Water clock

Study the flow of a water clock by measuring the mass of water collected over time with the FizziQ balance.

How do you measure time without a clock or watch? Long before the invention of mechanical clocks, Egyptian, Greek, and Chinese civilizations used the clepsydra, a water clock whose principle is simple: water flows through a hole and the amount collected indicates elapsed time. But is the flow rate constant? By measuring the mass of water collected over time with the FizziQ connected balance, you can discover that the answer is no — and understand why using Torricelli's theorem.

Activity overview:

The student builds a water clock from a pierced plastic bottle and measures the water flow in real time using the FizziQ connected balance. They observe that the flow rate decreases as the water level drops, then model the m(t) curve.

Level:

High School

FizziQ

Author:

Duration (minutes) :

30

What students will do :

- Measure the mass of water collected over time using a connected sensor
- Plot and interpret the m(t) curve to identify the non-linearity of the flow
- Understand the relationship between hydrostatic pressure and flow velocity
- Apply Torricelli's theorem to model the outflow
- Discuss the limitations of a water clock for uniform time measurement

Scientific concepts:

- Torricelli's theorem
- Hydrostatic pressure
- Volume and mass flow rate
- Mass conservation
- Mathematical modeling (parabolic function)
- Uniform versus non-uniform flow

Sensors:

- FizziQ connected balance (force sensor via FizziQ Connect)

Material needed:

- Smartphone or tablet with FizziQ
- FizziQ Connect module with force sensor (balance)
- Plastic bottle with a small hole near the bottom
- Collection container
- Water
- Stopwatch

Experimental procedure:

Open FizziQ Connect and select 'External sensors'. Connect to the M5 Stack module via Bluetooth and select the mass (force) measurement.
Fill the plastic bottle with water and place the collection container on the balance. Tare (zero) the balance.
Position the pierced bottle above the collection container so water flows through the hole into the container on the balance.
Start recording in FizziQ Connect. The mass of collected water is recorded continuously.
Let the water flow until the bottle is nearly empty. This typically takes 3 to 10 minutes depending on the hole size.
Stop the recording and examine the mass versus time graph.
Observe that the curve rises steeply at first (high flow rate when the bottle is full) then flattens (flow rate decreases as the water level drops).
The curve is not a straight line, confirming that the flow rate is not constant. Try fitting a parabolic function to the data.
Calculate the flow rate (slope of the mass curve) at several different times. Verify that the flow rate decreases over time.
Compare your data with the theoretical model from Torricelli's theorem: the flow velocity depends on √h, so the mass curve follows a parabolic law.

Expected results:

The m(t) curve shows a characteristic shape: it rises rapidly at the beginning (high flow rate when the bottle is full), then its slope decreases progressively until the bottle empties. The curve is well fitted by a parabolic function, consistent with Torricelli's theorem. The flow rate at the start is typically 2-5 times higher than at the end.

Scientific questions:

- Why does the flow rate decrease over time? What is the connection to pressure?
- What mathematical function best models the m(t) curve?
- What shape should a container have to produce a constant flow rate?
- How did ancient civilizations compensate for the non-uniform flow of water clocks?
- What are the main sources of error between theory and experiment?
- How does the hole diameter affect the emptying time?

Scientific explanations:

Scientific analysis of the water clock

The water clock is one of the oldest time-measuring instruments. Its operation relies on the flow of fluid through a small orifice driven by gravity.

Torricelli's theorem

The flow velocity through the orifice depends on the water height h above the hole. According to Torricelli's theorem, the velocity is:

v = √(2gh)

where g ≈ 9.81 m/s² is the gravitational acceleration and h is the height of the water column.

Flow rate and collected mass

The volume flow rate through the orifice of cross-section s is Q = s × v = s × √(2gh). As the water flows out, the height h decreases, so the flow rate decreases continuously.

Mathematical modeling

By combining mass conservation with Torricelli's theorem, one can show that for a cylindrical container of cross-section S, the water height follows:

h(t) = (√h₀ - (s/S) × √(g/2) × t)²

where h₀ is the initial height. The mass collected in the container therefore follows a parabolic law as a function of time.

T = (S/s) × √(2h₀/g)

Consequence for time measurement

Since the flow rate is not constant, a simple free-flowing water clock does not measure time uniformly. To solve this problem, ancient civilizations designed conical or funnel-shaped containers.

Sources of error

Discrepancies between the theoretical model and experimental measurements can arise from several factors: surface tension at the orifice, viscous losses, vortex formation, and air intake through the hole.

Extension activities:

Frequently asked questions:

Q: The FizziQ balance shows zero values during acquisition. What should I do?
R: This may occur if the sensor is not properly connected. Check the Bluetooth connection and restart the M5 Stack module.

Q: The water flows too fast to get good data.
R: Use a smaller hole or a thicker bottle. You can also partially cover the hole with tape to reduce the flow rate.

Q: My curve does not look parabolic.
R: Ensure the bottle is cylindrical (constant cross-section). A bottle with a tapered neck will produce a different curve shape.

Q: How accurate is Torricelli's theorem in practice?
R: The theorem gives an idealized velocity. Real flow is reduced by a discharge coefficient (typically 0.6-0.7) due to viscous losses and contraction of the jet.