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SimplePendulum&HarmonicMotion

Q: Why does a pendulum take longer to swing at large initial angles?

The linear approximation assumes sin θ ≈ θ, which holds true only for small angles (< 15°). At large angles like 90°, sin θ < θ, meaning the restoring force near the extremes is weaker than linear spring physics predicts. This weakens acceleration and elongates the period by ~18%.

Q: What does the phase-space portrait of a pendulum represent?

A phase-space portrait plots angular velocity (ω = dθ/dt) on the y-axis against angular position (θ) on the x-axis. For undamped motion, it forms closed elliptical orbits representing energy conservation. With air friction, the orbit spirals inward toward the resting origin (0, 0).

Q: How does planetary gravity change the pendulum's oscillation frequency?

Period is inversely proportional to the square root of gravity (T ∝ 1/√g). On the Moon where gravity is 1/6th of Earth (1.62 m/s²), a 1-meter pendulum takes nearly 5 seconds per swing compared to 2.01 seconds on Earth.

Explore harmonic motion and large-angle pendulum physics. Adjust string length, bob mass, release angle, and air damping to visualize phase-space trajectories and measure exact oscillation periods.

Launch Interactive Lab Read the theory

Simple Pendulum & Harmonic Motion interactive Physics simulation illustration

Runge-Kutta RK4 Non-Linear OscillatorPhase-Space (ω vs θ) • Photogate Laser Sensor • Energy Conservation

Interactive Experiment Guide

Use this studio like a real-time physics workbench

Start with fundamental scientific principles, launch the simulation, and verify mathematical predictions against real-time outcomes.

DisciplinePhysics

Simulation ModeInteractive Numeric Engine

Governing ModelRunge-Kutta RK4 Non-Linear Oscillator

DeploymentIn-Browser WebAssembly / GPU

Scientific Foundation

What is simple pendulum & harmonic motion?

A simple pendulum consists of a point mass m suspended on a massless string of length L. The exact non-linear equation of motion is d²θ/dt² = -(g/L) sin θ - γ(dθ/dt). For small angles (θ ≤ 15°), sin θ ≈ θ yields Simple Harmonic Motion with period T0 = 2π√(L/g). For larger angles, exact elliptic integrals demonstrate period elongation T ≈ T0(1 + ¼sin²(θ0/2)).

Interactive Simulation Flow

Experiment Execution & Governing Equations

Launch the simulation workspace, adjust parameters in real time, and observe the immediate response in the telemetry and graphical indicator loops.

Small-Angle Approximation & Large-Angle CorrectionT_0 = 2\pi\sqrt{\frac{L}{g}} \quad \text{and} \quad T \approx T_0\left(1 + \frac{1}{4}\sin^2\frac{\theta_0}{2}\right)

Frequently Asked Questions