DON'T BE alarmed: the equation reprinted above has nothing to do with nuclear fission, economic forecasting or new paradigms. Rather, it is concerned with one of life's simpler pleasures: the act of getting a playground swing to go higher.
As with most things physical, the pleasure is simpler than the explanation of it. This equation, in fact, is just a small part of a five-page article in the American Journal of Physics setting forth a new and controversial theory of the forces involved in pumping a swing. It came about because Robert Case, a physics professor at Grinnell University, ceased to believe in the standard explanation of swing-pumping -- one that had been taught to generations of students. It had to do with something called "parametric instability": at the bottom of the swinger's swoop he or she "pumps the swing by rocking back and forth, raising and lowering the center of mass." That didn't jibe with what Prof. Case observed on the playground. "The children . . . were rocking more at the ends of the arc than they were at the lowest point," he noted.
He and a student, Mark Swanson, produced a paper ("The Pumping of a Swing From the Seated Position") arguing that it was the swinger's rocking forward near the front of the arc and backward near the back that produced the motion by making the swing into a "driven harmonic oscillator." The paper, submitted to the physics magazine in June 1988, initially "attracted a barrage of criticism," according to The Wall Street Journal, but was accepted for publication after a year and finally appeared last May.
Children with a precocious command of the physical sciences may wish to study it in detail with an eye toward maximizing their performance the next time they go out to play on the harmonic oscillator. Most others will find it more efficacious just to shout, "Hey, pa, give me a push."