Physics Calculators

Find the range, maximum height, flight time and impact speed of a projectile from its launch speed and angle.

Calculate the fall time, drop height and impact velocity of an object falling freely under gravity.

Calculate the terminal velocity of a falling object from its mass, drag coefficient, air density and area.

Calculate the centripetal force and acceleration needed to keep an object moving in a circle.

Calculate the linear momentum of a moving object from its mass and velocity, plus its kinetic energy.

Calculate impulse from force and time, and the resulting change in velocity of a given mass.

Calculate the torque of a force about a pivot from the force, lever arm length and angle.

Find angular velocity in rad/s from rpm, Hz or degrees per second, from linear speed and radius, from an angle turned in a given time or from the period of one turn — with every unit, the linear speed and centripetal acceleration at your radius, and force, angular momentum and kinetic energy for a mass.

Calculate the period and frequency of a simple pendulum from its length and local gravity.

Calculate the energy stored in a stretched or compressed spring, plus the spring force, from its stiffness and displacement.

Calculate the observed frequency of a sound when the source or observer is moving, using the Doppler effect.

Calculate the gravitational attraction between two masses using Newton's law of universal gravitation.

Calculate the pressure at depth in a fluid from its density, depth and the pressure at the surface.

Calculate the buoyant force on a submerged object from the fluid density and displaced volume using Archimedes’ principle.

Calculate the mechanical advantage of a lever, inclined plane or pulley system and the effort force it saves.

Find the horizontal range of a projectile launched from level ground, plus the angle that maximises distance.

Find the impact velocity of an object dropped from rest after a given fall time or drop height, ignoring air resistance.

Calculate the rotational kinetic energy of a spinning body from its moment of inertia and angular speed.

Work out the gravitational potential energy stored by lifting a mass to a height near the Earth’s surface.

Find the final velocity in a perfectly inelastic collision where two masses stick together, using conservation of momentum.

Solve any one of voltage, current or resistance from the other two using Ohm’s law, with power included.

Calculate electrical power from voltage, current and resistance using P = VI, I²R and V²/R.

Find a wave’s speed from its frequency and wavelength, plus its period, using v = f·λ.

Solve for density, mass or volume from the other two, with results in SI and common units.

Calculate pressure from a force spread over an area, with results in pascals, kPa, bar, psi and atmospheres.

Calculate the work done by a force acting at an angle over a distance, using W = F·d·cosθ.

Find the electrostatic force between two point charges from their charges and separation using Coulomb’s law.

Find acceleration from a change in velocity over time, plus the distance covered while speeding up.

Calculate force from mass and acceleration using Newton’s second law, with the result in newtons, kilogram-force and pounds-force.

Convert rotational power and speed (RPM) into torque — the twisting force at an engine or motor shaft.

Calculate the centre-seeking acceleration of an object moving in a circle from its speed and radius.

Calculate the gravitational field strength (g) at a distance from a mass using Newton’s law of gravity.

Find the escape velocity from Earth, the Moon, Mars and other bodies — or from any custom mass and radius.

Calculate the settling (terminal) velocity of a small sphere in a viscous fluid using Stokes’ law.

Find the length a simple pendulum needs for a chosen swing period — including the classic one-second (seconds) pendulum.

Find a spring’s stiffness (spring constant k) from the force applied and how far it stretches, plus the stored elastic energy.

Calculate the heat energy needed to change a substance’s temperature from its mass, specific heat and temperature change.

Calculate the efficiency of a machine or process from its useful energy output and total energy input.

Find the critical angle for total internal reflection between two media from their refractive indices.

Find the change in momentum of a mass, the impulse delivered and the average force from the contact time — the impulse–momentum theorem worked out.

Compute the final velocities of two objects after a one-dimensional elastic collision, where both momentum and kinetic energy are conserved.

Resolve the weight of a mass on a ramp into its along-slope and normal components, add friction and find the net force and acceleration down the incline.

Find the friction force between two surfaces from the coefficient of friction and the normal force, with the normal force taken from the object’s weight.

