Classical Mechanics
Projectile motion, forces, energy, collisions, and rotational dynamics
Vortex Ring (Smoke Ring)
Self-propelling toroidal vortex shown as a 2D cross-section: counter-rotating Lamb–Oseen pair drifts at V_self ≈ Γ/(4πa).
Rayleigh–Taylor Instability
Heavy fluid above light: Fourier modes grow as exp(σt) with σ = √(A g k); mushrooming fingers reach down.
Faraday Waves
Vertical parametric drive of a thin liquid layer: Mathieu sub-harmonic instability locks stripes, squares or hexagons.
Airfoil Streamlines (Joukowski)
Potential flow over a Joukowski airfoil: stream-function bands with the Kutta condition fixing circulation.
Projectile Motion
Launch projectiles with adjustable angle, velocity, and gravity. Trace parabolic paths with real-time graphs.
Trebuchet
Counterweight-driven beam: torque launches a projectile at a chosen release angle. Explore range vs masses and arm lengths.
Chain Sliding Off Table
Uniform chain on a smooth edge: hanging weight pulls, friction on the tabletop resists. Watch when it slips and how acceleration grows with s.
Block Stack & Friction
Pull the bottom block in a vertical stack: spring-like links with friction caps show who slips — compare with rigid-stack estimates.
Bicycle Stability (2D)
Side view: roll dynamics with fork trail and gyroscopic wheel torques — see speed, trail, and ω = v/R vs lean.
Spring Pendulum
2D elastic pendulum: swing and spring stretch together — energy swaps between modes; try chaotic-looking presets.
Bridge Resonance (1-D mode)
Damped modal oscillator with harmonic drive: sweep ω near √(k/m) — presets for cadence-like and low-ζ peaks.
Rising Bubble (Archimedes)
Ideal-gas bubble expands as it rises (p = p_atm + ρgy); buoyancy vs weight + Stokes drag — depth and velocity graphs.
Belt Drive & Slip
Two pulleys: tension difference limited by μ and wrap θ (e^{μθ}); load torque vs τ_max and simple slip on ω₂.
Free Fall
Drop objects of different masses with optional air resistance. Prove Galileo right.
Uniform vs Accelerated Motion
Compare constant velocity and accelerating objects side by side.
Relative Motion
Boat crossing a river, plane in wind. Vector addition visualization.
Circular Motion
Object on a string with centripetal acceleration and force vectors.
Force Diagram Builder
Place objects, add force vectors, see net force and resulting acceleration.
Newton's Cradle
Conservation of momentum and energy visualized with swinging balls.
Inclined Plane
Adjust angle and friction. See force components and resulting motion.
Inclined Plane: Work & η
W = F·s, friction and gravity work, ΔU, efficiency η = ΔU/W_F; slide-down preset.
Friction Simulator
Static vs kinetic friction with adjustable coefficient.
Slide to Stop
Initial velocity on a rough table: constant μ_k g decel, time and distance to rest.
Atwood Machine
Two masses on a pulley. Adjust masses to see acceleration and tension.
Elevator Physics
Person on a scale in an elevator. See apparent weight change.
Energy Conservation
Ball on a roller coaster track with real-time KE, PE, and total energy bars.
1D Force Field
U(x) presets, F = −U′, bead on potential, phase (x,v) and E(t).
1D Collisions
Elastic and inelastic collisions with momentum and energy tracking.
2D Collisions
Billiard-ball style collisions with adjustable angles.
Torque & Balance
Beam on a fulcrum. Place weights to balance or tip.
Lever Classes
1st / 2nd / 3rd class: arms, τ about fulcrum, MA vs Torque & Balance.
Blocks & Tackle
n supporting strands, same rope tension T, ideal MA = n, F = T.
Simple Pendulum
Adjust length, mass, and gravity. Observe period and damping effects.
Conical Pendulum
Steady cone: ω(θ,L), T and mg vectors, T_rev vs simple-pendulum T₀.
Physical Pendulum (Rod)
Thin uniform rod: pivot along L, I and T(δ), equivalent length L_eq.
Pendulum Collision
Two bobs: hit along normal, e elastic; θ¨ between hits vs 1D collisions.
Yo-Yo Dynamics
Unwinding string: a = g/(1+I/mr²), T, α, optional friction torque τ.
Satellite Yo-Yo Despin
L = const: ω_f = ω_0(I+2mr_i²)/(I+2mL²) as tethers pay out from r_i to L.
Rubber Sheet & Ball
Sheet height ∝ −Σm/r; ball rolls along −∇h — embedding metaphor, not GR.
Slinky Drop (Springs)
1D chain: release top anchor; bottom lags until stress wave — toy masses & k.
Huygens Pendulum Sync
Two pendula + κ(θ₁−θ₂) on shared beam; phases drift toward lock.
Ballistic Pendulum
Bullet hits block: embedded vs e; ω₀ = v/L, θ_max, energy graph.
Center of Mass System
2–4 bodies or rod: R_cm, V_cm; explosion with ΣΔp = 0; |P| and |V_cm| graphs.
