Abakcus
← All DIY cards

DIY · Card 33 · Engineering · Time

How to Make a Water Clock That Times Exactly One Minute

Two plastic cups, a pinhole and three-quarters of a cup of water. Tune the hole until the cup runs dry in exactly sixty seconds, and you have built the oldest kind of clock there is.

Time
30 minutes
Difficulty
Grown-up
For the pin
Mess
Wet table

01The brief

Build a clock from water that measures exactly one minute.

180 mL

of water, three-quarters of a cup: the clock’s whole supply of time

4×

as much flow from a hole twice as wide, because flow goes with the hole’s area

3,400

years ago, Egyptian priests timed the night with a stone water clock

Before springs, pendulums or quartz, people told the time by letting water run out of a pot. Fill it, let it drip, and the falling level tells you how long has passed. It sounds simple, and it is, until you try to make it exact.

Your challenge is one minute, no more and no less, from three-quarters of a cup of water. You will find that the hole has to be tuned to a fraction of a millimetre, and that water does not drain the way you might expect: fast at first, then slower and slower. Like the jar barometer, it is a measuring instrument you build and then calibrate yourself.

02Kit

What you need

Two cups, a pin and a stopwatch to check yourself against.

Tick off what you have

0 of 5 ready

03Method

Eight steps, from a pinhole to a minute

Make a hole, set up the cups, time the water, tune the hole, then mark the seconds.

Step 1 of 8 · Minute 0

Gather the kit

You need two plastic cups, at least one of them see-through, a drawing pin, a measuring cup, water, a waterproof marker, a stopwatch, and two books or a strip of card to hold the top cup up.

Why
One cup lets water out at a steady drip; the other collects it so you can see time pile up. The stopwatch is the clock you are trying to copy.
Watch for
Two cups, a pin and a timer, and a lot of water to come.

If it doesn’t work

The water won’t come out at all.

Why

The hole is too small, and surface tension holds the water back.

Fix

Widen it slightly with the pin, and check the hole isn’t blocked by a flap of plastic.

It takes a different time on each run.

Why

The amount of water or the cup’s level changes from run to run.

Fix

Measure the water exactly, and make sure the top cup sits flat and doesn’t move.

Water runs down the outside of the cup.

Why

The drip clings to the plastic and creeps sideways.

Fix

Make the hole in the very centre, and wipe the outside of the base dry before you start.

04Physics

Why water slows down as it runs out

Water leaves the hole because the water above it presses down. The deeper the water, the harder it presses, and the faster it squirts out. In the 1640s Evangelista Torricelli, a student of Galileo, worked out how fast: as fast as a drop would be moving if it had fallen from the surface down to the hole.

So a full cup gushes and an almost empty one dribbles. The time it takes to run dry depends on the area of the hole, the width of the cup and the depth of the water. Try to make it exactly one minute:

0:00

The clock on screen runs four times faster than real time.

Empties in

92 s

Off by

+32 s

Too slow. A slightly bigger hole lets water out much faster, because the flow goes with the hole’s area, the square of its width.

Mark the level every ten seconds and you find the marks aren’t evenly spaced. Egyptian clock makers met the same problem and answered it with shape: their stone vessels were wider at the top, so the fast early flow drained a thicker layer and the level fell more evenly. With the right curve, the marks come out perfectly even. Try the shapes:

0 s11 s23 s34 s45 s57 sfulltime →a steady fall

The marks bunch up: the biggest gap is 2.3 times the smallest. When the cup is full, the deep water pushes it out fast; near the bottom it only dribbles, so the level falls more and more slowly.

Both widgets use Torricelli’s law with a typical loss through a small hole in thin plastic, and stop when only a couple of millimetres of water are left, where real cups start to drip unevenly.

05The big idea

A clock is a
steady leak.

Every clock does the same two things: something runs down at a steady rate, and something counts how much has run. Water, sand, a falling weight, a swinging pendulum, a vibrating crystal of quartz.

The whole history of timekeeping is the search for a steadier leak. Your cup slows down as it empties; the clock makers of Egypt, Greece and China spent centuries finding ways to make it stop slowing.

06The math

Torricelli, a hole and a square root

1 · Torricelli’s law

v = √(2 g h)

Water leaves a hole under a depth h at the speed a stone gains falling from h.

2 · The flow

Q = C a √(2 g h)

A hole of area a passes a volume Q each second. C, near 0.6, accounts for the jet narrowing as it leaves.

3 · Time to empty

t = (A ⁄ C a) √(2h ⁄ g)

For a straight-sided jar of floor area A, filled to depth h.

Take a straight jar 7 cm across, floor area 38 cm². Three-quarters of a cup fills it 4.7 cm deep. With a 3 mm hole, a = 0.071 cm², the formula gives t = (38 ⁄ (0.62 × 0.071)) × √(2 × 0.047 ⁄ 9.8) ≈ 85 s. Too slow. Because t goes as 1 ⁄ a, and a as the width squared, a hole 3 × √(85 ⁄ 60) ≈ 3.6 mm across should run for about a minute.

The square root in the formula is why the marks bunch up. In a straight jar, √h falls at a steady rate, so after half the time three-quarters of the depth has already gone, because √(1⁄4) = 1⁄2. The last quarter takes the whole second half. To make the level fall at a steady rate, the vessel’s radius must grow as the fourth root of the height, r ∝ h^¼: wide at the top, narrowing towards the hole.

