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Physics Internal Assessment: Experiment Report
An Increasing Flow of Knowledge:
Investigating Torricelli’s Law, or the Effect of Height on Flow Rate
Candidate Name:
Chunyang Ding
Candidate Number:
Subject: Physics HL
Examination Session: May 2014
Words: 4773
Teacher: Ms. Dossett
If you sit in a bathtub while it is draining, you may notice that the water level seems to
drop quickly initially, but then slows down as the height of the water decreases. Although you
could attribute this time distortion to being impatient and wanting the bathtub clean, it is also
possible that there is a physics explanation for this. This paper will investigate the correlation
between the height of water and the flow rate of the water. Our hypothesis is that as the height of
the water increases, the flow rate of the water will increase linearly.
In order to perform this experiment, we will use a two liter bottle and drill a small hole in
the side. We will fill the bottle with water and allow the water to drain from the bottle. The
independent variable is the height of water above the hole in the two liter bottle, measured in
meters. The dependent variable is the flow rate of water out of the bottle, measured in milliliters
per second. However, these two variables are very difficult to measure directly with a high level
of accuracy. Therefore, for practical purposes, we shall measure the amount of water poured into
the two liter bottle as the independent variable, and the amount of water drained in the duration
of 5.0 seconds as the dependent variable. We can easily convert these values to the units required
for our lab.
One important control for this lab is that the walls of the two liter bottle are very similar
to a perfect cylinder. Otherwise, we could not convert the volume of water in the bottle to the
height of water above the drilled hole. Therefore, instead of drilling the hole at the bottom of the
two liter bottle, where there are irregular shapes created by the “feet” of the two liter bottle, we
will drill the hole roughly 5 cm above the bottom of the two liter bottle, at a region where the
two liter bottle approximates a perfect cylinder.
We choose to measure 100 mL differences for water volume differences in order to get a
full range of values between 500 and 1700 mL. In addition, we determined that at 1700 mL, the
flow rate allows for the water to drain nearly 100 mL over the 5 seconds. If we used a value that
was less than 100 mL as the difference between successive conditions, we would find that one
trial would “overlap” into another trial. This is a fundamental error in the lab, but it is
unavoidable provided the equipment that we were given. More on this error will be discussed
later.
We choose use a period of 5.0 seconds for several reasons. Most importantly, this period
allows us to collect a wide range of data for amount of drainage, between 27 and 96 mL. If we
used a longer duration, such as 10.0 seconds, we would find that the amount of water drained
would be roughly between 50 and 200 mL, which would overlap into the other trials far too
much. The reason why we did not choose a value less than 5.0 seconds, such as 1.0 seconds, is
because such a small value greatly amplifies the problem of human reaction time. Human
reaction time is roughly 0.20 seconds, so using 1.0 seconds would result in a 20% error
minimum. In addition, the values collected would fall between 5mL and 25 mL, which would be
too small to be significant.
Finally, we chose to use 6 trials for each condition, primarily because we were concerned
with the errors associated with liquids. As it is possible for a couple of droplets of water to spill
here and there, disrupting the accuracy, we use more trials so that there is more ability to gather
uncertainty data. Using 6 trials would exceed the 5 trial minimum required for statistically
significant data.
Materials:
Drill
Drill bit with diameter of ( )
Two Liter Bottle
100 mL Graduated Cylinder
Large bucket (Capacity greater than 2000 mL)
250 mL Beaker
Water
Timer App for iPhone 4 (Big Stopwatch by Yuri Yasoshima)
String
Scissors
Ruler ( )
Procedure:
1) Prepare the drill with the drill bit of diameter ( )
2) Drill a hole in the two liter bottle roughly 10 cm from the bottom of the bottle. There
should be a thin line at the location, and the plastic should be relatively straight above the
line.
3) Fill the two liter bottle with water such that the water level is right at the hole. Measure
how much water is currently in the bottle by using the 100 mL graduated cylinder.
Record this number.
4) Wrap string around the circumference of the two liter bottle. Use the scissors to cut off
this piece of string.
Illustration:
Fig 1: Experimental Setup
Fig 2: Schematic of Two Liter Bottle
Data Collection and Processing :
Volume Lost over 5 seconds( ± 1mL)
Volume of
Water (mL) Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Trial 6
Error in Measuring
Volume of Water (mL)
Volume of
Water
Below
Hole (mL)
Error in
Volume
Below
(mL)
Circumference
of Two Liter
Bottle (cm)
Error in
Circumference
of Bottle (cm)
An explanation for the large error in the circumference of the bottle: The string that was
used to determine the circumference of the two liter bottle was found to be rather elastic.
Therefore, the maximum that the string could be stretched was by an additional 0.4 cm.
