A. Introduction

Time can be measured using periodic or continuous physical processes. Before the development of modern atomic clocks, various methods were used for timekeeping, such as sundials, pendulum clocks, and water clocks.

A water clock measures time through the controlled flow of water. In this experiment, you will investigate the physical principles governing water flow and apply them to construct a water clock.

B. Experimental Components

The following equipment is provided:

1. Tank A (lower tank, equipped with a valve and two hole plugs)

2. Tank B (upper tank, equipped with a hole plug, for water supply)

3. Tank support

4. Water collection container

5. Two water bottles (2 L each, not shown in the picture)

6. Siphon pump

7. Stopwatch

8. Beaker

9. Ruler

10. Spirit level (bubble level)

11. Adjustable support base

12. Towel (not shown in the picture)

*The towel is a souvenir, please take it with you after the exam is over.

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Adjusting the support base level

There are four screw pedals below the plate of the adjustable support base. You can change the height of each pedal by rotating it.

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Siphon pump

If the water container fills up with a lot of water, you can remove it using a siphon pump. Insert the straight hose into the water, and put the flexible hose into the water tank. Then, by squeezing and releasing the red handle, you can transfer the water into the water tank.

NOTE:

Ensure that the adjustable support base and water tanks are level before starting any measurements, so that you can measure the heights correctly.

Empty the collection container regularly to prevent floating of the apparatus.

Dry your hands before writing on the answer sheet or handling other experimental equipment.

Inform a supervisor immediately if there is leakage, malfunction, or you lose the hole plug.

C. Experiment

Water discharge through a valve

1. Place the adjustable support base in the water collection container

2. Adjust the screw pedals to ensure the base is level.

3. Place the tank support onto the adjustable support base.

4. Place tank A on the tank support.

Tank A has three outlets: one 5 mm diameter hole, one 4 mm diameter hole, and one 4 mm diameter valve.

In part A, use only the valve.

A1  ^0.50 Measure the internal dimensions of Tank A: its width, its length, and the spacing between the height markings on the side of the tank. Note that the height markings begin at the level of the centre of the valve and the holes.

A2  ^1.00 Fill Tank A with water to a height of 5.0 cm. Open the valve fully. Measure the time taken, with uncertainty, for the water level to decrease from 5.0 cm to 4.0 cm. Use the markings on the tank to measure the height.

A3  ^1.50 Repeat the measurements in A.2 for at least four different initial water heights greater than 3.0 cm (at initial heights below 3.0 cm, the water takes too long to drain out). For each measurement, record your experimental data and calculate the average time taken for the water level from each initial height to decrease by 1.0 cm.

A4  ^1.00 Find the average speed $v_v$ at which water flows out as the water level decreases by 1.0 cm from each initial height.

NOTE : You may use the calculator provided.

A5  ^1.00 Assume that the average speed satisfies

$$v_v = C_1 h^x$$
Plot a suitable graph to determine the exponent $x$. What is the exponent $x$ obtained from your graph?

Note that it is better to use the centre value between the initial and final heights to determine the exponent $x$.

Water discharge through a hole

In this part, use the smaller hole in tank A, which has a diameter of 4 mm.The valve used in part A should be closed.

B1  ^1.50 For at least five different initial heights $h$, measure the time taken for the water level to decrease by 1.0 cm.

B2  ^0.50 Using the same method as in A.4, determine the average speed $v_h$ of the water leaving the hole for each initial water height.

B3  ^1.00 Compare the average flow speed through the valve $v_v$ and through the hole $v_h$.

Assuming that $v_v = C_2 v_h$, determine the constant $C_2$.

Construction of a water clock

Construct a water clock using Tanks A and B.

Arrange the tanks as follows:

Tank A (with the valve) is placed below.
Tank B (with the 5 mm hole) is placed above.

Tank B supplies water to Tank A.

Tank A discharges water through the valve, which is fully open.

For stable operation, Tank A must be kept full (or overflowing) by the water supplied from Tank B.

C1  ^1.00 Let $h_c$ be the minimum water height (measured from the centre of the hole) in Tank B required to keep Tank A full or overflowing. Design an experiment to measure the value of $h_c$. Record your experimental data and determine $h_c$.

C2  ^1.00 Tank B must be refilled periodically to keep the water clock operating. Design an experiment to determine the maximum time interval between refills. Draw your experimental setup and write down what you measure.

Perform your experiment to determine the maximum time interval between refills required to maintain continuous operation of the water clock.