Car on a Curve
Flat: v_max = √(μgR), F_c vs μmg. Banked ideal: tan θ = v²/(gR). Top view + wedge.
Quarter-Car Suspension
¼-vehicle vertical model: sprung vs unsprung masses, Kₛ, Cₛ, tire Kₜ, sinusoidal road — RK4 time histories.
Capstan (Rope on Cylinder)
T₂ = T₁ e^{μφ}: μ, wrap angle φ, top view + T₂/T₁ vs φ graph.
Spring Work (F–x)
F = kx graph, shaded W = area = ½kx²; U(x) curve vs Spring-Mass dynamics.
Double Pendulum
Mesmerizing chaotic motion with path tracing.
Spring-Mass System
Simple harmonic motion with adjustable spring constant and damping.
Coupled Oscillators
Two masses, three springs: normal modes ωₛ, ωₐ and beats.
Forced Oscillator
Driven damped harmonic oscillator: transients, resonance curve A(ω).
Buoyancy Simulator
Drop objects in water. Float or sink based on density.
Bernoulli Flow
Pipe narrows: speed up, pressure down. Continuity + Bernoulli.
Hydraulic Press
Pascal: same pressure, large piston → large force. F₂ = F₁·A₂/A₁.
Rocket Propulsion
Variable mass: thrust ṁu, Tsiolkovsky Δv, vertical launch with gravity.
Angular Momentum
Two masses on a rod: I = 2mr², change r or m and watch ω adjust to keep L constant.
Rolling & Sliding Disk
No-slip v = ωR vs sliding: translational vs rotational KE, disk vs hoop inertia.
Gyroscope Precession
Gravity torque τ = mgd, spin L = Iω, steady precession Ω ≈ τ/L — schematic 3D view.
Coriolis Effect
Puck on a rotating platform: curved path in the rotating frame vs straight line inertial.
Wall Bounce
2D box: frictionless walls, restitution e, optional gravity. Trails and kinetic energy graph.
Oblique Wall Impact
One tilted segment: e, angles to normal n, open boundaries vs closed Wall Bounce.
Rutherford Scattering
Repulsive 1/r² orbit sketch vs impact parameter and energy.
Ideal Line Vortex
v_θ = Γ/(2πr) arrow field; circulation and potential flow cartoon.
Forearm Lever (class 3)
Elbow fulcrum, load at hand, muscle moment arm — torque estimate.
Communicating Vessels & Manometer
Hydrostatic balance, U-tube ΔP = ρgΔh, and inclined-tube vertical head.
Simple Machines
Wedge, wheel & axle, and screw: ideal mechanical advantage vs geometry.
Arch & Wedge Statics
Compressive thrust in a stone arch and wedge force resolution (schematic).
Boids (Flocking)
Separation, alignment, cohesion on a torus; cursor as predator cue.
Particle Life
Six types, pairwise matrix forces on a torus — clusters, worms, foam presets.
Poincaré Section (Double Pendulum)
(θ₁, ω₁) when sin θ₂ crosses 0 with ω₂>0; RK4, chaotic return map.
Catenary Cable
Uniform chain between level anchors: y ∝ cosh(x/a); sag, arc length, tension directions.
Magnus Effect (Ball)
Same v₀ and θ with vs without spin: toy a = (kωv_y, −g − kωv_x); range comparison.
Fluid Surface (Accel / Spin)
Linear tank: tan α = a/g; rotating bucket: paraboloid sketch vs rpm.
Water Hammer (1D)
Linearized P,V waves; valve closes; Joukowsky ΔP ≈ ρaV hint.
Foucault Pendulum (Sketch)
Ω_eff = Ω_E sin|λ| with time scale; top view rotating swing line.
Tippe Top (Schematic)
Offset COM, friction at rim, spin — qualitative flip when μ is high enough vs spin.
Kapitza Pendulum
Pivot shakes vertically: fast driving can stabilize the inverted equilibrium — parametric pumping in θ̈ + (g/L) sin θ ≈ (Aω²/L) cos(ωt) sin θ.
Rattleback (Celt)
Chiral hull couples spin and rocking in a toy ODE: one rotation sense reverses after growing wobble.
Maxwell Wheel
Disk on a thin axle unwinds string: shared linear and rotational kinetic energy from falling potential, a ≈ g / (1 + I/(mr²)).
Brachistochrone Race
Beads race down a line, a circular arc, and a cycloid with v = √(2gy); the cycloid wins the descent-time minimum.
Coupled Pendulum Chain
N pendula with neighbor springs in θ: watch traveling waves and reflections after a center kick.
Sliding Ladder
Frictionless wall and floor: θ̈ = −(3g/2L) cos θ; released from rest, wall contact is lost when sin θ = (2/3) sin θ₀.
Angle of Repose
1D sandpile lattice: avalanches relax until slopes stay below μ = tan α — compare the green guide line.
Cart & Inverted Pendulum
Horizontal force on a cart with a hinged rod: coupled (M+m)ẍ and mL²θ̈ integrated for manual balancing.
Woodpecker Toy
Stick–slip sleeve on a rod: static hold, spring loading, then kinetic slip — a qualitative hop cycle.