# 1 The Relationship Between Length and Spring Constant of an

``` 1
The Relationship Between Length and Spring Constant of an Elastic Bungee Cord:
Static and Dynamic Cases
Jenny Wang &amp; McCauley Massie – PHYS113-01 – October 20, 2014
I. Introduction
In this experiment, data is collected to investigate the relationship between the
unstretched length of an elastic bungee cord and the spring constant of the bungee cord. Because
the cord is elastic, it possesses certain spring-like qualities, so data is analyzed using Hooke’s
Law, which is characterized by the following equation:
𝑭𝒔𝒑𝒓𝒊𝒏𝒈 = −𝒌𝒙
(1)
This law describes the restoring force of a spring when it is stretched, stating that there is a linear
relationship between this force (Fspring) and the displacement (x) of the spring. At equilibrium, the
restoring force of the spring should also be equivalent in magnitude to the force applied on the
spring by the hanging mass. The magnitude of this applied force is given by the equation for
weight:
𝑾 = 𝒎! |𝒈|
(2)
In his equation, mh is the mass of the hanging mass in Kg, and g is acceleration due to gravity,
9.81m/s2. When the system is not at equilibrium, the spring must exert a force that tries to restore
equilibrium. In this experiment, the force applied on the cord will remain constant. The
relationship between the unstretched length of the cord and the stretch, or displacement, of the
cord is used to investigate the relationship between the spring constant (k), which is related to the
force in equation 1, and the length of unstretched cord. It is hypothesized that the k constant will
remain constant as the length of unstretched elastic bungee cord increases.
II. Methods
One end of an elastic bungee cord forms a loop so that a hanging mass of 150 g can be
attached. The weight of the hanging mass, given by equation 2, will constitute the force acting on
the bungee cord. The bungee cord is tied to form another small loop at a distance from the first
loop. This distance is the “unstretched” bungee cord length, and it is varied throughout the
experiment. The second loop is attached to a force sensor attached to the top of a L-shaped rod
clamped to a table. The force sensor connects to the Capstone computer program, which records
the restoring force of the bungee cord as it is stretched. Measuring tape is also suspended from
the L-shaped rod adjacent to the force sensor. This is used to measure the displacement of the
cord.
1
Figure 1: Experimental Set-Up. A length of bungee cord is tied
such that each end has a small loop. One end is attached to a
force sensor mounted on an L-shaped rod clamped to a table. A
hanging mass (mh) is attached to the other end. The
“unstretched” length and the displacement (x) of the bungee cord
are measured using a tape measure.
2
The length of unstretched bungee cord is measured from the bottom of the loop attached
to the force sensor to the top of the loop that holds the hanging mass, without the hanging mass
attached. For each length of unstretched string, the static stretched bungee cord length was
measured with the spring and hanging mass system at equilibrium. In this case, the force applied
to the bungee is equivalent to the restoring force of the bungee cord. This measurement was
taken from the bottom of the loop attached to the force sensor to the top of the loop where the
hanging mass is attached.
In addition, the dynamic stretch for each length of unstretched bungee cord was also
measured. In this condition, the hanging mass is dropped from rest from the point where the
bungee cord is just on the verge of being stretched. Because it is difficult to determine the
maximum amplitude of the cord in this condition with the naked eye, the frame-by-frame video
analysis function of the CoachMyVideo (CMV) app is used to determine the dynamic stretched
bungee cord length. Again, this measurement was taken from the bottom of the loop attached to
the force sensor to the top of the loop where the hanging mass is attached.
In both the static and dynamic cases, the displacement, or stretch, of the bungee cord can
be calculated by subtracting the unstretched bungee cord length from the stretched bungee cord
length. This stretch, which is related to the restoring force in Equation 1, can be used to calculate
the spring constant, k. Because this experiment seeks to investigate the relationship between the
unstretched length of the bungee cord and the spring constant, the length of the unstretched
bungee cord is varied, and the subsequent stretch of the bungee is used to determine the spring
constant in both static and dynamic cases.
