Could Cycloidal Pendulums Make Peter and Miles Better Spider-Men?
A very brief dive(or swing?) into isochronism
A conversation between Peter and Miles in the PlayStation game “Marvel’s Spider-Man”:
Peter: If a pendulum is 40m long and is attached at a 45 degree angle, how far would the pendulum fall at its lowest point?
Miles: 11.72m, but why does that matter?
Peter: It matters a lot if you’re the pendulum
Over the past month, I’ve been reading James Stewart’s Calculus: Early Transcendentals, comparing it to the textbooks I read when I took Calculus in my undergrad math program. In today’s reading, I encountered cycloids.
Stewart describes a cycloid as “the curve traced out by a point P on the circumference of a circle as the circle rolls along a straight line. Think of the path traced out by a pebble stuck in a car tire.” A helpful illustration is shown in the picture labeled Figure 15.
It turns out that curves which form part of an inverted arch of a cycloid(like the illustration labeled Figure 18) solve the tautochrone problem, that is, “no matter where a particle P is placed on an inverted cycloid, it takes the same time to slide to the bottom”.
Why does this matter? Well, according to the 17th century Dutch physicist Christiaan Huygens, if you managed to build a pendulum bob that traced out a cycloidal arc, the time taken for one complete oscillation would be constant and would not depend on the amplitude of the swing, “whether it swings through a wide arc or a small arc”. This phenomenon is referred to as isochronism, a nice property to have in timekeeping devices.
However, does this work in practice? I’m still very new to the concept of cycloids so my knowledge is, as yet, very limited. However, what I’ve found so far hasn’t been very promising. For instance, Proof Wiki states, “the technique proved impractical, as the energy losses caused by the mechanics of the system compromise the pendulum’s ability to swing reliably for a practical length of time.” Bummer.
However, there is hope, not in the timekeeping world, but in the web-slinging world. Peter Parker and Miles Morales are both comic book characters whose web swings closely mirror the swings of an actual pendulum. So theoretically, if they(in the air) ever needed to gang up on a villain(on the ground) at the same time, it would help if their web swinging trajectories traced an arc of an inverted cycloid.
This is why mathematicians make the best superheroes. And therein lies the truth.
Originally published at https://mathirl.substack.com.