# How does a pendulum clock work?

## What is a pendulum?

A pendulum is a rod hanging vertically from its top end (or a weight called a bob hanging from a string) that swings from side to side due to the force of gravity. As Italian scientist Galileo Galilei (1564–1642) discovered, the clever thing about a pendulum is that it always takes the same amount of time to make one complete swing. In theory, the only thing that affects how fast a pendulum swings is its length and the strength of gravity. For relatively small swings, the time (T) it takes to make one complete swing (known as the period) is given by this little equation:

Mathematical equation for the period T of a pendulum. T is equal to two times pi times the square root of (l divided by g).

Here, l is the length of the pendulum and g is a measure of gravity’s strength (what we call the acceleration due to gravity). You can see from this equation that you need to quadruple the length of a pendulum to double the time it takes to make a swing.

## How does a pendulum work?

#### The pendulum works by converting energy back and forth

A pendulum works by converting energy back and forth, a bit like a rollercoaster ride. When the bob is highest (furthest from the ground), it has maximum stored energy (potential energy). As it accelerates down toward its lowest point (its midpoint, nearest the ground), this potential energy is converted into kinetic energy (energy of movement) and then, as the bob climbs up again, back to potential energy. So as the bob swings (oscillates) back and forth, it repeatedly switches its energy back and forth between potential and kinetic. Something that works this way is called a harmonic oscillator and its movement is an example of simple harmonic motion, though we won’t go into those things here.

#### Friction and resistance

If there were no friction or drag (air resistance), a pendulum would keep on moving forever. In reality, each swing sees friction and drag steal a bit more energy from the pendulum and it gradually comes to a halt. But even as it slows down, it keeps time. It doesn’t climb as far, but it covers the shorter distance more slowly—so it actually takes exactly the same time to swing. This handy ability (technically called isochronism, which just means “equal amounts of time”) is what makes a pendulum so useful for timekeeping.

#### Huygens pendulum clock

Galileo figured that out straight away and though he never actually managed to build a complete pendulum clock. he came quite close (here’s a model of the 1642 pendulum clock he was designing just before his death); it was left to another brilliant scientist, Dutchman Christiaan Huygens (1629–1695), to finish the job in the 1650s.

## How does a pendulum clock work?

#### The easiest way to build a clock from scratch

Suppose you want to build a clock from scratch in the simplest way possible with the fewest number of parts. You could start with a dial and some hands and move them around the face with your finger, counting seconds to yourself and moving the hands accordingly. You move the second hand once a second, the minute hand once every 60 seconds, and the hour hand once every 60 minutes (3600 seconds). Some clock! That’s going to get tedious quite quickly, so what about automating things? You could mount the hands on a little axle driven by what we’ll call “timekeeping gears,” so that the second hand automatically turns the minute hand at 1/60 of its speed, and the minute hand, likewise, turns the hour hand at 1/60 of its speed. Then all you have to do is count seconds, turn the second hand, and the rest of the job is done for you.

#### Method for automatically driving pointer

But, hang on, that’s still pretty tedious. What we really need is some way of powering the hands automatically. You could wrap a piece of string around the axle and attach a weight to it. As the weight falls, it will pull the axle around, turn the second hand, and that will drive the rest of the clock. The only trouble is, the weight is going to fall really quickly and the second hand will whizz around too fast so the clock won’t keep time. Okay, let’s introduce another set of gears—we’ll call them “power gears” (to avoid confusing them with the timekeeping gears)—that will take power from the falling weight and transform it so that, as the weight falls, the second hand advances exactly one position on the dial in one second. But that still won’t work because the weight is going to accelerate as it goes down, like any falling object. In other words, the clock is going to get faster and faster until the weight hits the ground with a smack!

#### Adjust the speed of weight drop

What we need to add is a mechanism that regulates how fast the weight can fall, allowing the whole timekeeping mechanism to advance so the second hand moves one second on the dial (and only one second) in a time of one second. That’s what the pendulum does. As it swings from side to side, it rocks a lever called an escapement that locks and then unlocks the part of the mechanism driven by the falling weight. (Think of it this way: the mechanism is locked and the escapement releases it so it can move—in other words, lets it escape—once per second.) It’s this repeated locking and unlocking that makes the tick-tock sound you can hear. Since (in theory, at least) a pendulum of a certain length always takes the same amount of time to swing back and forth, the pendulum is what keeps the clock to time. The escapement mechanism that the pendulum regulates also (cleverly) keeps it moving back and forth by repeatedly giving it a slight nudge—an extra injection of energy to counteract friction and drag.

##### That’s not exactly how pendulum clocks work; it’s a very simplified approximation of what’s going on that’s reasonably easy to follow.

Escapement in a pendulum clock. Photo by Anders Sandberg. Animation showing how a simple lever escapement works in a falling weight pendulum clock.

## A summary of how clocks work

#### In summary, then, the key parts of a pendulum clock are:

1. A dial and hands that indicate the time.
2. A weight that stores (potential) energy and releases it to the clock mechanism as it falls, very gradually, over the course of a day (or several days, if you’re lucky). Winding the clock raises the weight back up, storing more potential energy to power the mechanism.
3. A set of power gears that take energy from the falling weight and use it to drive the clock mechanism at the correct speed. If we use a really heavyweight and the right power gears, the weight will store enough energy to drive the clock for days without us having to wind it up. (Remember the law of conservation of energy here: the longer the clock runs, the more energy it uses; a clock with a heavier weight can store more potential energy so, generally speaking, it’s going to run for longer without winding than one with a lighter weight.)
4. A set of timekeeping gears that drive the different hands around the clockface at different speeds. These are usually finer and more precisely made than the power gears.
5. A pendulum and escapement that regulate the speed of the clock and keep it (more or less) constant.

#### In practice, clocks have lots of other bits, pieces, parts, and features that horologists (master clockmakers) like to refer to—with splendid honesty—as “complications.”

Grandfather clock pendulum Grandfather clock dial A large grandfather clock
Photo: 1) When most people think of a pendulum clock, this is what they picture in their mind: a grandfather clock (also called a longcase clock). 2) Clocks like this have a long pendulum that makes only a relatively narrow sweep back and forth. 3) Grandfather clocks often have very ornate and beautifully painted dials.

### Some drawbacks of pendulum clocks

Clemens Riefler pendulum clock made in Germany c1904. NIST photo.

Can you see any problems with pendulum clocks? Me too! A pendulum of a certain length takes the same amount of time to move back and forth if the strength of gravity stays the same. But what if the length of the pendulum changes? That could certainly happen if it’s a metal pendulum and the room temperature goes up or down (say, between winter and summer), because metals expand in the heat and contract in the cold.