DISCOVERY PROJECT

Torricelli's Law

OBJECTIVE To obtain an expression for Torricelli's Law by fitting a quadratic function to data obtained from a simple experiment.

Evangelista Torricelli (1608─1647) was an Italian mathematician and scientist. He is best known for his invention of the barometer. In Torricelli's time it was known that suction pumps were able to raise water to a limit of about $9$ meters, and no higher. The explanation at the time was that the vacuum in the pump could support the weight of only so much water. In studying this problem, Torricelli thought of using mercury, which is $14$ times heavier than water, to test this theory. He made a onemeter-long tube sealed at the top end, filled it with mercury, and set it vertically in a bowl of mercury. The column of mercury fell to about $76$ cm, leaving a vacuum at the top of the tube. In an impressive leap of insight, Torricelli realized that the column of mercury is held up not by the vacuum at the top of the tube, but rather by the air pressure outside the tube pressing down on the mercury in the bowl. He wrote:

I claim that the force which keeps the mercury from falling … comes from outside the tube. On the surface of the mercury which is in the bowl rests the weight of a column of fifty miles of air. Is it a surprise that … [the mercury] should rise in a column high enough to make equilibrium with the weight of the external air which forces it up?

The device Torricelli made was the first barometer for measuring air pressure.

Another of Torricelli's discoveries, based on the same principle but applied to water pressure, is that the speed of a fluid through a hole at the bottom of a tank is related to the height of the fluid in the tank. The precise relationship is known as Torricelli's Law.

I. Collecting the Data

In this exploration we use easily obtainable materials to conduct an experiment and collect data on the speed of water leaking through a hole at the bottom of a cylindrical tank. To do this, we measure the height of the water in the tank at different times.

You will need:

• An empty $2$-liter plastic soft-drink bottle
• A method of drilling a small ($4$ mm) hole in the plastic bottle
• Masking tape
• A metric ruler or measuring tape
• An empty bucket

Procedure:

This experiment is best done as a classroom demonstration or as a group project with three students in each group: a timekeeper to call out seconds, a bottle keeper to estimate the height every $10$ seconds, and a record keeper to record these values.

1. Drill a $4$ mm hole near the bottom of the cylindrical part of a $2$-liter plastic soft-drink bottle. Attach a strip of masking tape marked in centimeters from $0$ to $10$, with $0$ corresponding to the top of the hole.
2. With one finger over the hole, fill the bottle with water to the $10$-cm mark. Place the bottle over the bucket.
3. Take your finger off the hole to allow the water to leak into the bucket. Record the values of height of the water $h$ $\left({t}\right)$ for $t$ = $10$, $20$, $30$, $40$, $50$, and $60$ seconds.

Time (s)	Height (cm)
$10$
$20$
$30$
$40$
$50$
$60$

II. Testing the Theory

Torricelli's Law states that when a fluid leaks through a hole at the bottom of a cylindrical tank, the height $h$ of fluid in the tank is related to the time $t$ the fluid has been leaking by a quadratic function:
$$h \left({t}\right) = at^2 + bt + c$$

where the coefficients $a$, $b$, and $c$ depend on the type of fluid, the radii of the cylinder and the hole, and the initial height of the fluid.

1. Use a graphing calculator to find the quadratic function that best fits the data. $$h \left({t}\right) = \color{red}{\fbox{   }}t^2 + \color{red}{\fbox{   }}t + \color{red}{\fbox{   }}$$
2. Graph the function you found together with a scatter plot of the data in the same viewing rectangle. Does the graph appear to fit the data well?

III. Another Method

Torricelli's Law actually gives more information about the form of the quadratic function model. It can be shown that the function defining the height of the fluid can be expressed as

$$h \left({t}\right) = \left(C - kt\right)^2$$

where the coefficients $C$ and $k$ depend on the type of fluid, the radii of the cylinder and the hole, and the initial height of the fluid.

1. Let's find $C$ and $k$ for the experiment we conducted in Part I.
   1. Use the fact that the height $h(t)$ of the water is $10$ cm when the time $t$ is $0$ to solve for $C$ in the above equation.

      $C =$ __________

   2. Use the height $h(t)$ you obtained in the experiment when the time $t$ is $60$ seconds and the value of $C$ you obtained in part (a) to solve for $k$ in the above equation.

      $k$ $=$ __________.

   3. Use the results of parts (a) and (b) to find an expression for $h(t)$. $$h \left({t}\right) = \left(\color{red}{\fbox{   }} - \color{red}{\fbox{   }}t\right)^2 $$
2. Expand the expression for $h(t)$ that you obtained in 1(c). $$h \left({t}\right) = \color{red}{\fbox{   }}t^2 + \color{red}{\fbox{   }}t + \color{red}{\fbox{   }}$$ How does this model compare with the model you got in Part II?