Find the tension in the rope or ropes holding a hanging mass — a single vertical rope, two symmetric ropes at an angle, or a mass in an accelerating lift.

Compute the upward buoyant force on a submerged or floating object from Archimedes’ principle, plus its apparent weight and whether it floats or sinks.

Find the volumetric flow rate through a pipe from its diameter and flow speed, and use the continuity equation to get the speed in a second, narrower section.

Use Bernoulli’s equation to find the pressure at a second point in a flowing fluid from the pressure, speed and height at a first point.

Combine up to three resistors in parallel into a single equivalent resistance, and compare it with the series total of the same resistors.

Find the energy stored in a capacitor and the charge it holds from its capacitance and the voltage across it.

Find the magnetic force on a straight current-carrying wire in a uniform magnetic field from the field strength, current, wire length and angle.

Use the thin lens equation to find the image distance, magnification and image height from a lens’s focal length and the object distance.

Find average power from the work done and the time it took — plus the equivalent in horsepower.

Convert revolutions per minute into angular velocity in rad/s, and find the linear speed at a given radius.

Compute the rotational inertia of a point mass, thin ring, solid disk, solid sphere or rod about its centre.

Find the outward centrifugal force on a rotating mass from its speed and radius, and the g-force it feels.

From input teeth, output teeth and input speed and torque, get the gear ratio, output speed and output torque.

Convert a sound intensity in W/m² to a decibel level relative to the threshold of hearing.

Find the frequency a listener hears when a sound source and observer move, using the classic Doppler equation.

Find how much a rod, rail or pipe lengthens when heated, from its length, temperature change and expansion coefficient.

Find the steady-state heat flow through a wall or slab using Fourier's law of conduction.

Find the secondary voltage and current of an ideal transformer from its turns ratio and primary values.

Find the time constant τ = RC of a resistor–capacitor circuit — or the R or C for a target τ or cutoff frequency — with kΩ/MΩ and µF/nF/pF units, the capacitor voltage and current at any time, time to any charge level, rise time, stored energy, a charge curve and the worked steps.

From launch speed and angle, find a projectile’s maximum height, total time of flight and horizontal range.

Find the speed a satellite needs to stay in a circular orbit around any central body from its mass and orbital radius.

Compute how long an orbit takes from the central mass and the semi-major axis using Kepler’s third law.

Find the gravitational acceleration at the surface of a planet or moon from its mass and radius.

See what you would weigh on the Moon, Mars, Jupiter and other worlds based on their surface gravity.

Find how long light takes to cross a distance — from a room to another star — at 299,792,458 m/s.

Convert a redshift z, a pair of observed and rest wavelengths, or a velocity into recession velocity (relativistic and v = cz), Hubble-law distance and flat ΛCDM comoving, luminosity and angular-diameter distances, lookback time and the age of the universe at that redshift.

Find the event-horizon radius of a black hole of a given mass — the size to which any mass must be crushed to become one.

Find the minimum kinetic energy an object needs to escape a planet’s gravity — and the matching escape velocity.

Find the orbital radius and altitude a satellite needs to have a chosen orbital period — including geostationary orbit.

Compute a star’s total power output from its radius and surface temperature using the Stefan–Boltzmann law.

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

Find a rocket’s change in velocity from its specific impulse and wet-to-dry mass ratio using the Tsiolkovsky equation.

Find a material's Young's modulus from an axial load, cross-section and stretch — plus the stress and strain behind it.

Find the maximum sag of a simply supported beam carrying a central point load from its span, load, modulus and moment of inertia.

Find the gauge and absolute pressure at a depth in any fluid from its density, the depth and gravity.

See how a hydraulic press multiplies force between two pistons, and how far the output piston moves, using Pascal’s principle.

Find the power factor, phase angle and reactive power of an AC load from its real (watts) and apparent (VA) power.

Find the impedance, reactances and phase angle of a series RLC circuit at a given frequency from R, L and C.

Find the resonant frequency of an LC tank circuit from its inductance and capacitance.

Convert a power or voltage ratio into decibels of gain — or a loss — with the correct 10 log or 20 log rule.

