This simulator models a nonlinear pendulum — a point bob on a massless, inextensible string swinging from a fixed pivot under gravity and viscous damping. Adjust length, release angle, damping, gravity and mass, then watch the swing, its angle-versus-time trace and its energy exchange update together from the same underlying model.
• A real-time 3D view of the pivot/support frame, suspended bob and string, an angle arc with vertical reference, kinetic/potential energy indicator bars and the swing path, with home view, focus-selected-part, auto-rotate, expand and toggleable labels. • Five live controls: pendulum length (0.5-4 m), release angle (-80° to 80° from vertical, released from rest), viscous damping rate, gravitational acceleration (1.62-15 m/s²) and mass. • Run/pause the simulation, advance by 0.1 s or 1 s single steps, and a playback-speed selector from 100x slow motion to 1 minute per second. • Reset laboratory (fresh trial from rest) and a live sequence narrative describing what is happening. • Nine live metrics: current angle, angular velocity, signed tangential velocity, kinetic energy, potential energy above the bottom, mechanical energy, string tension, the small-angle reference period and the measured full period. • A Curves & measurements tab with an energy-exchange chart, an angular phase-portrait chart, the full model equations and snapshot measurements. • An Experiments tab with four guided scenarios (small-angle reference, large amplitude, energy decay, length scaling), a model-verification bench of independent automated checks, and a timestamped event log with a copyable trial report. • A Learn & assess tab with guided lessons, a two-question knowledge-check quiz and a written scope/reference statement.
The pendulum's true equation of motion is θ'' = −(g/L) sin θ − γθ', where the restoring term depends on sin θ rather than θ itself. Replacing sin θ with θ is only a small-angle approximation, valid for modest release angles — at large amplitudes such as 75°, the measured period runs noticeably longer than the small-angle prediction 2π√(L/g), which this simulator lets you confirm directly by comparing a 5° release against a 75° release at the same length and gravity.
Energy exchanges between kinetic and potential form as the bob swings: potential energy referenced to the lowest point converts to kinetic energy during descent, and in the undamped model their sum stays constant. Adding viscous damping introduces a term proportional to angular velocity that removes mechanical energy over successive swings without changing the pendulum's mass-independent ideal motion for a given damping rate.
The model panel shows θ'' = −(g/L) sin θ − γθ', tangential speed v = Lθ', kinetic energy K = ½mL²θ'², potential energy U = mgL(1−cos θ), string tension Tstring = m(g cos θ + Lθ'²), and the small-angle reference Tsmall ≈ 2π√(L/g). The measured full period only becomes available after two successive same-direction center crossings, so it has no value for a pendulum released and left at rest with zero amplitude.
This is a point-bob, massless-string, fixed-pivot, constant-gravity teaching model. The nonlinear dynamics are integrated with RK4 at steps of at most 0.005 s, and the swing path uses a fixed visual radius for display — the length control changes the underlying model and the labeled measurements, not the drawn arc's visual size.
The exact restoring term in the pendulum equation is proportional to sin θ, not θ. The small-angle approximation 2π√(L/g) replaces sin θ with θ, which only holds closely for modest amplitudes. At a large release angle such as 75°, the nonlinear sine term makes the true, measured period longer than the small-angle reference — a comparison you can run directly in the Experiments tab.
At the bottom of the swing. Potential energy relative to the lowest point is zero there, so all of the mechanical energy has converted to kinetic energy, giving the bob its maximum speed. At the extremes of the swing the bob is momentarily at rest, with all energy stored as potential energy.
No. For the specified damping model, mass cancels out of the ideal angular equation of motion, so a heavier or lighter bob follows the same angle-versus-time curve at a given length, angle and damping rate. Mass does change the absolute kinetic and potential energy values and the string tension, since those scale directly with mass.
This is a point-bob, massless-inextensible-string, fixed-pivot teaching model with constant gravity. It excludes string mass, pivot friction beyond the modeled viscous damping term, air resistance beyond that same damping term, and any elastic stretching of the string. Release is restricted to within ±80° from vertical and dynamics are solved numerically with RK4 at small fixed steps.