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Galileo and Pendulums

Mathematical equations can be beautiful, or they can be ugly and messy. When we find a simple, elegant equation, we can rightly say it’s beautiful. Most people are familiar with the simple and elegant equation on the left. E=MC etcIt’s sometimes called Einstein’s equation and it expresses with admirable mathematical succinctness the relationship between energy, matter and the speed of light — a deep and astonishing fact of nature expressed in three letters.

Not so many people are acquainted with the equation on the right. It’s a humble formula, having nothing to do with the speed of light, atomic power plants, or thermonuclear bombs. Pendulum equationIt’s about pendulums. This equation expresses the relationship between the time it takes a pendulum to swing back and forth, the length of pendulum and the acceleration of a falling body due to gravity. The equation is associated with Galileo, and to some it’s as beautiful as the one about mass and energy.

According to one of Galileo’s students, Galileo was attending a religious service in Pisa when he noticed that a breeze caused very slight, very slow back-and-forth motion of a chandelier suspended in the great cathedral. Galileo’s mind was not focused on the sacred service being performed that day; instead he kept looking at the slow and gentle motion of the chandelier and he noticed that even though the breeze stopped and the back-and-forth distance traveled by the pendulum shrank, yet the time it took the chandelier to make the back-and-forth trip seemed to remain constant. There were no clocks back in 1582 — he’d invent one later – so he timed the swinging of the chandelier by the regular beating of the pulse in his wrist. He was right; no matter the distance traveled, the time it took was always the same.

Galileo by DomenicoLater, Galileo experimented with pendulums and discovered that the remarkably regular period of the pendulum (the uniform time it took to make a full back-and-forth sweep) was proportional to the square root of the length of the pendulum. The pendulum bob (the weight at the end of the pendulum) had no effect on the length of time or its regularity. The only things that mattered were the pendulum’s length and, of course, gravity that caused the pendulum to swing once it was released.

If you want to express time in, say, seconds, and you know that it’s the result of a mix of length and gravity, you have to compose an equation such that the units of length get removed and only seconds remain. If the acceleration of gravity is expressed as, say, centimeters per second per second, and the pendulum’s length is expressed in centimeters, then by dividing the pendulum’s length by the acceleration of gravity you get rid of the centimeters and are left with seconds per seconds. And if you take the square root of that, you re left simply with seconds. So the relationship between the  time T and the length l and gravity g looks like the proportional formula below. Pendulum equation proportionalThe two times pi, as in the  in the equation at the top right, turns the proportional expression into a true equation — but that involves a mathematical maneuver that Galileo didn’t get around to. The simplicity of the equation, with or without the 2pi constant, is striking, as is the curious fact that the swing of a pendulum is constant, even as the length covered by that swing becomes less and less. The recognition of that fact resulted in the invention of the pendulum clock, a timepiece that endured as the best method of marking time from Galileo’s era into the 20th century.

(By the way, the portrait of Galileo is by the artist known as Domenico, the son of the more famous Tintoretto. Galileo was himself the son of the famous musician Vincenzo Galilei, a performer, composer and theorist of music. From his father, Galileo learned  the mathematics of musical harmony and with that as a start it’s not surprising that when deep into the physics of the natural world he declared that the language of God’s creation was mathematics. )


We especially enjoyed the Twitter wit who reported that the United States Geological Survey rated the earthquake in Washington, DC,  as a magnitude 5.8, but Standard & Poor’s rating agency downgraded it to a 4.0.

Sexual Failure & the Social Network

Two social scientists have made an astonishing discovery. Or maybe Phrenological head. Jealousy must be in there someplace.not. Benjamin Cornwell at Cornell and Edward Laumann at the University of Chicago have published a study of erectile dysfunction and the social network. Or, as the title of the study says, “Network Position and Sexual Dysfunction: Implications of Partner Betweenness for Men.” (You probably never knew betweenness was a word; it probably wasn’t a word before these researches got to work.)

Their study showed that when a wife had more contact with her husband’s friends than he did, the chance that her husband would have trouble making love to her increased. Or as the researchers put it: “Men who experience partner betweenness in their joint relationships are more likely to have trouble getting or maintaining an erection and are also more likely to experience difficulty achieving orgasm during sex.”

Isn’t this called jealousy? Did this come as a surprise to the researchers? And isn’t that line in quotes a rather weird sentence? And do people ever achieve orgasm not during sex?

The two scientists also discovered that while this effect was most apparent for men in their 50s and early 60s, it just sort of disappeared for men in their 70s and 80s. Want to guess why? Don’t even try. Here’s the answer: “Older men’s greater focus on close, kin-oriented relationships increases their likelihood of adopting new definitions of masculinity that emphasize conveying experience and mentoring rather than independence and autonomy, and under these circumstances partner betweenness is less likely to trigger erectile dysfunction.” Now you know.

Just some flowers, damnit!We’re putting these flowers here just because we need a break from this all this heavy thinking. Besides, we like the way they look. The flower is called called foxglove and it belongs to a larger family called digitalis. You may have heard of digitalis as a medication for certain heart problems, specifically atrial fibrillation, a rapid and irregular heartbeat.  An extract from the foxglove plant was used as a remedy that disorder as far back as the late 18th century.  The name digitalis is appropriate, for the elongated bell shaped flower do fit neatly, glove like, over your finger tips — over your digits.  Get it? As for the name foxglove; well, there are lots of theories about that name but no one theory is agreed upon. And, anyway, we said we were posting this photo to escape heavy thinking.

More Notes

Tim Carmody, in his excellent piece, "How Haiti Became Poor", notes that President Trump's racist policies and vulgar language have sullied the word "shithole" which used to be one of the all-time great swear words. He's right. It's another terrible power this careless President wields.