Find the magnification of a lens and the image size from the object and image distances.

Solve the mirror equation for image distance and magnification of a spherical mirror from its focal length and object distance.

Find how far and how long an object slides to a stop under kinetic friction from its speed and the friction coefficient.

Find the terminal velocity of a falling object from its mass, cross-section, drag coefficient and the fluid density.

Find the ideal (frictionless) banking angle of a road or track curve from the design speed, turn radius and gravity.

Find the aerodynamic drag force on a moving object from its speed, frontal area, drag coefficient and the fluid density.

Estimate how far a projectile travels with quadratic air drag, and compare it to the ideal vacuum range, by numerical integration.

Find the elastic energy stored in a spring from its stiffness and stretch, plus the spring force and the launch speed it can give a mass.

Find the moment of inertia of a common shape and its rotational kinetic energy from the mass, radius and rotation speed.

Find angular momentum for a spinning rigid body (L = I·ω) or a particle moving around a point (L = m·v·r).

Find the torque needed to angularly accelerate a rotating body (τ = I·α), plus the equivalent force at a given radius.

Find the oscillation period, frequency and angular frequency of a mass on a spring in simple harmonic motion.

Find the average power a transverse wave carries along a string, plus its energy per period and per wavelength.

Find the harmonic frequencies and wavelengths of a standing wave on a string fixed at both ends from its length and wave speed.

Find the maximum kinetic energy of ejected electrons, the stopping voltage and the threshold wavelength from the light wavelength and a metal’s work function.

Find the quantum (matter-wave) wavelength of a moving particle from its mass and speed using λ = h ÷ (m·v).

Launch a projectile from an elevated point and find its range, flight time, peak height and impact speed for any launch angle.

Combine two one-dimensional velocities to find how fast one object moves as seen from the other, and whether they close or separate.

Combine two legs of a trip to see why the overall average speed is not the average of the two leg speeds.

Measure the bounciness of a collision from drop and bounce heights, and read off the impact and rebound speeds.

Find the balance point of two masses on a line from their sizes and positions — the weighted average that is the center of mass.

Relate a wheel’s linear speed to its spin using v = ωr, and see the speed of the top, center and contact point.

Find the work a force does when it pushes at an angle to the motion, using W = F·d·cosθ.

Estimate the mechanical power you produce climbing stairs from your mass, the height gained and the time taken.

Find a machine’s efficiency from its useful output and energy input, and see how much is wasted.

Split a swinging pendulum’s energy into potential and kinetic at any angle, and find its speed there.

Split a spring-mass oscillator’s energy into potential and kinetic at any displacement, and find its speed, period and frequency.

Find the pitch a listener hears when the sound source, the observer, or both are moving, using the full Doppler formula.

Maximum theoretical efficiency of a heat engine between two temperatures, plus the Carnot COP for a refrigerator or heat pump working between the same reservoirs.

Heat needed to melt, freeze, boil or condense a mass of water, ice or other substances — Q = mL — with time at a given heating power.

Heat flow through a slab from Fourier’s law — conductivity, area, thickness and temperature difference — with U-value, R-value and daily energy loss.

Electric field strength and potential at a distance from a point charge, and the force on a test charge placed there — Coulomb’s constant, SI units.

Lorentz factor γ from a speed (as a fraction of c, km/s or m/s), with relativistic length contraction, mass-energy and kinetic energy per kilogram.

How much time passes for a stationary observer when a moving clock records a given interval — special-relativity time dilation at any speed.

Convert mass to rest energy (or energy back to mass) in joules, kilowatt-hours, tonnes of TNT and Hiroshima equivalents — E = mc².

Mach number from speed and air temperature (or altitude), the local speed of sound, and the flight regime — subsonic to hypersonic.

Estimate the age of a sample from its remaining carbon-14 fraction (or activity), using the 5,730-year half-life — or predict the fraction left after a given age.

Inductance of an air-core or iron-core solenoid from turns, length, coil diameter and core permeability, with stored energy at a given current.

