### Classical Mechanics

In 1656, a Dutch physicist and mathematician Christiaan Huygens completed the world’s first pendulum clock. A lifelong bachelor, Huygens built clocks of many shapes and sizes and collected his clocks in a single room.

One day he noticed that whenever he reset two identical clocks mounted on the same beam, within an hour, they would develop what he called an “odd sympathy”—the motion of their pendulums would synchronize.

In Huygens’s day, most people would have attributed this odd behavior either to God or to Satan, but as a rationalist, Huygens went searching for another reason. In this quiz, we’ll do the same. First, we'll experiment with the universal dynamics of a single, simple pendulum.

# Huygens' Clock Puzzle

For hundreds of years, Huygens’s pendulum clock was the world’s most precise timekeeper. But why was his invention so effective at keeping time?

We will learn later in this course that a pendulum is an example of a harmonic oscillator—a simple pattern of motion that is everywhere you look in nature, from atomic vibrations to cosmic radiation.

When you set a pendulum into motion, the time it takes to move back and forth is called its period of oscillation. Use the visualization below to release the pendulum from different heights.

How does the pendulum's period change as you increase the release height?

# Huygens' Clock Puzzle

Similar to the cellular automata you encountered earlier, an update rule generates the speed and position of the pendulum at any moment. This update rule is expressed as a differential equation called the equation of motion.

In this course, we will mostly skirt the prickly business of solving differential equations. In this quiz, the visualizations you're interacting with are solving the differential equations for you. But by the end of this course, setting up and solving the equation of motion for any mechanical system happen like clockwork!

# Huygens' Clock Puzzle

If a pendulum were all there is to Huygens's clocks, the "odd sympathy" he noticed would never spontaneously arise. Like most interesting phenomena in classical mechanics, his observation is the result of an interaction, and it turns out there is more to Huygens's clocks than meets the eye.

The challenge of building a clock with a pendulum for timekeeping is that no matter how perfect your design is, the pendulum will not swing regularly for more than a few minutes. Friction at the point of attachment of the pendulum removes energy from it during each cycle.

The genius of Huygens’s invention was the addition of a mechanical apparatus that used gravity to replace the energy dissipated by friction. The so-called escapement gives the pendulum a sharp push once per period. If you listen to a pendulum clock, you can hear the escapement mechanism working—the kick it applies to the pendulum is the source of a clock's characteristic tick and tock.

# Huygens' Clock Puzzle

A pendulum clock with a properly functioning escapement can keep good time over a period of days. However, the strength of the "kicks" supplied by the escapement has to be precisely tuned to the frictional losses. Kick too hard, and the pendulum swings out of control breaking the clock; kick too gently, and eventually the pendulum stops swinging altogether.

Adjust the size of the escapement's kick on the pendulum below. In what range does it need to be to stabilize the oscillation indefinitely?

Note: The size of the kick is expressed as the change in the velocity of the pendulum after it's applied.

# Huygens' Clock Puzzle

When Huygens hung two clocks side by side on a heavy beam and set them running, they were not in direct contact, yet over time, they started moving in a related way.

In fact, this kind of emergent behavior is common when simple oscillators exert a small, but persistent, influence over each other. This idea is the basis for a variety of phenomena, from phase transitions in matter, to avalanches and forest fires, to the sensory-processing capabilities of the brain. The common thread is interaction.

Which choice is likely the strongest interaction between the two clocks shown?

# Huygens' Clock Puzzle

Our horologists at Brilliant have programmed this visualization of the dynamics of the two pendulum clocks firmly attached to the same beam.

Their escapements have been tuned so that they will keep good time for several days. Watch for the synchronized state (and speed up the simulation if you get tired of waiting).

Which choice describes the "oddly sympathetic" motion that Huygens wrote about?

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