Equation of the Day #4: Measurement Standards
Standard Meter Pendulum
The word meter (or metre for those who live across the pond/in Canada) comes from the word for "measure" in Greek/Latin (e.g. speedometers measure speed, pedometers measure steps, &c.), but the meter I'm talking about is the International System (SI) unit of distance. The original definition of the meter was one ten-millionth of the distance from the Earth's equator to the North Pole at sea level (not through the Earth). The first person to measure the circumference of the Earth was the Greek mathematician/astronomer/geographer Eratosthenes (and he was accurate to within 2% of today's known value) circa 240 B.C., so this value was readily calculable in 1791 when this standard was accepted.
In 1668, an alternative standard for the meter was suggested. The meter was suggested to be the length a pendulum needed to be to have a half-period of one second; in other words, the time it took for the pendulum to sweep its full arc from one side to the other had to be one second. The full period of a pendulum is
So, when L = 1 m and T = 2 sec, we get what the acceleration due to gravity, g, should be in meters per second per second (according to this standard of the meter). It turns out that g = pi2 meters per second per second, which is about 9.8696 m/s2. This is very close to the current value, g = 9.80665 m/s2 which are both fairly close to 10. In fact, for quick approximations, physicists will use a g value of ten to get a close guess as to the order of magnitude of some situation.
So, you may be wondering, why is it different nowadays? Well, among a few other changes in the standard meter including using a platinum-iridium alloy bar, we have a new definition of the meter: the speed of light. Since the speed of light in a vacuum is a universal constant (meaning it is the same no matter where you are in the universe, unlike the acceleration due to gravity at a point in space), they decided to make the distance light travels in one second a set number of meters and adjust the meter accordingly. Since the speed of light is 299,792,458 meters per second exactly, this means that we have defined the meter as the distance light travels in 1/299,792,458th of a second.
This is all nice, but it's not a very intuitive number to work with. After all, we humans like multiples of ten (due to having ten fingers and ten toes), so why not make a length measurement of the distance light travels in one billionth (1/1,000,000,000th) of a second (a.k.a. nanosecond)? That seems a bit more intuitive, don't you think? It turns out that a light-nanosecond is about 11.8 inches, or about 1.6% off of the current definition of a foot. In fact, one physicist, David Mermin, suggests redefining the foot to the "phoot," or one light-nanosecond, since it's based off of a universal constant while the current foot is based off the meter by some odd, nonsensical ratio.