Gustavo Larramendi
Bungee II Report
Section 06
Predicting the Stretch of a Bungee Cord Based on its Length
Introduction:
The purpose of the study is to see whether it is possible to make a mathematical model
based on the CWE theorem that can be used to predict the stretch of a bungee cord based on
its length. If a valid relationship is found between the length of the bungee cord and how much
it will stretch, it will be possible to constrain stretch in a future bungee jump design that will
allow the jumper, a raw egg, to get as close to the ground as possible without being damaged.
To achieve our goal two experiments were conducted. One experiment was a static experiment
used to determine the K value that will be used in our final model. This experiment was based
on measuring the bungee cords displacement and modeling it using Hook’s Law. The other
experiment was a dynamic experiment that measured the stretch of the bungee cord at
different lengths to come up with an equation for the final model.
Important Equations:
Hook’s Law (Eq.1): Fspring = - kx (Static Experiment)
Where “Fspring” is the force of the spring, “k” is the spring constant for the bungee cord,
and “x“ is the displacement of the bungee cord.
CWE Theorem (Eq. 2): (PE - KE)top = (PE – KE)bottom (Dynamic Experiment)
Where “PE” is the potential energy and “KE” is the kinetic energy.
CWE Theorem (Bungee) (Eq. 3): mgh = ½ kx2 (Dynamic Experiment)
Where “m” is the mass of the jumper, “g” is gravity (9.81 m/s), “h” is the full length of
the stretched string, “k” is the spring constant of the bungee determined in the static
experiment, and x is the stretch of the bungee cord.
Relationship between the Bungee Cord Length and the Stretch (Eq. 4): x = h - L (Dynamic
Experiment)
Where “x” is the stretch, “h” is the full length of the bungee stretched, and “L” is the unstretched length of the bungee cord.
Figure 1: Set up (Static Experiment)
Figure 2: Set Up (Dynamic Experiment)
Methods:
The static experiment consisted of hanging a bungee cord from a hook that was
suspended in the air (see diagrams above). First the length of the un-stretched bungee cord
was measured using a tape measure. Once the un-stretched length was determined weight was
added to the hanging end of the bungee cord and was left until the cord stopped elongating.
Once the cord stopped elongating the length of the stretched cord was measured and the
displacement of the bungee cord was determined by finding the difference of the stretched and
un-stretched lengths. The bungee cord was measured from where it was tied on the hook to
where it was tied on the weight. The experiment was run using three different bungee cord
lengths that were carefully measured. Those lengths were 13.6cm, 24.2cm, and 34.6cm. For
each length the experiment was run at 5 different weights. Those weights were 50g, 100g,
200g, 250g, 300g. The resulting data was then compiled and analyzed in Microsoft Excel.
The dynamic experiment also consisted of hanging a bungee cord on a hook that was
suspended in air (see diagrams above). A tape measure was set up next to the hanging bungee
cord close enough to where it could be captured in a frame by frame recording of each trial.
First, just like the static experiment, the un-stretched length of the bungee cord was measured
using a tape measure. Then weight was added to the hanging end of the bungee cord and held
up to the same height by the researchers thumb and pointer finger as the hook the bungee was
hanging on. The weighted end of the bungee was then released very carefully to prevent any
extra forces from effecting the drop. The drop was recorded by an IPad equipped with a frame
by frame picture application. After the weight started to rebound back up toward the hook the
weight was caught and removed to prevent any change in the bungee’s stretch properties,
which could skew future trials. The recording of the trial was also stopped. The frame by frame
pictures were then used to determine when the bungee cord was at its most stretched length
during the trial. This was done by finding the captured frame that showed the weight at its
lowest point during the trial. The measurement of the fully stretched bungee cord was then
taken by using the tape measure and measuring from where the bungee cord is tied to the
weight. To get the stretch of the bungee cord Eq. 4 was used. For this experiment five different
bungee lengths were used. Those lengths were 10.2cm, 19.3cm, 24.2cm, 29.9cm, and 39.3cm.
Three trials were run at each bungee length and a constant weight was used throughout the
duration of the experiment. That weight was 150g and was chosen because it falls in the range
of weights that the egg in our future bungee jump will weigh.
Results:
Static Experiment:
Table 1A:
Mass (g)
50
100
200
250
300
Length With Mass Attached (cm)
13.6
24.2
34.6
17.40
30.03 43.10
21.30
38.50 54.70
37.40
50.05 73.20
45.20
66.00 94.30
overstretching
79.90 114.10
* First row of bungee lengths is the length of the un-stretched bungee cord
Table 1 shows the data collected from each of the trials of the static experiment.
The lengths that are bolded are the three un-stretched lengths. The last trial for the unstretched length of 13.6cm had to be aborted because the cord was overstretching and looked
like it could have possibly broken. The uncertainty of the measurements were + or - .005cm
(.00005m) because the tape measure only accurately recorded to the hundredth of a
centimeter.
Graph 1A:
length vs displacement
Avereage Displacement (m)
0.8
y = 2.1692x + 0.0068
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Un-streatched Bungee length (m)
Graph 1A shows the trend of the average displacement of the bungee cord while
increasing the initial un-stretched length of the bungee cord. Assuming that Hook’s Law (Eq. 1)
is a good approximation of the force of the bungee the slope of the trend line would give us our
static “k” value. A linear regression was run to give us the uncertainty of our static “k” value,
which turned out to be + or - 0.023. This makes the static “k” value 2.169 + or – 0.023. The
percent uncertainty was calculated to be 1% making the determined “k” value very precise.