Power dissipated in a resistor or component from any two of voltage, current and resistance, with energy per hour and a resistor wattage-rating check.

X-ray diffraction with Bragg’s law — solve for the diffraction angle, d-spacing or wavelength given the other two and the order n, with the 2θ value diffractometers report.

Angles of the diffraction maxima for a grating — from lines per mm and wavelength, every visible order, the maximum order, angular dispersion and the spot spacing on a screen.

Wavelength shift, scattered photon energy and electron recoil energy for Compton scattering at any angle — with the Compton wavelength and the maximum (back-scatter) shift.

Energy, orbital radius and electron speed for any level of a hydrogen-like atom, plus the photon emitted or absorbed between two levels — wavelength, frequency and spectral series.

Wavelengths of hydrogen spectral lines from the Rydberg formula — pick a series or two levels, get wavelength, wavenumber, frequency and photon energy, plus the series limit.

Frequency, period and gyroradius of a charged particle circling in a magnetic field — electrons, protons, alpha particles or a custom ion — with the relativistic correction.

How fast electrons actually drift in a wire — from current, wire gauge and material carrier density — with current density, time to travel a metre and a comparison with the signal speed.

Electromagnetic skin depth in a conductor from frequency, conductivity and permeability — copper, aluminium, steel or custom — with the AC resistance rise of a round wire.

Quality factor of a resonator from bandwidth, from series/parallel RLC values or from energy decay — with damping ratio, ring-down time and half-power bandwidth.

Series RLC circuit at any frequency — resonant frequency, impedance magnitude and phase, reactances, current, Q, bandwidth and the voltages across L and C at resonance.

Ideal transformer relationships — turns ratio from voltages or windings, secondary voltage and current, reflected impedance and power, with efficiency and regulation adjustments.

Resistance of a wire from length, diameter or AWG gauge, material and temperature — ohms total and per metre, voltage drop and power loss at your current.

Orbital period from semi-major axis (or the reverse) around the Sun, Earth, Moon, Jupiter or any central mass — with orbital speed, geostationary check and the mass of the primary from an observed orbit.

Delta-v from wet mass, dry mass and specific impulse — or the propellant needed for a target Δv — with mass ratio, propellant fraction, exhaust velocity and burn time at a given thrust.

Pascal’s principle for a hydraulic press or jack — output force from input force and piston diameters, mechanical advantage, pressure, and how far the load rises per stroke.

Radiant power from a hot surface — emissive power, total watts from an area, net exchange with the surroundings and Wien’s peak wavelength, for any temperature and emissivity.

Peak wavelength or temperature of a black body from Wien’s law — colour and spectral region, peak frequency, and comparisons with the Sun, a light bulb and the cosmic microwave background.

Capacitive reactance Xc = 1 ÷ (2πfC) for any capacitance and frequency — with the current at a given voltage, phase relationship and a frequency sweep table.

Inductive reactance XL = 2πfL for any inductance and frequency — with current at a given voltage, impedance including winding resistance, quality factor and a frequency sweep.

Find motor torque from power and rpm (T = 9,549 × kW ÷ rpm), or power or speed from the other two, or torque from volts, amps, power factor and efficiency — in N·m, lb·ft, kgf·cm and more, with a gearbox, pulley line pull, synchronous speed and slip, and a torque-versus-speed table.

Convert rotational speed to linear or surface speed from wheel, pulley, roller, spindle or tyre diameter — m/s, km/h, mph, ft/min (SFM) and m/min — find the rpm needed for a target speed or the diameter that gives it, read a tyre size such as 205/55R16, allow for a gear or pulley ratio and slip, and see speed across a range of rpm.

Push and pull force of a double- or single-acting air cylinder from bore, rod and pressure (bar, psi, kPa or MPa) — or the bore or pressure you need for a load — with a load factor, spring return, air consumption per cycle and per minute including tubing, piston speed and a force table for every standard ISO bore.

Water flow through a nozzle or orifice from diameter and pressure — Q = 29.84 × Cd × d² × √P (gpm) with metric output — for fire hoses, pressure washers, irrigation and hydrants, with reaction force and velocity.