III. Results
In this experiment, the stretch of the bungee cord increased linearly with the length of the
unstretched bungee cord. Consequently, the spring constant decreases as the length of the
unstretched bungee cord increases.
Figure 2: Table Showing Static Stretched Bungee Cord Length (in cm), Average Dynamic
Stretched Bungee Cord Length (in cm), and Average Force (in N) for each Unstretched Cord
Length (in cm). All of the data in this table are directly measured in the experimental design. For
each unstretched cord length, three trials of dynamic stretch bungee cord length and force were
conducted, and the resulting values were averaged.
Unstretched
Cord Length
(cm, &plusmn;0.2)
Static
Stretched
Cord Length
(cm, &plusmn;0.2)
5.5
11.8
18.0
30.0
38.5
13.6
28.2
45.9
92.7
115.4
Average
Dynamic
Stretched
Cord Length
(cm, &plusmn;0.2)
38.73333
62.36667
86.33333
131.7
194.6
Average
Force,
Fspring
(N, &plusmn;0.1)
2.37
2.55
2.46
2.36
2.52
Figure 2 provides the data obtained by direct measurement in the experiment. As the
unstretched bungee cord length increases, the static stretch increases. Moreover, the average
dynamic stretch increases with the unstretched bungee cord length at a greater rate. The average
3
restoring force of the bungee cord remains about constant regardless of unstretched bungee cord
length.
Figure 3: Table Showing the Spring Constant derived from the Static Stretch (in cm) of the
Bungee Cord and the Restoring Force Exerted by the Bungee Cord. The static stretch was
calculated by subtracting the unstretched bungee cord length from the static stretched cord
length. The restoring force of the bungee, Fspring (in N) was calculated using Equation 2. The
spring constant, k (in N/m), was then calculated using Equation 1 for each static stretch
measurement obtained from each unstretched cord length condition.
Fspring
Spring
Static
(N)
Constant,
k
Stretch, x
(N/m)
(cm, &plusmn;0.2)
1.47
0.181
8.1
1.47
0.0896
16.4
1.47
0.0527
27.9
1.47
0.0234
62.7
1.47
0.0191
76.9
Figure 3 shows the derived values of static stretch, or displacement, of the bungee cord
for each unstretched bungee cord length. For each of these values, the restoring force applied by
the spring remains constant. This figure also shows the derived spring constant for each of the
displacement values. The results for this experiment indicate that the spring constant decreases as
the static stretch increases.
Figure 4: Table Showing the Spring Constant, k, derived from the Dynamic Stretch (in cm) of
the Bungee Cord and the Restoring Force (in N) Exerted by the Bungee Cord. The dynamic
stretch was calculated by subtracting the unstretched bungee cord length from the dynamic
stretched cord length. The spring constant, k (in N/m), was then calculated using Equation 1 for
each dynamic stretch measurement obtained from each unstretched cord length condition.
Fspring
Spring
Dynamic
(N)
Constant,
k
Stretch, x
(N/m)
(cm, &plusmn;0.2)
2.37
0.0713
33.23
2.55
0.0504
50.567
2.46
0.0360
68.33
2.36
0.0232
101.7
2.52
0.0161
156.1
Figure 4 shows the derived values of dynamic stretch, or displacement, of the bungee
cord for each unstretched bungee cord length. For each of these values, the restoring force
applied by the spring remains fairly constant. This figure also shows the derived spring constant
for each of the displacement values. The results for this experiment indicate that the spring
constant decreases as the static stretch increases. In general, the spring constants in the dynamic
condition are smaller than the spring constants in the static condition.
4
Figure 5: Linearized Graph Showing the Relationship between Unstretched and Static
Stretched Cord Length. The equation y=2.941(&plusmn;0.2)x gives the linear regression of the graph,
indicating that the unstretched and static stretched cord length are positively, linearly related by a
factor of ~3.
