# Path of a swinging object on a rope [closed]

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?

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