Pendulums and Periodicity

Physicists in the 16th and early 17th centuries had no clocks to measure seconds, so they used what was available. Many used their pulse. But an exciting experiment would cause their heart rate to go up and ruin the measurement!

When a pendulum moves away from its central (equilibrium) position, it experiences a gravitational force that tends to restore the pendulum’s central position. But the pendulum’s momentum makes it swing back and forth. Friction will cause the movement to stop eventually unless a source of energy, such as a falling weight, drives the motion.

not measure even smaller units of time with enough precision. Clocks that were accurate enough to measure time in seconds could only be made after Galileo (1564–1642) discovered the physics of the pendulum. As discussed in the sidebar on this page, the pendulum is important because in most situations its swing has a rate that is nearly constant and depends only on the pendulum’s length. This precise “ticking” can be the basis of an accurate clock.

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Galileo realized that the swing of a pendulum makes a good oscillator. A pendulum is a bar or a weight attached to a string that is connected to a pivot point at the top, as shown in the fi gure on page 106. When the pendulum is not moving, it hangs straight down, but a little push makes the pendulum swing—the force of gravity causes it to fall back and forth. The farthest that the pendulum reaches during its swing is called the amplitude. The period is the time required for the pendulum to make one full swing. The period is important because clocks can only be accurate if their ticking involves oscillatory (back-and-forth) motion with an unvarying period. Frequency is another term used to describe an oscillation, and it equals the number of back-and-forth cycles per second. The unit of frequency is the Hertz, which is one cycle per second. (The Hertz is named after Heinrich Hertz [1857–94], a German physicist who studied electromagnetic radiation and its frequencies.) Fre- quency is the inverse of period, so an oscillation having a period of

0.25 seconds has a frequency of 1/(0.25) = four cycles per second. Pendulums make good clocks because they have regular, constant periods. At small amplitudes, the period depends on the length of the pendulum but not on the amplitude. (At higher amplitudes, this is not true, which is undesirable because it can cause the period to vary.) Although Galileo put a lot of thought into the mechanism of pendulum clocks, the fi rst one was built by Dutch physicist Christiaan Huygens (1629–95) in 1656.

Pendulum clocks need something to give the pendulum a little boost (such as a falling weight in many “grandfather” clocks that use pendulums), otherwise the swing would gradually stop due to friction. Clockmakers also have to worry about thermal expan- sion; the period of a pendulum depends on its length, and if the period varies, so does the clock. Pendulums in the most accurate clocks are often made of material that does not expand much with temperature.

The pendulum solved many of the problems of time measurement. With pendulum clocks, physicists could conduct accurate experiments, and people could know exactly what time of the day it was. When a church or school met at 9:00 in the morning, there was no excuse for being late.

There was one problem, however, that pendulum clocks could not solve. This problem was important because it affected com- merce, the development of trade, and the exploration of the world.

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The weight of the three cylinders in this grandfather clock provide the force to keep the pendulum swinging. (Kyle Kirkland)

The problem pendulums could not solve involved the measurement of a ship’s location at sea. What sailors needed was an accurate clock that worked on the shifting, rolling deck of a ship—a place where the motion was too severe for pendulum clocks to function well.

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Why was time needed to determine a ship’s location? Any spot on the surface of Earth can be identified by its latitude—the north- south distance from the equator—and its longitude, the location along the east-west direction. Latitude was not a problem, at least in the Northern Hemisphere, because navigators could use the North Star, whose height above the horizon varies with latitude. Longitude was the reason why time became involved.

Longitude can be determined by using the position of the stars, but the problem is that stars except the North Star move during the night. (The North Star, Polaris, is almost directly above Earth’s axis of rotation—the North Pole—so the planet’s spin does not change this star’s position in the sky.) To find longitude from the stars, navigators needed to not only measure the position of the stars but also know the time of night they were making the measurement. Just knowing the day of the year was not good enough, the exact time mattered. This required an accurate clock.

The problem was severe enough that in 1714 England’s gov- ernment offered a reward of £20,000 to anyone who could make

a clock accurate enough to permit longitude measurement. (Today this amount of money in American currency would be worth more than a million dollars.) The money inspired a huge number of attempts, all of which failed in some way or another, but finally a carpenter and clockmaker named John Harrison succeeded. A new and improved set of timekeeping pieces resulted.

The clocks of Harrison and the clockmakers that followed him were based on either springs or wheels. These items had been used earlier, but Harrison had to make adjustments in order to increase their accuracy. Wheels, and the foliots mentioned earlier, fail to

be perfectly periodic. Springs can be used to run a clock—some- what like a falling weight—but the force exerted by a spring varies depending on its compression. These are the problems that Har- rison and others had to solve. They did so with elaborate and com- plex mechanisms, all of which ensured that the ticks of the clock were precisely equal, or, in other words, that the clock’s motion was steady and periodic.

Springs became common in small clocks such as wristwatches, but today clocks and watches that need to be wound (in order to

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compress the spring) are much less popular. Oscillations used to drive clocks include alternating current electricity, which in the United States oscillates at 60 cycles per second, and the vibration of an instrument called a tuning fork. Musicians use tuning forks to tune their instruments because tuning forks oscillate at a precise frequency. This precise frequency is also what makes tuning forks useful to timekeepers, since the periodic motion is an excellent basis for a clock.

In the 1950s, electric watches began to appear, run by batter- ies. Today many wristwatches have a quartz crystal (a material found in white sand) that oscillates at a precise frequency when it is controlled with electricity. The best quartz crystal clocks are accurate to within a few milliseconds (thousandths of a second) per year.

Physicists of the 20th century began experimenting with tiny pieces of matter and found even better oscillators—atoms. Atoms emit energy in the form of electromagnetic radiation at remarkably precise frequencies. Light is electromagnetic radiation of a certain frequency (in the range of 425,000,000,000–750,000,000,000 Hertz), but electromagnetic radiation exists at a lot of different frequencies (radio waves, microwaves, and X-rays are other examples of electromagnetic radiation). The radiation emitted under certain circumstances by cesium atoms is quite stable and has been used as the basis of “atomic” clocks, which can be accurate to within a few milliseconds per 1,000 years. Atoms are useful in clocks not only because of their precise periodicity but also because they are not affected by mechanical and thermal disturbances that disrupt pendulums, crystals, and springs. The radiation of the cesium atom is so precise that in 1967 scientists used it to define the second—a second is officially the duration of 9,192,631,770 oscillations of the cesium atom.

Pendulums and Periodicity