The Relationship Between the Spring Constant and Number of Bungee Strands
Introduction
In this experiment, we sought to determine the relationship between the “spring constant”
of the bungee cord and the number of strands of bungee. In the end of our experimentation, this
will allow us to relate the length of the bungee cord to the number of bungee strands. Although
our experiment did not use a spring, our process used a spring as an approximation of the
bungee’s stretch properties, and our work was based on Hooke’s Law, which relates the force of
a spring with the amount the spring is stretched or compressed (Equation 1).
Equation 1: 𝐹 = 𝑘𝑥
where 𝐹 is force, 𝑘 is the spring constant, and 𝑥 is the displacement of the spring.
Methods
In our experiment, we measured the displacement of the bungee at different mass
increments. We kept the total length of the bungee constant, but varied the amount of bungee
strands, or the amount of times the bungee was looped on itself. This allowed us to find the
relationship between the “spring constant” of the bungee and the number of strands. Hooke’s
Law effectively approximates our bungee’s properties, but because the bungee does not behave
like a spring in every way, we need to determine this relationship for our bungee experiment. In
the classroom, we measured out a set length of bungee and hung it from a metal hanger, which
was connected to the table by a clamp (Figure 1). For our entire experiment, we kept this total
length of the bungee constant at 0.5 m, but varied the amount of times we looped the bungee on
itself. Thus, while the total amount of bungee used did not change, the length of the cord with no
weight added decreased as we increased the number of bungee strands. When the bungee was
doubled on itself, for example, this distance decreased by half. For each number of bungee
strands, we hung different masses and measured the displacement of the bungee from
equilibrium. By using a variety of masses, we varied the gravitational force on the bungee, and
thus observed the different amounts of stretch in the bungee cord. We used a meter stick to take
each of these measurements. In each trial, we allowed the bungee to stop oscillating and come to
rest before we took our measurements.
Metal Hanger Bungee cord
Mass
Direction of bungee’s stretch with added mass
Figure 1: Diagram of bungee cord with hanging mass
Results
From our data, we found an exponential relationship between the number of bungee
strands and the k-value of the bungee.
Weight (N)
F, ± 0.01 N
0.00
1.47
1.52
1.57
1.62
1.67
1.72
1.77
1.82
1.86
Displacement (m)
x, ± 0.01 m
0.00
0.23
0.25
0.26
0.27
0.28
0.30
0.31
0.33
0.34
Figure 2: Weight and displacement data for 2 strands of bungee. For each trial, we varied
the gravitational force on the bungee and measured the bungee’s displacement. Force was
calculated by multiplying the hanging mass by gravitational acceleration, 9.81 m/s2.
We then plotted the Weight vs. Displacement from Figure 2 to find the k-value of our
data. Based on Equation 1, the slope of our graph is 𝑘.
Weight vs. Displacement Weight (N) 1.95 y = 3.529x + 0.657 1.85 1.75 1.65 1.55 1.45 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 Displacement (m) Graph 1: Weight vs. Displacement. The slope of our data’s linear fit is the 𝑘 value for 2
strands of bungee.
The slope of the line is the calculated spring constant of our bungee cord. The calculated
uncertainty of the spring constant was 0.28 N/m, or 28%. The calculated Y-intercept of our data,
which is the displacement of the bungee with 0 kg hanging mass, was 0.08 m with an uncertainty
of 0.076 N, or 7.6%. As previously mentioned, though, the actual Y-intercept of our data was 0
m because there was no displacement of the bungee with 0 kg of hanging mass.
We repeated this process for the same total length of bungee, but varied the number of
strands. We did this same procedure 4 times in total and found the 𝑘 value for each number of
bungee strands.
Number of Strands
1
2
3
4
k-value
1.465
3.529
7.228
11.881
Figure 3: 𝒌 values calculated as the number of strands was varied. We repeated the same
procedure above for each number of strands and found the 𝑘 value from the slope of the Weight
vs. Displacement graph.
We then plotted the above data to find the relationship between the number of bungee
strands and the spring constant value of the data.
Spring Constant (N/m) Number of Strands vs. Spring Constant 14.000 k-­‐value= 0.647(# strands)2 + 0.259(# strands) + 0.524 12.000 10.000 8.000 6.000 4.000 2.000 0.000 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Number of Strands Graph 2: Relationship between 𝒌 value and the number of bungee strands. Notice
that the relationship is not linear, but exponential.
Within the data for each set number of bungee strands, our 𝑘 value was constant; note the
constant slope in Graph 1. However, as we varied the number of bungee strands, the 𝑘 value did
not change linearly as an ideal spring would.
Discussion
Our results verify that our bungee cord does not behave like an ideal spring. The
relationship between the spring constant and the number of strands of bungee is exponential.
Our data seems largely reasonable—consistently within each set number of bungee strands, the
relationship between the spring constant and the gravitational force was linear. Similarly, once
these spring constants were plotted against the number of bungee strands, the graph showed a
clearly exponential relationship between the two variables. Based on the measured displacement
and the calculated displacement of our bungee, we calculated the % error of the displacement to
be 8.0%. The uncertainty of our calculated spring constant was also very large, at 0.28, or 28%.
These values stem from a few main sources. First, the elastic in our bungee could have stretched
throughout our measurements; thus, those measurements could be slightly incorrect. Next,
because of a lack of small mass increments in the lab, we could not perform as many trials as we
wanted; though we gathered enough data to see the general trend and relatively accurate
calculated values, more data points could have further improved this accuracy. Finally, because
we opted to take our own measurements instead of using a computer program, our results again
lost a small amount of accuracy. None of these sources of uncertainty would have changed the
overall results of our experiment, but they could have slightly affected the numerical values that
we calculated.
Conclusion
Our experiment sought to determine the relationship between the “spring constant” of our
bungee cord and the number of bungee strands. Based on our results, we conclude that as the
number of bungee strands increases, so does the spring constant, but the relationship between
these two variables is exponential rather than linear. This relationship goes to show that our
bungee cord does not behave like an ideal spring. The next step of our bungee experiment is to
find a relationship between the spring constant and the total length of bungee cord. We will
repeat the same procedure as above, but for different lengths of bungee cord. In the end, this will
allow us to relate the spring constant to both the length of bungee cord and the number of bungee
strands. This relationship will allow us to solve for the correct number of bungee strands and the
total length of bungee cord necessary to ensure that our egg successfully survives the drop.