance M and the series-aiding and series-opposing total inductance of two magnetically coupled coils from their self-inductances and coupling coefficient.

Size the capacitor kVAR and capacitance (µF) needed to raise a single-phase load’s power factor to a target, and see the reduction in apparent power and line current.

Compute the output pulse width of a 555 timer in monostable (one-shot) mode from the timing resistor and capacitor.

Find the maximum power delivered to a load, the matching load resistance and the transfer efficiency from a source’s Thévenin voltage and internal resistance.

Convert a three-resistor network between its wye (star, Y) and delta (Δ) equivalents to simplify circuit analysis.

Compute the uniform magnetic field B at the centre of a Helmholtz coil pair (spacing equal to the radius) from the turns per coil, current and coil radius.

Use Ampère's force law to find the force per metre and the total force between two parallel current-carrying wires from their currents, separation and length, and tell whether they attract or repel.

Find the output voltage, divider ratio and total series capacitance of a capacitive voltage divider from two series capacitors and the input voltage.

Compute the back-EMF, electrical input power and winding copper loss of a DC motor from the supply voltage, armature current and armature resistance.

Find the electric field strength and electric potential produced by a single point charge at a given distance.

Compute the dimensionless Froude number from flow velocity, characteristic length and gravity to classify supercritical/subcritical flow or hull speed regime.

Compute the motional EMF generated by a conductor moving through a magnetic field from the field strength, rod length and velocity, with optional induced current and force.

Compute the dimensionless Weber number from fluid density, velocity, characteristic length and surface tension — the ratio of inertial to surface-tension forces.

Compute radiation pressure, force and beam energy density from light/EM intensity and surface reflectivity (solar sails, lasers).

Find the magnetic dipole moment of a current loop from turns, current and area, plus the torque and potential energy in an external field.

Compute the voltage gain, input resistance and output resistance of a BJT emitter follower (common-collector) voltage buffer.

From an EM wave’s electric-field amplitude, compute the magnetic-field amplitude, Poynting vector (average intensity) and energy density.

Compute the undeflected charged-particle speed v = E/B in crossed electric and magnetic fields, and the electric/magnetic force balance.

Compute the Cherenkov radiation cone angle and threshold speed when a charged particle exceeds the speed of light in a medium.

Compute the radiation resistance R_r = 20π²(L/λ)² of a short dipole antenna from its length and wavelength to gauge radiating efficiency.

Compute a Darlington pair’s overall current gain β_tot = β₁·β₂ + β₁ + β₂, total V_BE and collector current from the two transistors’ betas.

Compute the Schwarzschild gravitational redshift z = 1/√(1 − r_s/r) − 1 and the matching frequency/wavelength ratio from mass and emission radius.

Compute the instantaneous radiated power of a non-relativistic accelerating charge via the Larmor formula P = μ₀ q² a² / (6π c), or invert for acceleration.

Compute the London penetration depth λ_L = √(m / (μ₀ n_s e²)) from superfluid density — the exponential decay length of magnetic field inside a superconductor.

Compute the fusion triple product n·T·τ_E from density, temperature and energy confinement time, and compare against the approximate D–T ignition threshold.

Compute the gravitational-wave chirp mass M_c = (m₁ m₂)^{3/5}/(m₁+m₂)^{1/5} from component masses, or invert from frequency chirp.

Compute the Kerr/Schwarzschild innermost stable circular orbit radius r_ISCO from black-hole mass and dimensionless spin, or invert for mass.

Compute the sound intensity level L = 10·log₁₀(I / I₀) in decibels with I₀ = 1×10⁻¹² W/m², or invert level back to intensity.

Compute the AC Josephson frequency f = 2e·V/h = K_J·V from bias voltage, or invert to voltage from frequency — the SI volt realisation and SQUID scale.

Compute the Alfvén wave speed v_A = B / √(μ₀·ρ) in a magnetised plasma, with optional Mach number and transit time.

Compute the planar-diode space-charge-limited current density J = (4ε₀/9)·√(2e/m)·V^{3/2}/d² (Child–Langmuir law).

Compute the Rabi frequency Ω = μ·E/ħ (or γ·B) from dipole moment and drive field, plus the π-pulse duration t_π = π/Ω.

Compute the natural-convection Grashof number Gr = g·β·ΔT·L³/ν², with optional Rayleigh number Ra = Gr·Pr.

Compute straight rectangular fin efficiency η_f = tanh(mL)/(mL) with m = √(2h/(k·t)), plus actual heat dissipation.

