Mathematical pendulums are among the most fundamental oscillating systems studied in physics. This interactive simulation allows you to explore how pendulum motion depends on length, gravity, and initial displacement. The model uses the full nonlinear pendulum equation and integrates the motion using a fourth-order Runge–Kutta method for improved numerical accuracy.

Brief Theory #

A mathematical pendulum consists of a point mass suspended from a massless, inextensible string.

For small oscillation angles, the period is given by:

T = 2π√(L/g)

where:

• T is the oscillation period,
• L is the pendulum length,
• g is the gravitational acceleration.

Increasing the pendulum length increases the oscillation period.

Increasing gravitational acceleration decreases the oscillation period.

This simulation uses the full nonlinear equation of motion:

θ’’ + (g/L)sin(θ) = 0

The equation is numerically integrated using the fourth-order Runge–Kutta (RK4) method, providing accurate results even for larger oscillation amplitudes.