L1: foundations
Resistance, then reactance
PHYThe ratio of voltage to current is the whole of the first tool. The second adds a component that has no resistance and limits the current anyway. The third asks what that costs at each frequency.
Physics
Physics keeps discovering that unrelated things obey the same equation. A pendulum, a spring and a tuned circuit are one problem wearing three costumes, and you can drive all three here.
Newest in this category
- Screw Thread Explore screw lead angle, mechanical advantage, efficiency and self-locking.
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Gear Train Ratio & Speed Simulator
Four gears, two ways to arrange them. Set the tooth counts and see which of them reach the output ratio and which are only along for the ride.
- Snell's Law Refraction Simulator Visualize light refraction at a boundary with adjustable refractive indices and total internal reflection.
Chapters
Short sequences rather than single tools. Each step hands the next one something specific, so they are worth doing in order.
L2: school
Energy instead of forces
PHYTwo tools and one shortcut: when all you want is speeds and heights, you can skip the forces entirely and let energy keep the books.
L2: school
Bending light
PHYA lens is a boundary that somebody curved, and no page on this path introduces physics the first one does not already contain. What changes is how much is asked of it: one surface, then an ideal lens, then a piece of glass.
L2: school
Simple machines: trading force for distance
PHYFive machines and one bargain. None of them creates energy, and every one of them lets you choose how to spend it: a smaller force over a longer path, or the reverse. Each step hands the next a specific piece of that bargain, and by the end you can read any machine by asking two questions about it.
L2: school
Circles and waves are the same motion
MATHThree tools, and the point of the sequence is that they are not three subjects. A sine wave is what uniform circular motion looks like from the side, and a swinging pendulum is what it looks like when gravity supplies the turning.
L2: school
Throwing things
PHYThree tools, and the third one breaks the first two. Constant acceleration has a closed-form answer, so the first two steps are algebra. Air resistance is not constant, and from there the only honest method is to integrate it step by step.
L2: school
Water out of a tank: one speed, spent two ways
PHYTwo tools built on a single result. Water leaves a hole at exactly the speed a stone reaches falling the same depth, and everything else follows from where that speed is spent: sideways it is a projectile, downwards it is the tank emptying itself.
L2: school
From one spring to a standing wave
PHYOne restoring force, followed as far as it will go. A mass on a spring oscillates, a row of them passes the oscillation along, two of those meeting decide whether to add or cancel, and a boundary at each end throws away every frequency that does not fit.
Problems solved in full
- Lateral acceleration for an interceptor flying 1500 m/s with navigation ratio N = 3
- Why the bottom half of a barrel takes 2.4 times as long as the top
- The period of two clean oscillations from release at rest
- Unrolling one turn of a screw thread into a triangle
- Freezer ice for a 250 g mug of tea at 80 °C
- A fighter 90 km out with a radar cross-section of 5 m²
- A resonant peak of 1.0002 and an infinite gain margin
- The dipole arrangement with two charges 4 units apart
- Why your arm needs 350 newtons to hold a 50 newton weight
- Four ropes, eight hundred newtons, and where the missing 141 joules go
- Resonant frequency of 100 Ω, 10 mH, 2.53 µF in series
- A 1 kg car released from rest at 5 m on the track
- A mismatched 915 MHz antenna on a 50 Ω line
- Two holes, one wet spot
- Why the winch handle travels 3.77 metres to raise 63 centimetres of rope
- A 5 m simply supported steel beam carrying 10 kN at midspan
- An engine taking in 1000 J between 600 K and 300 K
- 1 kg ball on a 1 m string swung in a vertical circle
- Frequency ahead and behind a 700 Hz siren moving at 30 m/s
- Output that has reached 63.21% one time constant into a step input
- Drop a stone from 100 m with g = 9.81 m/s²
- Two 150 N/m springs joined end to end, stretched 0.25 m in total
- One mole of gas occupying 22.4 litres at one atmosphere
- The one orbit in which a satellite hangs motionless
What you find out here
- Plane sections stay plane, until the beam gets short Beam Deflection Calculator
- You cannot destabilize this loop with gain Bode/Nyquist Control Lab
- Use Celsius here and the engine looks half again as good Carnot Engine Simulator
- Ten times better data buys only a little more time Chaos Bouncing Balls Lab
- Some paths through this field are completely free Electric Field Visualiser
- Halving the detection range costs a 16-fold cut in cross-section Radar Physics Simulator
Articles written from these tools
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A machine that holds its own load is always under half efficient
Self-locking and "wastes more than half of what you put in" turn out to be the same inequality written two ways. That is why a bolt needs no ratchet, and why turning one is such miserable work.
