Interactive Math & Science Lessons
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.
Short sequences rather than single tools. Each step hands the next one something specific, so they are worth doing in order.
L1: foundations
PHYOhm’s law, as its own page says, describes one resistor at a steady temperature, on direct current, with nothing else in the loop. Each page after it puts back one of the things that sentence left out: a second component, the wire, the battery, a circuit that will not reduce, and a voltage that changes.
L2: school
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
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
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
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
PHYA capacitor has no resistance and limits the current anyway, and what it sets is a time rather than a number of ohms. The second page shows that the time belongs to any system with one store and one drain. The third adds a second store, and the time becomes a frequency.
L2: school
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
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
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.
L2: school
PHYRay optics says light goes straight and an edge casts a sharp shadow. Each page here draws that answer as a ghost beside the wave answer, and what stands in the light's way changes from page to page: two surfaces, one slit, a round aperture, and finally an opaque disc whose shadow is bright at its centre.
L2: school
PHYThe path a light ray takes, a sliding bead follows and a thrown ball flies is the one along which a small change makes no first-order change in a total: time for the first two, action for the ball. Every page lets you bend the path and watch the total barely move, and two of them find a path whose total is stationary without being least.
L2: school
PHYA spinning body carries a vector, its angular momentum, and nearly everything surprising about rotation is that vector refusing to change. Pull your arms in on a turntable and you spin faster; a wheel turns sideways instead of falling; a racket flips; a pendulum's plane stays put while the floor turns under it.
L2: school
PHYOhm's law describes one kind of part. The five here store energy in a voltage or a current, conduct in one direction, amplify, and amplify precisely, and each is unreliable on its own: a transistor's current gain β varies threefold between samples of the same type. The circuit around a component is what makes it predictable, and every page shows how.
L2: school
PHYLight arrives in packets, and the same constant h then gives every particle a wavelength. The rest follows from taking that wave seriously: made to fit a box, it has a lowest energy that is not zero; it leaks through walls it cannot climb; and squeezed into a small space it spreads over many wavenumbers, which is the uncertainty principle.
L3: undergraduate
PHYSeven pages on one question: which part of a structure is actually carrying the weight, and in what form. It starts inside a single beam, where the middle does almost nothing, and ends with a truss that has more bars than equations and still says exactly when it will fall.
L3: undergraduate
PHYThree equations built on one operator, the Laplacian, that differ only in what time does. Heat forgets, the sharpest detail first. A wave carries a disturbance away, and on a sheet never quite finishes arriving. Laplace's equation has no time in it at all, and it is where heat forgets its way to: every point the average of its neighbours.
L3: undergraduate
GEOA forecast is a model run forward from a starting state nobody knows exactly, so forecasters run it many times and let the disagreement say how far to trust it. The second page opens the model and finds a way to fail that has nothing to do with chaos. The third goes back to Lorenz's equations of 1963, where the run's position is soon lost and the shape it draws never is.
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.
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.
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.
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.
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.
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.
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.
