Simple Pendulum Motion - Interactive Simulation

Interactive simulation of simple pendulum motion with real-time visualization of angle, angular velocity, and energy conservation

Current Angle: 0.00°
Angular Velocity: 0.00 rad/s
Time: 0.00 s

Angle vs Time

Angular Velocity vs Time

Energy Conservation

Kinetic: 0.00 J
Potential: 0.00 J
Total: 0.00 J

Parameters

Physical Equations

Nonlinear Equation: θ'' + (g/L)sin(θ) = 0
Small Angle Approximation: θ'' + (g/L)θ = 0
Period: T = 2π√(L/g)
Energy: E = ½mL²θ'² + mgL(1-cosθ)
Theoretical Period (T): 2.01 s

What is a Simple Pendulum?

A simple pendulum consists of a mass m hanging from a string of length L and fixed at a pivot point. When displaced from its equilibrium position and released, it oscillates back and forth under the influence of gravity.

Small Angle Approximation

For small angles (θ << 1 rad), sin(θ) ≈ θ, and the motion becomes simple harmonic with period T = 2π√(L/g). This approximation is valid for angles less than about 15°.

Energy Conservation

In the absence of damping, the total mechanical energy of the pendulum is conserved. As the pendulum swings, energy continuously transforms between kinetic energy (maximum at the bottom) and gravitational potential energy (maximum at the turning points).

Damping Effect

In real systems, air resistance and friction cause damping, which gradually reduces the amplitude of oscillation. This is modeled by adding a damping term proportional to angular velocity: θ'' + b·θ' + (g/L)sin(θ) = 0.