Tightening torque from target preload, or clamp force from a known torque, for metric coarse/fine, UNC and UNF bolts: property class or SAE grade, nut factor K or thread and head friction coefficients, preload scatter, bolt stretch and a torque chart.

Spring rate (stiffness) of a helical coil spring from wire diameter, mean coil diameter, active coils and material — load and stress at a given deflection, spring index, solid height and natural frequency.

Basic rating life of a rolling bearing from dynamic load rating C, equivalent load P and speed — L10 in million revolutions and hours for ball (p = 3) and roller (p = 10/3) bearings — with reliability adjustment and the load for a target life.

Minimum solid shaft diameter for a transmitted power and speed from allowable shear stress — pure torsion and the ASME combined torsion-plus-bending equivalent — with actual stress, angle of twist and hollow-shaft option.

Belt pitch length for two pulleys from diameters and centre distance — or centre distance from a standard belt — speed ratio, belt speed, wrap angle and standard A/B/C belt numbers with the equivalent metric SPZ/SPA/SPB length.

Section modulus, moment of inertia and plastic modulus for rectangular, round, tube, box, I-beam, channel and T sections about either axis — then bending stress, shear stress and deflection for simply supported, cantilever and fixed beams under point or uniform loads, with moment, shear and deflection diagrams.

Critical buckling load of a column from length, end conditions (pinned, fixed, cantilever), cross-section and material — Euler load, slenderness ratio, whether Euler or yielding governs (Johnson parabola) and the safe load with a factor of safety.

Spur gear geometry from module (or diametral pitch) and tooth count — pitch, outside and root diameters, addendum, dedendum, circular pitch, tooth thickness, centre distance with a mating gear and the minimum teeth to avoid undercut.

Minimum thread engagement so the bolt breaks before the threads strip, using the FED-STD-H28 / Machinery’s Handbook shear areas of the internal and external threads — metric and UNC/UNF sizes, 6g/6H or 2A/2B limits, ISO and SAE bolt grades, 13 tapped materials or your own, the strength ratio J, stripping loads by depth and a check of the depth you have.

Interference fit design for a shaft in a hub: contact pressure from the interference (Lamé thick-cylinder equations), hoop stresses in hub and shaft, the axial force to press it and the torque it can transmit, thermal shrink-fit temperature, and ISO fit suggestions (H7/p6, H7/s6, H7/u6).

Assembly gap or overall dimension from a chain of toleranced parts — worst-case and statistical (RSS) stack-ups, the resulting min/max, how much tolerance you could relax under RSS, the sigma level of a required gap, and per-part contributions.

Calculate a helical compression spring's rate from wire diameter, coil diameter, active coils and material (music wire, stainless, chrome-silicon, phosphor bronze), then the deflection and force, solid height, spring index, Wahl-corrected shear stress against the allowable, and buckling — or find the coils needed for a target rate.

Calculate a helical torsion spring's rate in N·mm per degree (and per turn) from wire diameter, mean coil diameter, body turns and leg lengths, the torque at a given deflection angle, the bending stress with the inside-curvature correction against the allowable, the coil diameter reduction and the arbor size needed when wound up.

See how stall speed rises in a banked turn: load factor from bank angle, accelerated stall speed as Vs × √n, the stall speed at your actual weight, the reduced maneuvering speed VA below gross weight, and the bank angle at which your approach speed leaves no margin.

Correct a POH landing distance for the day: density altitude, headwind and tailwind components from runway and wind, dry or wet grass, runway slope and weight, then apply a safety factor and compare with the runway available to see whether to land, go elsewhere or accept a very tight margin.

Work out the holding pattern entry from your heading to the fix, the inbound course and turn direction, then the outbound heading and timing corrected for wind so the inbound leg is one or one-and-a-half minutes, the teardrop heading, and the fuel burned while holding.

Estimate the freezing level from surface temperature and lapse rate, the altitudes of −10 °C and −20 °C, the cloud base from the temperature–dew point spread, and the band of cloud where airframe icing is most likely, with the altitudes that keep you out of it.

