How is a parabola related to a circle?

Question

I know this seems like an incredibly general or even silly/random question, however, please allow me to explain why I'm asking this question.

Someone who, for example, didn't understand that trigonometric functions and circles are related, would be missing out on a whole genre of important possible insights about trigonometry and circles. I feel that I am missing a similar important understanding of a connection between parabolas and circles.

Here's why.

Note that if you plot a pendulum's position against time it would form a sine/cosine function. The force that sets a pendulum in motion is gravity. That same force causes a ball to fall in a parabolic path.

Circles -> sine functions -> pendulum -> gravity -> parabola.

Something else led me to this question, but this was my way of explaining it. I feel like I'm missing an insight that is deep, beautiful, and important.

Answer 1

From a geometrical point of view the link between circle and parabola is given by the conic sections. Indeed both of them can be obtained by intersection between a plane and a cone.

With reference to the pendulum and the projectile parabolic motion I'm not sure we can establish a link other than the underlying principle of mechanics and the related equations which contain always some kind of approximation (e.g. classical mechanics, relativity, etc.).

Answer 2

The motion of a pendulum swinging on a circular path is only approximately sinusoidal, and the approximation becomes very poor when the pendulum swings through a large angle. In order to make the approximation better, you must either limit the motion to a very small angle (so that the path of the pendulum bob barely curves, and it is practically indistinguishable from part of a parabola) or you must make it travel along a non-circular arc. So the analogy between pendulums and projectile motion is a bit flawed.

There is, however, a gravitational phenomenon that relates all the conic sections to each other: the two-body problem. This is a mathematical model in which two spherically symmetric bodies (for example, a sun and a planet) move under the influence of each other's gravity, assuming no other bodies influence their motion.

Depending on the masses of the bodies, the distance between them at a particular time, and their relative velocities at that time, the path followed by the planet can be a circle, an ellipse, a parabola, or a hyperbola.