Interactive visual
core
Balancing Act: Torque and Levers
force times lever arm decides which way it turns
A lever balances when m₁r₁ = m₂r₂. Torque is force times lever arm, and net torque over moment of inertia gives angular acceleration.
Physics
Interactive physics from forces and fields to quanta and curved spacetime, with the equations behind each model.
21 cheat sheets · 5 subcategories
Mechanics
Forces, oscillation, orbits, torque, waves and fluids.
Interactive visual
core
Buoyancy
the water pushes up with the weight of what you displace
Archimedes: a body is pushed up by the weight of fluid it displaces, so it floats with ρ_object/ρ_fluid of its volume under. Boats and crates in games.
Interactive visual
common
Damped Pendulum
a sine-shaped pull back to the bottom, minus friction
A swinging weight pulled by sin θ: its period is only 2π√(L/g) for small swings, and damping decays the arc. Grapples, lamps and ropes in games.
Interactive visual
common
Kepler's Laws
ellipses, equal areas, and T² ∝ a³
Orbits are ellipses with the star at one focus, sweep equal areas in equal times, and have T² ∝ a³. Place planets on rails exactly with Kepler's equation.
Interactive visual
common
Projectile Data Lab
small launch errors become a spread of landing spots
Repeat a launch with a little speed error and the landings form a distribution. Mean, spread and why R ∝ v² shifts the average: the maths of weapon spread.
Interactive visual
core
Projectile Motion
constant sideways speed, steadily changing vertical speed
Launch angle, speed and gravity fully determine a ballistic arc: x moves at constant speed while y accelerates downward.
Interactive visual
common
Solar System: Two-Body Orbits
everything orbits the shared centre of mass
Two gravitating bodies circle their shared barycentre: radii split by mass, period set by total mass. Plus a stable n-body integrator for games.
Interactive visual
core
Sound Waves
air molecules shuffle back and forth, the pattern travels
Sound is a longitudinal pressure wave: frequency sets pitch and wavelength, amplitude sets loudness. The physics behind 3D audio, delay and Doppler.
Interactive visual
core
Spring-Mass Oscillator
pull back in proportion, bleed energy, push in rhythm
Hooke's law, damping and a periodic push: the forced oscillator behind springy cameras, wobbly UI and suspension, and why driving at ω₀ explodes.
Electromagnetism
Fields, charged particles and electromagnetic waves.
Interactive visual
advanced
Charged Particle Motion
B bends, E pushes, together they drift
The Lorentz force curls a charge around magnetic field lines and crossed E and B make it drift sideways. Integrate it with Boris, not Euler.
Interactive visual
common
Electric Field Lines
add every charge's push, then follow the arrows
Coulomb fields add as vectors; tracing streamlines along the net field draws lines that leave positive charges, end on negative ones and never cross.
Interactive visual
advanced
EM Wave Propagation
leapfrog E and B on a staggered grid
Light is E and B regenerating each other. A 1D Yee grid (FDTD) leapfrogs them in time, slows them in glass and blows up past the Courant limit.
Thermodynamics
Heat flow, gases and statistical energy distributions.
Interactive visual
advanced
Boltzmann Energy Distribution
high energy is possible, just exponentially rare
At temperature T a state of energy E is weighted by e^(−E/kT). Metropolis sampling reproduces the distribution with a simple accept/reject rule.
Interactive visual
common
Heat Diffusion
each cell drifts toward the average of its neighbours
Heat flows down temperature gradients at a rate set by the Laplacian. The explicit grid update is four lines of code, and is stable only if αΔt/Δx² ≤ ¼.
Interactive visual
core
Ideal Gas Kinetics
pressure is just particles bouncing off walls
Temperature is average kinetic energy and pressure is momentum delivered to the walls. Bounce point particles in a box and PV = NkT falls out.
Quantum Mechanics
Wavefunctions, tunneling and wave packets.
Interactive visual
advanced
2D Schrödinger Wave Packet
a cloud of probability that spreads, scatters and interferes
A Gaussian wave packet in 2D drifts, spreads and splits at a potential bump; brightness shows |ψ|² and colour shows the quantum phase.
Interactive visual
common
Particle in a Box
walls force whole half-waves, so energy comes in steps
A particle trapped between hard walls can only hold standing waves, so its energy is quantized as n² and superpositions slosh at beat frequencies.
Interactive visual
advanced
Quantum Tunneling
a wave leaks through a wall it can't climb
A quantum packet hitting a barrier higher than its energy still partly gets through: transmission falls off exponentially with barrier width.
Relativity
Curved spacetime, gravitational waves and precession.
Interactive visual
advanced
Gravitational Waves
orbiting masses shake spacetime at twice their orbit rate
A compact binary radiates gravitational waves at twice its orbital frequency; chirp mass and distance set a strain of around 10⁻²¹ or smaller.
Interactive visual
common
Mass Effect on Spacetime
more mass, deeper well, tighter orbits
Mass curves spacetime: picture a gravity well whose depth follows −GM/r, with the Schwarzschild radius 2GM/c² marking where light can no longer escape.
Interactive visual
advanced
Mercury's Perihelion Precession
relativity keeps the ellipse from quite closing
General relativity adds a small 1/r⁴ pull that makes an orbit's closest point creep forward each lap: about 43 arcseconds per century for Mercury.