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You can derive the pendulum formula without knowing any physics. Almost.
Dimensional analysis hands you the form of a law from the units alone: which quantities matter, and how. What it cannot hand you is the pure number in front, and knowing exactly where the method stops is what makes it useful.
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Ten degrees on the fire is worth 8.2 joules. Ten degrees off the river is worth 16.7.
Carnot efficiency depends on both temperatures, but not equally: dropping the cold side beats raising the hot side by a factor of T_H/T_C. Awkwardly, the cold side is usually a river you do not control.
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Four tools on this site are running the same equation. Its parameter has four names.
RC, 1/λ, 1/k and the membrane time constant are one quantity in four disguises. Learn it once and you own it in four subjects. The conversion people get wrong is half-life.
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Resonance is a phase relationship, not a peak
The amplitude peak of a driven oscillator shifts with damping and can vanish entirely, but the phase lag passes through exactly 90 degrees at the natural frequency for every damping value. Phase is the definition; the peak is a symptom.
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The pendulum formula quietly assumes a small swing
T = 2π√(L/g) is not the period of a pendulum. It is the period of a pendulum that barely moves, and the error grows to nearly 16% by the time the swing reaches 85 degrees.
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Why no piano is in tune
Twelve perfect fifths land 23.46 cents above seven octaves. That gap cannot be closed by any tuning, because it is a fact about prime numbers rather than about music, and every keyboard you have played is a negotiated settlement over it.
Every tool on this shelf
Mechanics & Motion
Beam Deflection Calculator
Calculate deflection, bending moment, and shear force for cantilever and simply supported beams
Efflux Jets from a Tank
Compare the speed, range and flight time of jets from three holes at different depths, and find the two that land in the same place.
Free-Fall Calculator
Calculate fall time, impact speed, and kinetic energy for an object dropped from a given height under any gravity.
Gear Train Ratio & Speed Simulator
Four gears, two ways to arrange them. Set the tooth counts and see which of them reach the output ratio and which are only along for the ride.
Inclined Plane, Screw & Wedge
A ramp, a screw and a wedge are one shape used three ways. Set the angle and the friction, and watch the advantage the geometry offers drift away from the one you actually get.
Kinematics (SUVAT) Calculator
Solve the SUVAT equations of motion for constant acceleration. Enter any 3 of displacement, initial velocity, final velocity, acceleration, and time to find the other 2.
Lever Classes & Mechanical Advantage
Set the two arms and the load, and watch the effort, the mechanical advantage and the distance the effort end has to travel. The class is decided by the geometry, not by the button you pressed.
Momentum & Collision Lab
Simulate 1D cart collisions to explore conservation of momentum, kinetic energy changes, and the coefficient of restitution.
Newton's Second Law (F=ma) Simulator
Push a block with friction and an optional incline, and watch Newton's laws play out: net force, F=ma, and a full free-body diagram, animated live.
Projectile Drag/Wind Simulator
Projectile motion with air drag and wind.
Projectile Motion Simulator
Simulate projectile motion with launch speed, angle, and launch height. See range, time of flight, and max height live.
Pulley Systems & Mechanical Advantage
Count the rope sections holding the load and you have the mechanical advantage. Change the efficiency and watch where the loss actually shows up.
