Presets
Playback
Parameters
You can also click or drag inside the scene: the pointer sets the release angle.
Model checks
The science
Equation of motion
For a ball of mass m on a rigid rope of length L the angle obeys d2θ/dt2 = -(g/L) sin θ - c dθ/dt, where c is a viscous damping rate. The simulation integrates this with a fourth-order Runge-Kutta step of 2 ms.
Tension and energy
Along the rope T = m (g cos θ + L ω2): the tension is largest at the bottom, where the ball is fastest. Without damping the sum KE + PE = ½ m L2ω2 + m g L (1 - cos θ) is constant; the phase portrait is a closed curve.
Period
For small angles T = 2π√(L/g). At any amplitude θ0 the exact period is T = 4√(L/g) K(k) with k = sin(θ0/2), where K is the complete elliptic integral of the first kind (computed here with the arithmetic-geometric mean). At 90 degrees the period is already 18 % longer.
Honest simplifications: the rope is massless and inextensible, the ball is a point, damping is linear in the angular speed (real air drag grows with speed squared), and this 2D page does not model the crates that the 3D scene knocks over.
What to try
- Use Ideal (no damping) and compare the exact period readout with the small-angle one.
- Use Near the top: the phase portrait develops a pointed shape and the swing slows near the apex.
- Double the ball mass: the period does not change, but the tension does.
- Switch to Moon: the period grows by the square root of the gravity ratio.