Dynamic Experiment:
Table 1B:
Unstretched
Trial 1 h
Trial 2 h
Trial 3 h
Length (cm) (cm)
(cm)
(cm)
Average h (cm)
Stretch (x)
10.20
43.40
43.60
43.20
43.40
33.20
19.30
79.30
83.50
82.40
81.73
62.43
24.20
101.20
101.10
101.30
101.20
77.00
29.90
123.80
125.10
125.40
124.77
94.87
39.30
165.40
165.20
163.50
164.70
125.40
Table 1B shows the fully stretched length of the bungee cord for each trial at each of the
different un-stretched bungee lengths. The stretch values (x) used for the data analysis was
calculated using Eq. 4 where the average fully stretched length for each of the un-stretched
bungee lengths (average h) is the “h” value and the un-stretched length is the “L” value. The
uncertainty for the measurements is + or - .005cm (.00005m) because the instruments we used
were only accurate to hundredth place.
Graph 1B:
(1/2)x2 (m)
mgh vs 1/2x2
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
y = 0.1346x1.9911
Series1
Power (Series1)
0
1
2
mgh (m)
3
Graph 1B takes the data from Table 1B and incorporates it into Eq. 3 to create an energy
model. The stretch data from Table 1B was converted from cm to m. The equation of the trend
line matches the CWE equation for the bungee cord showing that the equation can be used to
make an energy model of a bungee jump. Based on the CWE theorem for a bungee cord (Eq. 3)
the slope of this trend line gives us the “k” value for the equation. There is a problem with this
that will be discussed in the discussion section of this report. Since this “k” value is invalid no
uncertainty was calculated for it.
Discussion:
The first experiment that was conducted was the dynamic experiment. The dynamic
experiment was the original experiment that was planned but after realizing that the “k” value
was invalid we decided to supplement the dynamic experiment with a static experiment to get
a usable “k”value. The dynamic experiment was able to show that Eq. 3 successfully models the
bungee cord because the equation of the trend line matches Eq. 3. The problem with the “k”
value was that the “k” value calculated in the experiment was not constant. This makes it
impossible to define a relationship between the length of the bungee and how much it will
stretch. This happened because when we set up our experiment we defined h as x +L (Eq. 4).
The resulting equation is then mg(L + x) = ½ kx2 (Eq. 3). This means that when you change the h
value the k value can be effected.
Some possible sources of uncertainty for the dynamic experiment were that over time
continuously dropping the weighted end of the bungee cord could have led to changes in the
stretch properties of the bungee cord throughout the experiment. That would affect how much
the bungee cord would stretch and skew the data slightly. Also there was uncertainty due to
the precision of the instruments used. Another possible source of uncertainty was that the
weighted end of the bungee cord was probably dropped from slightly different locations each
time. This is due to the fact that the location of the hook the bungee cord was high and it was
difficult to see how well the weighted end of the bungee cord and the hook lined up. One
possible way to limit this uncertainty would be to stand on a chair or some kind of stable object
that would allow you to get a good angle for lining up the weight and drop point.
After the dynamic experiment was completed and the “k” value was found to be invalid
the static Hook’s Law experiment was conducted to find a solution to this problem. The static
experiment was able to produce a precise “k” value for the bungee that could be used in Eq. 3.
The “k” value was deemed precise because the percent uncertainty was only 1%. On further
reflection the static “k” value is most likely not the best representative value to use in a
dynamic model. The reason for that is that the value was derived from a static system rather
than a dynamic system, like the one it will be used for. The reason why a static experiment was
conducted rather than a Hook’s Law based dynamic experiment was that after realizing a
constant “k” value was needed to make the model work there was not enough time left to
construct and perform a dynamic experiment. A great supplement to this study would be to run
an experiment to determine the dynamic “k” value of the bungee cord and use that value to
model the relationship between the length of the bungee cord and how much it will stretch
given a mass and a height.
Some sources of uncertainty for the static experiment were that the continuous hanging
of different weights on the bungee cord could have caused the bungee cord’s stretch properties
to change throughout the experiment. This could skew the data because the cord would stretch
differently each time a trial was conducted, this source of uncertainty can be minimized by
removing the weight or holding the weight up immediately after measuring the stretched
length to limit the time the bungee cord is being stretched, which would therefor limit the
amount the stretch properties of the bungee cord would change. Also the precision of the
instruments used for measuring is another source of uncertainty. Another source of uncertainty
is that the support that the hook that the bungee cord was tied to looked like it slightly tilted
when weight was added to the bungee cord. This could skew the data even if the tilt is very
small because the measurement of the stretched length could be thrown off. To limit this
source of uncertainty make sure the support holding the hook is tightly fastened to the table
being worked on and that the support beam is stable.
The results of both of the experiments were combined to create a model that will help
in the egg bungee jump. The static experiment yielded a usable “k” value and the dynamic
experiment proved that Eq. 4 could be used to determine the relationship between bungee
length and the amount it will stretch given a “k” value for the bungee cord.
Conclusion:
The CWE theorem for a bungee cord (Eq. 3) accurately models the behavior of the
bungee cord during a jump. The method used in this study however was unable to successfully
determine a valid “k” value because it was not constant. If a constant “k” value is
experimentally determined, for example through a dynamic experiment based on Hook’s Law, it
is possible to determine a relationship between the lengths of a bungee cord and how much it
will stretch. This would make it much easier to determine the length of the bungee cord that
will give a jumper the best jump without getting damaged.