You know the Slinky, but have you ever wondered how it walks down stairs, or why it appears to hang in midair when dropped? Scientists have peered into these phenomena for decades, but now a Princeton professor has mathematically proved that the answers can be found in your junior high physics class.
Slinkies get their “walking” ability from two properties: wave motions and Newton’s laws. When a Slinky sits atop a staircase, gravity acts on the toy, keeping it still. Knock over the Slinky, and Newton’s second law comes into play. As middle school physics class may have taught you, this law states that providing force to an object increases its acceleration. Gravity begins to provide this force, as soon as the Slinky is cast down a stairway. This motion is sustained through a directional wave that ricochets throughout the coil and stops when the toy hits the bottom of the stairs. Contrary to how it looks, the Slinky doesn’t walk, it somersaults.
“This is all simple Newtonian physics. Force equals mass times acceleration,” said Bob Vanderbei, a mathematician at Princeton University. But the Slinky captivates both scientists and YouTube stars alike because when dropped from a great height, it appears to “float” for a split second. Scientists have been researching this phenomenon for more than 25 years using high-speed cameras, mathematical proofs and lots of Slinkies.
Vanderbei recently wrote an article for American Mathematical Monthly that quantified the reasons why the falling slinky acts this way.
“It’s the combination of the Slinky pulling upward and gravity accelerating downward, and those two effects cancel each other out,” Vanderbei said. Both of these effects can be explained by relatively simple algebra and some not so simple calculus.
You can think of the Slinky coils as being 98, loosely connected objects. When one coil hits another, the center of mass is transferred down the slinky as it begins to bunch together. You can see an interactive model of this on Vanderbei’s website.
“There is no floating effect,” said Michael Wheatland of the University of Sydney, who has also studied the Slinky. “The center of mass falls with acceleration g. It’s just that the coils above the center of mass fall faster, and the ones below slower.”
Because the falling slinky is governed by simple Newtonian physics, the force of gravity is really the only constant you need. This concept means if you dropped a slinky from a helicopter, barring the fact that there would be the wind from the rotors blowing in every direction, these falling slinky physics would still apply.
“The scale of things doesn’t matter, as long as we are here on Earth,” Vanderbei said. “Things don’t change if gravity is constant, but if we were extending [the Slinky] from the moon to the Earth, that would be a very different setup.”
In this scenario, the collapse of this 238,000-mile long Slinky would take hours or maybe even days at such a large scale, and that’s not to mention the effect of the moon’s gravity tugging on the Slinky too. Also, given that the Earth is spinning once per day, things would get messy. Good luck untangling a Slinky wrapped from Beijing to Beirut.
Vanderbei couldn’t really think of any practical applications for the falling Slinky, but he believes the toy is great for getting people interested in the field of mathematics.