Therefore, the error in the circumference of the bottle was determined to be 0.4 cm, rather than
the 0.1 cm that the measurement error of using the ruler would be.
graduated cylinder. The measurement error for the time for loss is equivalent to human reaction
time, or roughly 0.2 seconds. Therefore for the same case as above,
Therefore, the following data tables are created:
Flow Rate (mL/sec)
Volume
Above
(mL) Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Trial 6
Error in
measuring
Volume Above
(mL)
Error in Flow Rate (mL/sec)
Volume
Above
(mL)
Error Trial
Error Trial
Error Trial
Error Trial
Error Trial
Error Trial
As we study this data, we realize that the error from measurements is greater than the
standard deviation of the measured flow rates. IB guidelines recommend for the maximum error
in measurement to be used rather than using the standard deviation of the flow rate data.
Next, we want to process our data to find the average flow rate. Using the 118 mL data:
Our data tables are of the following:
Using this conversion factor, we calculate that
We determine the cross sectional area of the two liter bottle using the value for the
bottle’s circumference that we determined in step 5 (34.4 cm). From this, we know that
However, we also understand that. Therefore, converting from
square centimeters to square meters:
To determine the height of the water, we calculate for the 118 mL condition:
To determine the error in this calculation, we would take the percent error for the area
and for the volume, add them together, and multiply by the final pressure.
However, the error for the area is a result of squaring the circumference value, which
changes the percent error. Therefore, this error is evaluated to be
Therefore, the total error in height is calculated by:
This produces the following data table:
Height vs. Flow Rate
Height
(m)
Error in
height (m)
Flow
Rate
(mL/sec)
Error in
Flow Rate
(mL/sec)
Therefore, the following graph is produced:
We can tell that our data seems to fit a square root curve very nicely. Our x axis is the
height as measured in meters, while the y axis is the flow rate of the water out of the drilled hole
as measured in milliliters per second. The x and y intercepts are at (0, 0), which implies that
when there is no water above the hole, the flow rate will be zero milliliters per second. This
makes a lot of sense, as if there is no water to drain out of the two liter bottle, the flow rate would
be zero.
We also notice that the square root function has a domain of. This makes
sense, as if the height of the water is below the height of the hole, the flow rate would always be
zero. Although the “undefined” portion is questionable, it is not sufficient to ignore the trend.
The next step is to linearize the data. We will use the variable to represent this
linearized value. Our current regression indicates a square root relationship between the height of
the water and the flow rate of the water. Therefore, we can linearize the data by taking the square
root of the pressure (x-variable) as follows:
Using the 0.013 meter experiment, we have
Evaluating the error of this linearized data is done by
Therefore, our data table is:
Linearized Height vs. Flow Rate
Linearized
Height ( √
) Error in Height ( √
Flow
Rate
(mL/sec)
Error in
Flow Rate
(mL/sec)
Next, we determine the lines of maximum and minimum slope. The maximum slope is
determined from the maximum uncertainty values of the highest and lowest points, so that the
two points responsible for this calculation would be:
and ( ). Using our data, these
two points would be (0.112+0.0037, 5.30-0.42) and (0.288-0.0070, 18.80+0.97), giving us the
points (0.116, 4.88) and (0.381, 19.77). As the equation for slope is , we can process
by
y = 48.4z + 0.
R² = 0.
Max Slope
y = 56.1z - 1.
Min Slope
y = 42.3z + 1.
0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.
Flow Rate (mL/sec)
Linearized Height (m^0.5)
Flow Rate of Water vs. Linearized Height
For this graph, we notice that the z variable is the linearized height, as shown in the SI
units of
, while the y variable is the flow rate as measured in milliliters per second. We
see that our linearization is a very close fit. The correlation factor is 0.999, but more importantly,
the best fit line nearly passes through every single data point and the error regions associated
with those data points. In addition, the lines of maximum and minimum slopes are well within
the error regions. The large number of data points used for this experiment provides further
confidence in the validity of our data. For this data, it is not possible that the error regions alone
could have caused the trend that we see. Instead, the values clearly show a positive increase,
from (0.112, 5.30) to (0.388, 18.80). There is a constant increase between every data point which
supports the high correlation value of this graph. Finally, the minimum and maximum slopes are
both positive and are very close to the slope of the best fit line.
The slope in this graph is interpreted to mean how rapidly the flow rate increases as the
square root of the height increases. If the linearized height increases by 0.300 √ , the
increase in flow rate would be predicted to an increase of 14.52. These values correspond
with our data, as the increase between the minimum and maximum points for linearized
height,
, corresponds to the flow rate increasing by 13..
Our best fit regression line has the equation of. The reason for the
apparent mismatch of precision is because in our scenario, the number of significant figures is
more important that the actual precision of the number. All of these numbers are calculated
values, not directly measured values. Therefore, it is valid to use a constant three significant
figures in all of these equations.