Compute the gain G = 1 + 2R / R_g and output voltage Vout = G·(V+ − V−) of a three-op-amp instrumentation amplifier.

Compute the convective heat-transfer Nusselt number Nu = h·L/k, or invert for the convection coefficient h.

Compute the transient-conduction Biot number Bi = h·L_c/k and judge whether the lumped-capacitance approximation holds.

Compute the vortex-shedding / oscillatory-flow Strouhal number St = f·L/v, or solve for frequency from St.

Compute the Bond number Bo = Δρ·g·L²/σ (gravity vs surface tension) and the capillary length.

Compute the mass-transfer Schmidt number Sc = ν/D = μ/(ρ·D), with kinematic or dynamic viscosity input.

Compute the plasma beta β = 2μ₀·p/B² (thermal-to-magnetic pressure ratio), from pressure or n·k_B·T.

Compute the thermal or mass-transfer Péclet number Pe = L·v/α or L·v/D (= Re·Pr or Re·Sc).

Compute the Lewis number Le = α/D = Sc/Pr, comparing thermal and mass diffusivities.

Compute the natural-convection Rayleigh number Ra = g·β·ΔT·L³/(ν·α) = Gr·Pr.

Compute the mass-transfer Sherwood number Sh = k_m·L/D (mass analogue of Nusselt).

Compute the transient-conduction Fourier number Fo = α·t/L² (dimensionless time).

Compute the Eckert number Ec = U²/(c_p·ΔT), the ratio of flow kinetic energy to enthalpy difference — the key dimensionless group for high-speed flow and viscous dissipation.

Compute the Knudsen number Kn = λ/L (mean free path over characteristic length) to classify continuum, slip, transition or free-molecular flow regimes.

Compute the Stanton number St = h/(ρ·U·c_p) = Nu/(Re·Pr), the wall heat-transfer rate relative to the streamwise enthalpy transport.

Compute the Archimedes number Ar = g·L³·ρ_f·(ρ_p−ρ_f)/μ², gravity/buoyancy over viscous forces — the criterion for particle settling and fluidisation.

Compute the magnetic Reynolds number Rm = U·L/η_m = μ₀·σ·U·L to tell whether magnetic field lines are frozen into the flow or diffusion-dominated.

Compute the Hartmann number Ha = B·L·√(σ/(ρ·ν)), the MHD ratio of electromagnetic to viscous forces.

Compute the Brinkman number Br = μ·U²/(k·ΔT), the ratio of viscous dissipation to heat conduction.

Compute the Graetz number Gz = Re·Pr·D/x = ṁ·c_p/(k·x) for thermal entrance-length regime in ducts.

Compute the Damköhler number Da = τ_flow/τ_reaction, comparing reaction rate to flow / residence time.

Compute the Lundquist number S = L·v_A/η_m = τ_d/τ_A, the ideal- vs resistive-MHD criterion.

Compute the magnetic Prandtl number Pm = ν/η_m = μ₀·σ·ν, comparing momentum and magnetic diffusivities.

Compute the Elsasser number Λ = σ·B²/(ρ·Ω), comparing Lorentz to Coriolis forces in planetary dynamos.

Compute the Womersley number α = R·√(ω/ν) for pulsatile pipe flow (quasi-steady vs inertia-dominated).

Compute the Ohnesorge number Oh = μ/√(ρ·σ·L) = √We/Re for sprays, inkjet and droplet breakup.

Compute the capillary number Ca = μ·U/σ, comparing viscous forces to surface tension in multiphase flow.

Compute the Rossby number Ro = U/(Ω·L) or Ro = U/(|f|·L), inertia vs Coriolis.

Compute the Ekman number Ek = ν/(Ω·L²), viscous vs Coriolis, plus Ekman-layer scale.

Compute the Marangoni number Ma = |dσ/dT|·ΔT·L/(μ·α) for surface-tension-driven flow.

Compute the Galilei number Ga = g·L³/ν² (= Re²/Fr), gravity vs viscosity without density contrast.

Compute the Atwood number A = (ρ₁−ρ₂)/(ρ₁+ρ₂), the density contrast for Rayleigh–Taylor instability.

Compute the Taylor number Ta (centrifugal vs viscous forces) for Taylor–Couette onset and rotating shear.

Compute the Deborah number De = λ / t_c comparing material relaxation time to the observation time scale.

