Poe, Bessel, and Drones
What's a connection between Edgar Allan Poe and Friedrich Wilhelm Bessel?
Poe's story The Pit and the Pendulum describes a pendulum with a semicircular blade being lowered on a cable. This is an interesting variation on the usual harmonic oscillator because the length L of the pendulum is not constant but rather an increasing function of time L(t).
Using the small angle approximation sin() , the equation of motion is
L '' +L' ' + g = 0.
If we assume the length is increasing at a constant rate, then with a change of variables described in [1], this differential equation becomes Bessel's equation and its solutions are Bessel functions.
The authors of [1] state that their primary interest is whether a pendulum can have the motion that Poe describes. But as they indicate in their opening sentence,
The study of the motion of a steadily descending pendulum has practical applications (e.g., to the oscillations of a load being lowered by a crane).
A more recent application would be describing the motion of a package being delivered by a Zipline drone.
Related posts[1] Robert L. Borrelli, Courtney S. Coleman, and Dana D. Hobson. Poe's Pendulum. Mathematics Magazine, Vol. 58, No. 2 (Mar., 1985), pp. 78-83
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