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Physics-based simulation of a vibrating pendulum with a pivot point is shaking rapidly up and down. Surprisingly, the position with the pendulum being vertically upright is stable, so this is also known as the inverted pendulum.

Students can click and drag near the pendulum to modify its position. The anchor can also be moved. Enable the "show controls" checkbox to set gravity, frequency of oscillation, magnitude of oscillation, and damping (friction).

A regular non-vibrating pendulum is stable only when it is hanging straight down. In this simulation, the support pivot point of the pendulum is oscillating rapidly up and down. When the oscillation is rapid and of small amplitude, there is a second stable position, with the pendulum standing straight up, in a vertical upright "upside-down" position.

Some experiments to try:

  • Disturb the pendulum from its stable inverted position. Can it recover? How far can you disturb it?
  • What is the slowest frequency that still has the stable inverted position?
  • What is the range of amplitude that still has the stable inverted position?
  • For a faster or slower frequency, does the range of amplitude change?
  • Does the strength of gravity affect whether the inverted pendulum is stable?
  • How does friction (damping) affect the inverted pendulum?
  • Make the time step smaller to get more calculations per second, which will give a smoother graph.

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