Compute the Weissenberg number Wi = λ · γ̇ measuring elastic effects in viscoelastic shear flow.

Compute the Dean number De = Re · √(D / 2R_c) for secondary vortices and instability in curved-pipe flow.

Compute the Richardson number Ri = g′L / U² (or gradient form) for stratified-flow stability and mixing.

Compute the Stefan number Ste = c_p · ΔT / L_f comparing sensible heat to latent heat in phase change.

Compute the dimensionless Euler number Eu = Δp / (ρ·V²), comparing pressure difference to inertial force for pipe losses and drag.

Compute the particle Stokes number Stk = τ_p·U / L to judge whether suspended particles follow the flow or impact surfaces.

Compute the Kolmogorov microscales η = (ν³/ε)^{1/4}, τ_η and u_η from kinematic viscosity and dissipation rate.

Compute magnetic pressure p_B = B²/(2μ₀) and magnetic energy density from magnetic field strength B for MHD and plasma work.

Compute the plasma ion-acoustic (Bohm) speed c_s ≈ √(k_B T_e / m_i) from electron temperature and ion mass.

Compute the upper-hybrid frequency f_uh = √(f_pe² + f_ce²) from plasma and electron cyclotron frequencies.

Compute the lower-hybrid frequency from ion/electron cyclotron and plasma frequencies for RF heating and wave propagation.

Compute the atmospheric/oceanic buoyancy (Brunt–Väisälä) frequency N = √((g/θ)·dθ/dz) from stratification.

Compute 1-D / 3-D thermal speed v_th = √(k_B T / m) or √(2 k_B T / m) from temperature and particle mass.

Compute the charged-particle E×B drift speed v_E = E×B / B² from electric and magnetic fields.

Compute the magnetic mirror ratio R = B_max/B_min and the loss-cone half-angle θ_lc = arcsin(√(B/B_max)) from field strengths.

Compute the ion inertial length d_i = c/ω_pi = √(m_i / (μ₀ n_i e²)) from ion density and mass number.

Compute the atmospheric scale height H = k_B T / (μ m_H g) = R_specific T / g from temperature, mean molecular weight and gravity.

Compute the gravitational free-fall time t_ff = √(3π / (32 G ρ)) from mass density — for molecular-cloud collapse and star formation.

Compute the Hill radius r_H = a (m / 3M)^(1/3) from primary mass, satellite mass and orbital semi-major axis.

Compute the unstable photon-sphere radius of a Schwarzschild black hole r_ph = 1.5 r_s = 3GM/c², and the matching mass or Schwarzschild radius.

Compute the Eddington luminosity L_Edd = 4πGMc/κ from mass and mean opacity (default electron-scattering κ = σ_T/m_p).

Compute the electron inertial length (collisionless electron skin depth) d_e = c/ω_pe = √(m_e/(μ₀ n_e e²)).

Compute the Fermi wave number, energy, velocity and temperature of a 3-D free-electron gas from number density.

Compute the fast magnetosonic speed c_ms = √(v_A² + c_s²) from Alfvén and sound speeds (perpendicular-propagation limit).

Estimate the Coulomb logarithm ln Λ ≈ ln(λ_D / b_min) from Debye length and impact parameter for plasma collision rates and Spitzer resistivity.

From density ρ and sound speed c_s compute the Jeans length λ_J = c_s √(π/(Gρ)) and Jeans mass M_J — the critical mass for gravitational collapse of a molecular cloud.

Compute magnetic rigidity R = p/|q| = B·ρ and convert among momentum, kinetic energy and gyroradius (cosmic-ray GV, accelerator design).

Estimate the Spitzer resistivity η ∝ T_e^(−3/2) ln Λ of a fully ionized plasma from electron temperature and Coulomb logarithm.

Compute the empirical Bohm diffusion coefficient D_B = (1/16) k_B T_e / (e B) for magnetically confined plasmas.

Compute the gravitational binding energy U = 3 G M² / (5 R) of a uniform sphere — stars, planets and cluster-scale estimates.

Estimate the virial temperature T_vir ∝ G M μ m_H / (k_B R) of a self-gravitating system from mass and radius.

Compute the classical electron radius r_e = e²/(4π ε₀ m c²) from charge and mass, or invert for mass or charge.

Compute the Unruh temperature T = ℏ a / (2π k_B c) of a uniformly accelerating observer, or invert for acceleration.

Compute the tokamak Greenwald density limit n_G = I_p / (π a²) from plasma current and minor radius.