Static Stretched
Cord Length (cm, &plusmn;0.2)
Unstretched vs. Static Stretched
Cord Length
140
120
100
80
60
40
20
0
y = 2.941(&plusmn;0.2)x
0
5
10
15
20
25
30
35
40
45
Unstretched Cord Length (cm, &plusmn;0.2)
Figure 5 shows that the total static stretched bungee cord length is positively and linearly
related to the unstretched bungee cord length. Each unstretched length of bungee cord stretches
about 3 centimeters when a force of about 1.5 N, calculated using equation 2, is applied to it.
Figure 6: Graph Showing the Relationship between Unstretched Cord Length (in cm) and
Average Restoring Force of the Bungee Cord in the Dynamic Condition (in N). For each
unstretched cord length, three trials of dynamic stretch were conducted and the resulting values
for the restoring force were averaged.
Average Force (N, &plusmn;0.1)
Unstretched Cord Length
vs. Average Force
5
4
3
2
1
0
0
5
10
15
20
25
30
Unstretched Cord Length (cm, &plusmn;0.2)
35
40
45
5
Like Figure 4, Figure 6 also shows that the average restoring force of the force in the
dynamic case remains constant regardless of unstretched bungee cord length. Figure 3 shows that
this is also the case for the static condition.
Figure 7: Linearized Graph Showing the Relationship between Unstretched Cord Length (in
cm) and 1/Spring Constant (in N/m) for the Static Condition. The equation y=1.3204(&plusmn;0.1)x
gives the linear regression of the graph, indicating that the unstretched cord length of the bungee
cord and the spring constant, k, of the bungee cord are inversely related in the static condition. In
other words, the unstretched cord length is directly related to 1/k.
1/Spring Constant (N/m)
Unstretched Cord Length vs. Static
Stretch 1/Spring Constant
60
50
40
30
20
10
0
y = 1.3204(&plusmn;0.1)x
0
5
10
15
20
25
30
35
40
45
Unstretched Cord Length (cm, &plusmn;0.2)
Figure 8: Linearized Graph Showing the Relationship between Unstretched Cord Length (in
cm) and 1/Spring Constant (in N/m) for the Dynamic Condition. The equation
y=1.5609(&plusmn;0.1)x gives the linear regression of the graph, indicating that the unstretched cord
length of the bungee cord and the spring constant, k, of the bungee cord are inversely related in
the dynamic condition. The unstretched cord length is directly related to 1/k.
1/Spring Constant (N/m)
Unstretched Cord Length vs. Dynamic
Stretch 1/Spring Constant
80
60
y = 1.5609(&plusmn;0.1)x
40
20
0
0
5
10
15
20
25
30
Unstretched Cord Length (cm, &plusmn;0.2)
35
40
45
6
Figures 7 and 8 show that, in both the static and dynamic cases, the spring constant is
inversely related to the unstretched length of the cord, meaning the spring constant decreases as
the length of unstretched bungee cord increases.
IV. Discussion
The results of this experiment do not confirm the hypothesis that the spring constant will
remain constant as the length of unstretched elastic bungee cord increases. In fact, the spring
constant decreases as the unstretched length of the bungee cord increases, so they are inversely
related. Even though these results do not support the hypothesis, they are still consistent with the
theory behind this experiment. Hooke’s law, characterized by equation 1, implies that the
displacement and the spring constant and inversely related. Since the force remains about
constant throughout each of the trials, it is absolutely reasonable to state that the spring constant
decreases as the displacement increases. Furthermore, the results imply that it is possible to
model the behavior of the elastic bungee cord using Hooke’s law even though the spring constant
changes depending on the length of the bungee cord. The changes in the spring constant merely
indicate that the bungee cord behaves differently at different lengths. This is no different from
the idea of connecting two springs that have the same properties. The new length of spring has
different qualities because it is twice as long.
