Graph (versus time):
Graph (versus angle):
Compare to (in orange):
Check here to show the free-body diagram

Purple is the true motion; orange is the comparison motion.

This is a simulation of a simple pendulum (a ball attached to a massless rod). If the damping is set to zero, the pendulum moves without resistance. The larger the damping, the larger the resistive torque.

Check out all the different graphs, and compare the zero damping case to the case with damping. You can also compare the real motion to the motion under the small-angle approximation - this is a ball for which the gravitational torque is proportional to the angle (an approximation) instead of what is actually true (and what happens for the purple ball), that the gravitational torque is proportional to the sine of the angle, measured from the equilibrium position. Note that the small-angle approximation is inconsistent with energy conservation!

If you view the free-body diagram, you will see the radial and tangential components of the force of gravity acting on the ball, the force the rod exerts on the ball, and the damping force (if damping is not zero).

Simulation written by Andrew Duffy, and first posted on 8-03-2017.
Updated version posted on 8-07-2017. Another update (graph colors) on 10-25-2017.
Another update with energy graphs in the small-angle approximation on 01-05-2018
Added custom buttons on 10-30-2018
10-26-2019: Updated ball and plot colors to purple and orange, and fixed a bug regarding plotting the small-angle energy when comparing to nothing.
10-28-2019: Fixed a bug associated with the step buttons.

Creative Commons License
This work by Andrew Duffy is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
This simulation can be found in the collection at

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