Find your true altitude when the altimeter is set wrong or the air is colder than standard: the error from an old or wrong altimeter setting at 1,000 ft per inch, the ICAO cold-temperature correction to minimum altitudes and approach fixes, and the terrain clearance you really have.

Bring an aircraft's centre of gravity back inside its limits: how much weight to move between two stations to shift the CG a given distance, how much ballast to add or remove at a station, and the new CG after changing fuel or passengers, using the weight-shift and weight-change formulas.

Convert a delta (Δ, triangle) network of three resistors to its equivalent wye (Y, star) network or back, with the Kennelly formulas and a terminal-resistance check.

Find the unknown resistance of a balanced Wheatstone bridge (Rx = R2·R3/R1) or the output voltage of an unbalanced bridge, with both midpoint voltages and the supply current.

Find the period and frequency of a physical (compound) pendulum — a rod, disc, ring or sphere swinging about a pivot — plus its equivalent simple-pendulum length and the pivot point for the shortest period.

Use Malus's law, I = I₀ cos²θ, to find how much polarized or unpolarized light passes through a polarizer and analyser at any angle, with a table of intensity from 0° to 90°.

Find Brewster's angle, θB = arctan(n₂/n₁), where reflected light is fully polarized — with the refraction angle, s-polarized reflectance, the critical angle and values for water, glass and diamond.

Convert radioactivity between becquerels and curies — Bq, kBq, MBq, GBq, TBq, Ci, mCi, µCi, nCi, pCi, dpm and rutherfords — with the same activity shown in every unit (1 Ci = 37 GBq).

Apply the Heisenberg uncertainty principle, Δx·Δp ≥ ħ/2: from a position uncertainty find the minimum momentum and velocity uncertainty of an electron, proton, neutron or any particle, and the matching kinetic energy.

Calculate the mass defect and nuclear binding energy of any nucleus from its proton number, neutron number and atomic mass, in MeV, MeV per nucleon, joules and kJ/mol — with presets from deuterium to uranium-238.

Use Newton's law of cooling, T(t) = Tₛ + (T₀ − Tₛ)e^(−kt), to find an object's temperature after any time, how long it takes to reach a target temperature, and a cooling curve table.

Find a lens's focal length and power in diopters from its refractive index and the radii of curvature of both surfaces with the lensmaker's equation — thin or thick lens, in air or any medium.

Young's double-slit calculator: fringe spacing Δy = λL/d, the angle and position of any bright or dark fringe, the highest visible order and a table of fringe positions on the screen.

Find the fundamental frequency and harmonics of an air column in a pipe open at both ends or closed at one end (resonance tube), with end correction, speed of sound from temperature and resonance lengths for a tuning fork.

Calculate the factor of safety from material strength (or failure load) and working stress, the margin of safety, utilization, the allowable stress for a target factor and typical design factors.

Calculate the log mean temperature difference (LMTD) of a counterflow or parallel-flow heat exchanger from inlet and outlet temperatures, plus effectiveness and the heat-transfer area for a given duty and U.

Hydraulic cylinder push and pull force from bore, rod and pressure, plus extend and retract speed and time from pump flow, oil volume per stroke and the hydraulic power needed.

Pipe friction pressure drop and head loss with the Darcy-Weisbach equation and the Colebrook friction factor — any fluid, pipe material and size, with Reynolds number, velocity and flow regime.

Hoop and longitudinal stress in a thin-walled pipe, tank or sphere under internal pressure, the Lamé thick-wall check, factor of safety and the ASME minimum wall thickness.

Flow rate through a Venturi meter or Venturi tube from inlet and throat diameters, measured differential pressure, fluid density and discharge coefficient — volume and mass flow and both velocities.

Hydropower output from flow rate and head (P = η·ρ·g·Q·H): electric power in kW and hp, net head after penstock losses, annual energy and its value, and the suitable turbine type.

Thermal stress in a restrained bar or pipe from a temperature change (σ = E·α·ΔT): stress, force, free expansion and strain for steel, stainless, aluminum, copper, concrete or PVC, checked against yield.