Unfortunately, the percent error of the results cannot be calculated because there is no
predicated value of the spring constant. Therefore, it is not possible to state whether or not the
results of the experiment are accurate to within the uncertainty of the experiment. The results in
this experiment could, however, be used as the predicted values for future experiments.
Similarly, the results of future experiments can be used to determine the accuracy of the results
of this experiment. Since the spring constant varies depending on the length of the elastic
bungee, future experimentation should seek to reproduce the value of the slope when 1/spring
constant is plotted against the unstretched bungee cord length. For example, in the static stretch
condition, a linearized graph, like Figure 7, should yield the equation y=1.3204x, where 1.3204
is the slope relating 1/spring constant to the unstretched bungee cord length. This value would be
used as the predicted value for a future experiment to determine the accuracy of the results of
that experiment. If the percent error between the predicted value and the experimental value is
less than the percent uncertainty, the results are accurate.
Still, the percent uncertainties for the results of this experiment are very small, indicating
that the results of this experiment are precise. For example, the percent uncertainty for the slope
quantifying the relationship between the static stretched and the unstretched length of the bungee
cord, depicted in Figure 5, is 0.07%. The percent uncertainty for the slope quantifying the
relationship between 1/spring constant and the unstretched bungee cord length in the static
condition, depicted in Figure 7, is 0.08%, and the percent uncertainty for the slope quantifying
the relationship between 1/spring constant and the unstretched bungee cord length in the
dynamic condition, depicted in Figure 8, is 0.06%.
There are many sources of uncertainty in this experiment. For the measurements of
length, there is uncertainty due to the change in angles of the camera when measuring the length
of stretch using the CoachMyVideo App. Uncertainty in the measure for restoring force of the
bungee cord can be attributed to the force sensor’s least count and the amount of bungee in the
loops at either end of the spring. Even though these loops were tied very small, they still stretch
to some degree, and they will behave differently from the single stranded length of bungee that
the experiment focuses on. In the dynamic stretch condition, there is come uncertainty that can
7
be attributed to the difficulty of eyeballing just exactly where the line between slack and
stretched bungee is.
V. Conclusion
The purpose of this experiment was to investigate the relationship between the
unstretched length of an elastic bungee cord and the spring constant of the bungee cord,
assuming that the behavior of the bungee cord can be modeled using Hooke’s law. The results of
this experiment state that the restoring force of the bungee cord is constant regardless of the
length of the unstretched bungee cord length as long as the force applied to the bungee cord is
constant. Consequently, as the displacement of the bungee cord increases in both the static and
dynamic stretch conditions, the spring constant of bungee cord decreases. These results not only
state this relationship, but they also indicate that the behavior of the bungee cord can be
successfully modeled using Hooke’s law. In addition, the static stretched bungee cord length
could be determined using the equation given by the linear regression in Figure 5 if the
unstretched bungee cord length is known. Similarly, the spring constant could be determined
using the equations given by the linear regression in Figures 7 and 8.
Since an important application of bungee cords is their use bungee jumping, it is
important to understand how the restoring force of the bungee cord is related to the force applied
to the bungee cord, such as the weight of the mass attached to it. If the restoring force of the
bungee cord is too great, the mass could incur damage do to a very quick deceleration. In
addition, the mass attached to the bungee cord could incur damage if the bungee cord stretches to
far, and the mass hits the ground. This experiment was able to describe the relationship between
the restoring force, the spring constant, and the displacement of the bungee cord. It was also able
to determine the relationship between static stretch and unstretched bungee cord length and
between dynamic stretch and unstretched bungee cord length when the mass is dropped from rest
at the point where the bungee cord is just about to be stretched. However, it does not describe the
relationship between dynamic stretch and unstretched bungee cord length when the mass is
dropped from rest at the maximum height, where the bungee attaches to the force sensor. This
information would be important to gather in order to predict the success of a bungee jump.
VI. Appendix
8
```