Principle

A common experiment in science education consists of setting a pendulum oscillating and measuring its half-period using an optical gate placed at its equilibrium position. In addition to providing a way to relate the period to the Earth's gravitational field, this experiment highlights one of the anharmonic effects of such an oscillator: the dependence of the period on the initial angle.

The simulation reproduces this experiment virtually, taking air resistance into account through a damping coefficient $\gamma$. The plotted curve corresponds to the theoretical trajectory for $\gamma = 0$.

Enregistrement.de.l.ecran.2026-10-12.a.17.12.28-compressed.mp4

Interactions

Keyboard controls

• Tab to stop the pendulum and clear the plotted points
• R to slow down the simulation (for increased precision)
• S to take a screenshot

Mouse controls

• Click and drag the slider to adjust the damping
• Click and drag the pendulum to set its initial state

Numerical Method

The pendulum's motion is modeled using the differential equation
$$\ddot{\theta}+\frac{1}{2}\gamma|\dot{\theta}|\dot{\theta}+\omega_0^2\sin(\theta)=0$$

where $\omega_0=\sqrt{g/\ell}$ is the pendulum's natural angular frequency, and $\gamma$ is a damping coefficient accounting for air resistance (assumed to be quadratic in velocity).

This differential equation is integrated numerically using the Verlet method, with a time step equivalent to 2 ms in slow-motion mode.