07Real life

Where flowing water kept the time

From Athenian courtrooms to Chinese clock towers, and still in every toilet cistern.

  • Athenian law courts

    In ancient Athens, speeches in court were timed with a klepsydra, ‘water thief’: a pot that drained through a spout into another. When a speaker’s water ran out, so did their time. Pausing the clock to read evidence meant plugging the spout.

  • Toilet cisterns

    The float valve that refills a toilet cistern and shuts off when it is full is the same idea Ctesibius used to keep his clock’s water at a constant depth. A rising float closes the inlet: one of the oldest feedback controls there is.

  • Su Song’s clock tower

    In 1090s China, the scholar Su Song built a clock tower about twelve metres high. A water wheel, filled cup by cup from a tank kept at a steady level, turned a bronze armillary sphere and set little figures ringing bells and gongs to sound the hours.

  • The elephant clock

    In 1206 the engineer Ismail al-Jazari described a water clock built into a model elephant. A bowl with a hole slowly sank in a tank inside it; every half hour it tipped, and the clock released a ball, sounded a cymbal and moved a little rider.

  • Hospital drips

    A drip feeding medicine into a patient’s vein runs on the same physics as your cup. The height of the bag above the arm sets the pressure, a narrow tube sets the flow, and nurses count drops through a little window to check the rate.

  • Hourglasses

    The hourglass appeared in Europe in the Middle Ages. Sand doesn’t behave like water: a deep pile of grains presses on its sides, not its bottom, so sand flows out at nearly the same rate however full the glass is. That makes it steadier than a simple water clock.

c. 270 BC

In Alexandria, around 270 BC, the engineer Ctesibius noticed what you have noticed: water drips out faster when the jar is full. His answer was a float valve. A float in the supply tank rose as it filled and closed the inlet, so the water there stayed at a constant depth and dripped out at a constant rate.

The steady drips filled a second vessel, where a rising float carried a pointer past the hours. Historians count it among the first machines to control themselves by feedback, sensing their own state and correcting it. His writings are lost; we know the design from later engineers who copied it. The float valve in every toilet cistern is its direct descendant, and a float that rises and sinks on command is the trick behind the Cartesian diver, too.

08Is and isn’t

Four things people believe about water clocks

Water runs out of a hole at a steady rate.

It slows down as the level falls. The flow goes as the square root of the depth, so a full cup gushes and a nearly empty one dribbles. That is why the marks on a simple water clock bunch together.

Twice as much water takes twice as long.

Not in the same cup. More water means deeper water, which flows out faster. In a straight-sided jar, doubling the depth makes the clock run only about 1.4 times as long.

A slightly bigger hole makes it a little faster.

The flow goes with the area of the hole, which is the square of its width. A hole 10 % wider lets water out about 20 % faster. Pinprick changes matter.

Water clocks were rough toys until real clocks came along.

The best regulated water clocks were among the most accurate timekeepers in the world for nearly two thousand years, until pendulum clocks arrived in the 1650s.

09Go further

Five more tests for your clock

  1. A

    Half and double

    Time the clock with half the water, then with double. Is the time half and double? Use the square-root rule to predict it first.

  2. B

    Two holes

    Make a second hole the same size beside the first. Does the clock run exactly twice as fast? Why might it not quite?

  3. C

    Five-minute clock

    Can you build a clock that measures five minutes? You’ll need a smaller hole, more water, or both.

  4. D

    Shape hunt

    Try the same hole in a straight jar, a plastic cup and a funnel. Which gives the most evenly spaced marks?

  5. E

    Sand versus water

    Fill the holed cup with dry sand instead of water. Does sand slow down as it empties, the way water does?

10Questions

Water clocks, quickly answered

How do you make a water clock that times one minute?

Make a tiny hole in the bottom of a plastic cup with a drawing pin and rest it above a clear cup. Pour in three-quarters of a cup of water and time how long it takes to run out. Widen the hole a little if it is too slow, or tape over it and make a smaller one if it is too fast, until it empties in exactly one minute.

How does a water clock work?

Water runs out of a small hole at a rate that depends on how deep it is. With the same amount of water and the same hole, it always takes the same time to run out, so the falling or rising water level can be marked to measure time. That is what an outflow water clock, or clepsydra, does.

Why does a water clock slow down as it empties?

Because the water flows out faster when it is deeper. The speed through the hole goes as the square root of the depth, a rule found by Evangelista Torricelli in the 1640s. As the cup empties the pressure falls, the flow slows, and the marks for equal times get closer together.

How did ancient water clocks keep steady time?

In two ways. Egyptian clock makers gave their vessels sloping sides, wider at the top, so the level fell more evenly. Later, around 270 BC, Ctesibius of Alexandria kept the water depth constant with a float valve, so water dripped at a steady rate into a second vessel whose rising float marked the hours.

What is the oldest water clock?

The oldest one that survives is an alabaster vessel found at the temple of Karnak in Egypt, made around 1400 BC in the reign of Amenhotep III. It has sloping sides, a small hole near the bottom, and scales inside for telling the hours of the night.

Safety · A grown-up should handle the drawing pin and put it away afterwards. Work on a tray or by the sink, and wipe up spills so nobody slips.

All DIY cards →

Keep wandering

A few more pieces in the same spirit — math, design, and slow attention.