Roller Coaster Energy Simulator
See kinetic and potential energy exchange on a customisable roller coaster track
Screw Thread
Explore screw lead angle, mechanical advantage, efficiency and self-locking.
Tank Drain & Torricelli Decay
Watch a tank empty under Torricelli's law, with the level falling as a parabola because the outflow follows the square root of the depth.
Wheel & Axle Mechanical Advantage
Two radii on one shaft. Set them and read the advantage, the effort, and how much rope each radius winds in a single turn.
Oscillations, Waves & Optics
Chaos Bouncing Balls Lab
Many-ball gravity and collision sandbox with twin-run divergence tracking.
Circular Motion Lab
Visualise centripetal force, centripetal acceleration, angular/tangential velocity, period, and frequency.
Doppler Effect Simulator
Visualize the Doppler shift with moving source wavefronts, Mach cones, and frequency calculations.
Double Pendulum Explorer
Interactive double pendulum chaos simulation.
Hooke's Law & Spring Lab
Explore force, displacement, potential energy, oscillation, and combinations of springs.
Lens and Imaging Studio
Move object and lens parameters to see real/virtual image formation with principal rays and lens equation updates.
SHM Resonance Lab
Explore mass-spring-damper dynamics with time response, phase portrait, and resonance curve.
Signal Sampling & Nyquist Theorem
Visualise the Nyquist theorem: see how sampling rate affects signal reconstruction and aliasing
Simple Pendulum / SHM
Simulate a simple pendulum with adjustable length, gravity, and initial angle. See period, frequency, and max velocity update live with a swinging animation.
Snell's Law Refraction Simulator
Visualize light refraction at a boundary with adjustable refractive indices and total internal reflection.
Standing Waves Explorer
Visualize standing waves with adjustable harmonic number, boundary conditions, and animated nodes/antinodes.
Thin Lens Ray Tracer
Trace rays through converging and diverging lenses with real-time image position and magnification.
Wave Interference / Two-Source
Explore two-source wave interference with adjustable wavelength, source separation, and fringe order in a live 2D pattern.
Thermodynamics
Carnot Engine Simulator
Explore the Carnot cycle: PV diagram, efficiency η = 1 − Tc/Th, and energy flow
Ideal Gas Law Calculator
PV = nRT with an interactive piston diagram and isotherm curves
Latent Heat: Ice in a Drink
Mix ice into water and read the temperature they settle at. Specific heat capacity, latent heat of fusion, and the 0 °C plateau where more ice changes nothing.
Electricity, Magnetism & Circuits
Electric Field Visualiser
Place point charges and see field lines, vectors, and equipotential contours
Inverse Square Law Explorer
See how gravity, light intensity, and electric force all decay as 1/r² — a geometric consequence of spreading over a sphere.
Ohm's Law Calculator
Solve V = IR for voltage, current, or resistance. Instant results with full formula.
RC Circuit Simulator
Visualize capacitor charging and discharging curves with adjustable R, C, and V₀.
RLC Resonance Analyser
Explore series RLC circuit resonance, Q factor, impedance, and frequency response
Smith Chart & RF Black Magic Lab
Interactive Smith Chart tool for RF impedance matching, reflection coefficient visualization, and understanding microwave black magic
Control Systems & Modern Physics
Missile Interception & Proportional Navigation
See how a guided interceptor homes in on a moving target. Compare pure pursuit with proportional navigation and watch the guidance equation a = N·Vc·λ̇ compute live, frame by frame.
PID Control Tuner
Tune PID gains and inspect step response.
Radar Physics Simulator
Simulate radar detection, Doppler filtering, PRF ambiguity, and range limits.
Special Relativity Explorer
Explore time dilation, length contraction, and the Lorentz factor - as velocity approaches the speed of light.
Other tools
Bode/Nyquist Control Lab
Explore gain, damping, and natural frequency with live Bode and Nyquist plots and stability margins.
First-Order Step Response Lab
Visualize how a first-order system responds to a step input: set gain, time constant, and step size, and watch the exponential rise toward its final value.