Compute the Thomson total cross-section σ_T = (8π/3) r_e² and differential cross-section from the classical electron radius.

Compute the synchrotron critical frequency ν_c = (3/2) γ² ν_g sin θ from electron energy and magnetic field.

Compute the Zeeman energy splitting ΔE = g μ_B B m_j from magnetic field and Landé g-factor.

Compute Landau level energies E_n = ħ ω_c (n + 1/2) of a 2-D electron gas in a magnetic field, and invert for B.

Compute the quantum-Hall plateau resistance R_H = (h/e²)/i from filling factor i, and the von Klitzing constant R_K.

Compute the BCS zero-temperature gap Δ(0) ≈ 1.764 k_B T_c from critical temperature, plus a finite-T approximation.

Compute the Casimir force F = −π² ħ c A / (240 d⁴) and pressure between two ideal parallel conducting plates.

Compute the e⁺e⁻ pair-production threshold 2 m_e c² in a nuclear field and 4 m_e c² in an electron field.

Compute gas-discharge breakdown voltage from gas type, pressure and gap using Paschen’s law.

Compute metal field-emission current density J and total current I from work function φ, electric field E and area A via the Fowler–Nordheim equation; inverse E from J supported.

Compute the solid-state Debye temperature Θ_D = ħ ω_D / k_B from Debye frequency, or from sound speed and number density; inverse frequency from Θ_D.

Compute E_n = n² π² ħ² / (2 m L²), standing-wave wavelength and transition energies for a 1-D infinite square well; electron, proton or custom mass.

Compute the Stokes–Einstein diffusion coefficient D = k_B T / (6 π η r) from temperature, solvent viscosity and hydrodynamic radius; inverse r from D.

Compute Scherrer crystallite size τ = K λ / (β cos θ) from X-ray wavelength, Bragg angle and FWHM peak breadth; shape factor K and unit conversion supported.

Compute the nuclear Coulomb barrier height V_C ≈ Z₁ Z₂ e² / [4πϵ₀ r₀ (A₁⅓ + A₂⅓)] and lab-frame threshold energy from charges and mass numbers.

Compute the Landau–Zener non-adiabatic transition probability P = exp(−2π|V|² / (ħ|α|)) for a two-level system under a linear sweep.

Compute the Gamow-peak energy E₀ = (E_G (k_B T)² / 4)^{1/3} and effective width for stellar nuclear reaction rates.

Compute thermionic emission current density from the Richardson–Dushman law J = A T² exp(−φ / k_B T).

Compute the Aharonov–Bohm phase φ = 2π Φ / Φ₀ acquired by a charged particle encircling a magnetic flux, and the fringe shift.

Compute the 12-6 Lennard–Jones potential V(r) = 4ε[(σ/r)¹² − (σ/r)⁶], force, and equilibrium separation.

Compute paramagnetic magnetization of local moments via the Brillouin function B_J(x) at given field and temperature.

Compute the half-value layer HVL = ln(2)/μ from the linear attenuation coefficient μ, plus TVL and transmission.

Compute the semiconductor band gap vs temperature with the Varshni equation E_g(T) = E_g(0) − α T²/(T+β).

Compute the Faraday rotation angle θ = V · B · L from the Verdet constant, magnetic field and path length.

Estimate the mass stopping power −dE/(ρ dx) of a heavy charged particle in matter with the Bethe–Bloch formula.

Compute the 3D Wannier–Mott exciton binding energy E_b = (μ/m_e) E_H / ε_r² and Bohr radius from reduced mass and dielectric constant.

Compute the plasma magnetic beta β = 2 μ₀ p / B² (thermal-to-magnetic pressure ratio), or invert for p or B.

Compute the Sagnac phase shift Δφ = 8πAΩ/(λc) and beat frequency of a rotating ring interferometer.

Compute the refractive index n(λ) of optical materials from the two- or three-term Sellmeier equation.

Compute molecular polarizability α from relative permittivity ε_r and number density, or invert for ε_r.

Compute yield strength σ_y = σ_0 + k_y / √d from grain size via the Hall–Petch relation.

Compute post-shock Mach number and pressure/density/temperature ratios across a normal shock from upstream Mach and γ.

Compute the Ginzburg–Landau parameter κ = λ/ξ from penetration depth and coherence length, and classify type-I vs type-II.

Compute impact parameter b, scattering angle θ and the differential cross section dσ/dΩ from the Coulomb (Rutherford) formula.