Convective heat transfer rate from a surface with Newton’s law of cooling (Q = h·A·ΔT): heat flux, thermal resistance and energy, with typical h values for air, water, boiling and condensation.

Calculate the speed of sound in air at any temperature, or in fresh water and sea water from temperature, salinity and depth — plus how long sound takes to travel a distance and the wavelength of a 1 kHz tone.

Calculate a hydraulic jump in a rectangular channel from the upstream depth and velocity — Froude number, sequent depth, downstream velocity, energy loss, jump length, critical depth and the type of jump.

Convert power between watts, kilowatts, megawatts, mechanical horsepower (hp), metric horsepower (CV/PS), BTU per hour, kcal/h and tons of refrigeration — e.g. hp to kW and kW to hp.

Convert force between newtons, kilonewtons, kilogram-force, gram-force, metric ton-force, pound-force, ounce-force, kips and dynes — e.g. N to kgf, kgf to N or newtons to lbf.

Convert torque between newton meters, pound-feet (ft-lb), pound-inches (in-lb), kilogram-force meters and centimeters, ounce-inches and kN·m — for torque wrenches, bolts and engines.

Convert volumetric flow rate between L/s, L/min, L/h, m³/h, m³/s, US and UK gallons per minute (GPM), gallons per hour and CFM — for pumps, plumbing, irrigation and ventilation.

Solve an Atwood machine: acceleration of two hanging masses over a pulley, string tensions, travel time and speed, with an optional solid-disc pulley mass for the rotational inertia.

Calculate the period, angular velocity, speed, string tension and orbit radius of a conical pendulum from the string length, cone angle and bob mass.

Find the speed of a bullet or projectile from a ballistic pendulum: block and projectile masses plus the rise height or swing angle, with momentum and the kinetic energy lost.

Combine up to three spring constants in series or parallel to get the equivalent stiffness, total extension under a load, force and stretch of each spring, stored energy and oscillation period.

Calculate how much the volume of a liquid or solid changes with temperature, ΔV = β·V₀·ΔT, with coefficients for water, fuel, ethanol, mercury, steel, aluminium and copper.

Air-standard Otto cycle: thermal efficiency from the compression ratio and γ, plus the temperature, pressure and volume at all four states, net work, heat rejected and mean effective pressure.

Calculate the COP of a heat pump, refrigerator or air conditioner from heat and power, compare it with the Carnot limit between the hot and cold temperatures, and get EER and tons of refrigeration.

Find the equivalent capacitance of up to four capacitors in series or parallel, plus the total charge and energy at a supply voltage and the voltage, charge and energy of each capacitor.

Equivalent inductance of up to four uncoupled inductors in series or parallel, with the inductive reactance at a frequency, the current in each inductor and the magnetic energy stored.

Calculate the time constant τ = L/R of a resistor–inductor circuit and the current, inductor voltage and resistor voltage at any time while the current builds up or decays.

Calculate the capacitance of a parallel-plate capacitor from plate area, gap and dielectric (C = ε₀εᵣA/d), plus the charge, electric field, surface charge density and stored energy at a voltage.

Magnetic flux density B around a long straight current-carrying wire (B = μ₀I/2πr) or at the centre of a circular loop or flat coil, in tesla, microtesla and gauss.

Magnetic field inside a solenoid, B = μ₀μᵣNI/L, from the number of turns, coil length, current and core permeability, with turns per metre, H field, flux through the core and a long-coil accuracy check.

Find the internal resistance of a battery or source from its EMF, terminal voltage under load and load resistance, plus current, lost volts, efficiency, short-circuit current and maximum power.

Calculate Poisson’s ratio ν from lateral and axial strain or from Young’s and shear modulus, and get the related shear modulus G, bulk modulus K and volume change for an isotropic material.

Plane-stress Mohr’s circle: principal stresses σ₁ and σ₂, principal angle, maximum shear stress, von Mises stress and the stresses on a plane rotated by any angle, from σx, σy and τxy.