With my lanyard in hand (weighted by my keys), a gentle swinging motion will put the keys in pendulum motion, swinging back and forth. Pendulum motion is relatively easy to model since it is sinusoidal. With a more forceful swing, the keys on the will instead travel in a circular motion. Circular motion can also be modeled sinusoidally.

If I swing my lanyard with medium power, it will start to take the path of a circle, but before the keys can reach maximum height, they begin to fall and start to model parabolic motion instead. I do not know how to model the path of my keys on my lanyard in this situation. I know it will transition from circular to parabolic motion once the vertical component of the velocity goes under zero, but I don't know what equation I could use. Any help?

  • $\begingroup$ "Sinusoidal" refers to position vs. time. "Circular" refers to x vs. y position in space. Your question is extremely poorly formulated. $\endgroup$ Commented Feb 4, 2016 at 20:30
  • $\begingroup$ This is a related, and possible duplicate of this question. $\endgroup$
    – fibonatic
    Commented Feb 5, 2016 at 10:46

1 Answer 1


Note that pendulum motion is only sinusoidal for small angular displacements; as you increase the amount of swing the harmonic approximation fails.

Lagrangian mechanics gives you a handle on all of the cases.

  • $\begingroup$ Hi Peter, this is a comment not an answer $\endgroup$ Commented Feb 5, 2016 at 6:40
  • $\begingroup$ This is exactly how I would answer a student asking this question. It is up to them to clarify their thoughts. $\endgroup$ Commented Feb 5, 2016 at 12:27
  • $\begingroup$ It's a very sparse post, to be sure, but it doesn't seem so obviously not-an-answer as to be deleted. $\endgroup$
    – David Z
    Commented Feb 5, 2016 at 14:45

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