By Steven Krolak

(NEW ALBANY, Ind.)—Subhranil De, associate professor of physics at IU Southeast, can defy gravity.

Sort of.

In a 2016 paper published in the European Journal of Physics, he devised a mathematical model to enable a cylinder to roll up a hill, accelerating toward the top and then decelerating as it rolled down the other side.

This year he’s back with another gravity-defying mind-bender pitting intuition against logic.

In a contribution to Mathematics Magazine, De devised another model that allows theoretical eggs to perch on their theoretical heads, with a sort of inverted reference to a toy from the 1970s called Weebles.

Along the way, he illustrated need-to-know concepts and demonstrated the fun side of physics.

**Wobbly Weebles and the triumph of logic**

Weebles were egg-shaped plastic toys that, like the cigar-shaped inflatable clowns a generation earlier, could not be induced to tip over, no matter how strongly they were nudged or, in the case of the clowns, punched.

“Weebles wobble, but they don’t fall down,” went the ad slogan that permeated many a childhood.

Probably most kids have always assumed that the reason Weebles couldn’t be tipped over was that they were loaded with some kind of ballast.

Yes and no.

There was indeed ballast, but it seems that it matters more precisely *how* the weight of that ballast is distributed, and consequently where the center of gravity of the Weeble is located in relation to the Weeble’s less pointed end, on which it tends to stand.

Consider the humble egg, a natural analog for the Weeble. Eggs can fall down. In fact they do precisely that, whenever they are placed on one of their ends, preferring to come to equilibrium on their sides (or on the floor, if you’re unlucky). This is because the egg’s weight is more uniformly distributed, so that the center of gravity is located more centrally rather than toward the end, or vertex.

De wondered what would happen if the distance of the center of mass were the same as the radius of curvature at the vertex.

The answer: a new theorem!

Wielding this new theorem, De constructs the model of a “doubly critical Weeble.” This egg-shaped Weeble can rest on *either* vertex—to all appearances, defying gravity. Like an ordinary egg-shaped Weeble, this theoretical Weeble can stand on its less pointed end, but if tilted, given a nudge, or released from any other position, interestingly enough, it would eventually stand up on the *more* pointed end.

De’s theorem is a higher order principle that builds on the established principle underlying the common type of Weeble.

Of course, the Weeble doesn’t really defy gravity so much as use gravity to create a sleight-of-hand that challenges us to understand it in a new way.

“When logic is telling me one thing, and intuition is telling me another thing, I am always drawn in,” De said. “But it always turns out that logic is right.”

**A debt to wooden wiener-dogs **

“I have always been intrigued by counter-intuitive mechanical systems,” De said.

Perhaps this fascination can be traced to a year of his childhood spent in Denmark, home of simple but often playfully creative wooden toys.

“I still have a wooden dachshund that rolls at an uneven speed, because the axles of the wheels are placed off-center,” De said.

For those Danish children who wonder why this works the way it does, the answer lies in a simple principle: the conservation of mechanical energy.

“If the gravitational potential energy increases, the kinetic energy must decrease since the sum of the two is a constant, and that is what leads to the change in speed in the dog,” De said.

Interesting, but not strictly counter-intuitive, since the dachshund’s potential energy increases as its height increases and consequently the center of gravity rises up. As a result, the kinetic energy decreases, causing the speed to decrease. When the dachshund loses height, it speeds up again.

However, for a Weeble, as well as De’s gravity-defying cylinder, his clever construct allows the center of gravity to go *down* when the object as a whole moves *up*, and vice versa.

There is much playfulness in De’s musings, but they are always geared to finding models that would lead to interesting or counter-intuitive phenomena that he has made mathematically possible, and using these to intrigue students and colleagues.

#### The physics of whimsy

The urge to wonder why things behave the way they do, and how they might behave if some of the factors affecting their behavior were altered, is not only a feature of childhood, but lies at the core of all scientific inquiry.

“It’s always good and useful to use intuition to teach physics,” De said. “But sometimes things are counter-intuitive, and that’s where they become even more engaging.”

They also offer a fun way to acquaint students with important core concepts.

For example, De’s cylinder paper involves the notions of neutral equilibrium, center of gravity, rolling without slipping, translational and rotational modes of kinetic energy of a rigid body, conservation of mechanical energy, incomplete elliptic integrals, static friction and the parallel axis theorem.

For its part, the Weeble paper deals with stable and unstable equilibrium, critical equilibrium, radius of curvature, differential calculus and the center of gravity (in addition to 1970s pop culture).

The papers may have begun in whimsy, “because the motivation was to play with interesting counter-intuitive mechanics,” as De observed.

But it is just that sort of impulse that intrigues and energizes De, and that helps bring principles of science to life.

“When anything like this is counter-intuitive, through engaging attention it can teach you the physics of it,” De said. “Nothing is useless.”