Does a Bungee Cord Behave Like an Ideal Spring?
Ashley Ooms
Introduction:
A bungee-cord possesses characteristics of a linear elastic spring. Thus, Hooke’s Law can
be used to model a bungee cord’s behavior:
Fspring=-kx
[1]
The restorative force, Fspring, is a function of k, a unique constant determined by the
spring’s stiffness, and x, the displacement from equilibrium. The restorative force acts against
the weight of the hanging object
W=mg [2]
where weight is determined by the mass of the object, m, multiplied by the acceleration
due to gravity, 9.81 m/s2. Since weight is the only force causing stretching of the bungee cord
and Fspring is the restorative force equal but opposite to the magnitude of the weight, Equation 1
can be rewritten as Newton’s Third Law Pairs:
mg=-kx
[3]
However, a bungee cord is not an ideal spring and should deviate from the above
equations’ predictions. By varying hanging masses and cord equilibrium lengths we can
determine if a bungee cord behaves according to Hooke’s Law. If not, we can quantify the
differences in behavior between a bungee cord and an ideal spring to accurately model the cord’s
characteristics.
Methods:
Metal stand on top of
flat table above
ground allowing free
movement of hanging
mass.
Bungee cord
tied at top of
stand and to
hanging mass
with smallest
knots possible.
To test whether a bungee cord diverges from ideal spring
behavior, a simple set up of a mass tied to the bottom of a
bungee cord hanging from a stand is used (Figure 1). Care
is taken to ensure the knots used to tie the cord to the stand
and to the hanging mass are as small as possible to prevent
extra stretching of the bungee cord which cannot be
Figure 1: Apparatus used to find k with
varying weights and equilibrium lengths.
The mass moves in the ĵ direction due to
gravity acting on the hanging mass.
quantitatively measured. Cord equilibrium lengths and
displacements are measured from knot to knot with a
measuring tape. A pencil is held perpendicular from the top
of the knot tied to the hanging mass to the measuring stick
to measure the stretch caused by the mass. We find the
relationship between weight and displacement by
changing the hanging mass and measuring the observed displacement. Eight different masses are
used. We determine the connection between the cord’s equilibrium length and k value by
keeping the hanging mass constant but varying the cord’s equilibrium length. Equilibrium length
is measured as the distance between the two knots without any mass hanging on the cord. This is
done with 7 different cord lengths. Equation 3 is used in both experiments to determine k.
Results
Our experiments generated results illustrating differences between a bungee cord and an ideal
spring. Figure 2 displays the raw data from the first experiment.
Hanging Mass
(kg)
.050 +/- .2
.10 +/- .2
.12 +/- .2
.14 +/- .2
.15 +/- .2
.17+/- .2
.20 +/- .2
.25 +/- .2
Weight (N)
.4905 +/- .2
.981 +/- .2
1.18 +/- .2
1.37 +/- .2
1.47 +/- .2
1.67 +/- .2
1.96 +/- .2
2.45 +/- .2
Equilibrium
Length (m)
.405 +/- .02
.405 +/- .02
.405 +/- .02
.405 +/- .02
.405 +/- .02
.405 +/- .02
.405 +/- .02
.405 +/- .02
Displacement (m)
K Value (N/m)
.09 +/- .02
.215 +/- .02
.285 +/- .02
.34 +/- .02
.415 +/- .02
.505 +/- .02
.582 +/- .02
.78 +/- .02
5.45 +/- .2
4.67 +/- .2
4.13 +/- .2
4.04 +/- .2
3.55 +/- .2
3.30 +/- .2
3.37 +/- .2
3.14 +/- .2
Figure 2: Table displaying raw data of the k value as a function of weight. Equation 3 is used to calculate
the k value. Equilibrium length is kept constant at .405 m.
If the bungee cord characterizes an ideal spring, then a graph plotting Table 2’s values of weight
versus displacement would show a linear line with a slope of k and a y intercept of 0.
Weight vs. Displacement
Figure 3: Graph of experimentally
measured weight vs displacement
values. A linear trendline is added
showing the theoretical k value of the
bungee cord to be 2.73 N/m.
Regression analysis finds a 4.4% error
in the slope and a 14% error in the yintercept.
3
y = 2.7289x + 0.3506
Weight (N)
2.5
2
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
Displacement (m)
While we do have a linear relationship between weight and displacement the slope found is 2.73,
a k value not close to any value we calculated experimentally with Equation 3 (Figure 3).
Additionally, we have a y intercept not equal to 0 which implies the bungee cord is not acting as
an ideal spring.
Next, we find the connection between the k value and the cord’s equilibrium length by
keeping the mass constant at 150 g. The following raw data (Figure 4) is generated:
Mass (kg)
Equilibrium Length
(m)
.13 +/- .02
.28 +/- .02
.31 +/- .02
.41 +/- .02
.62 +/- .02
.71 +/- .02
.82 +/- .02
.15 +/- .2
.15 +/- .2
.15 +/- .2
.15 +/- .2
.15 +/- .2
.15 +/- .2
.15 +/- .2
Displacement (m)
k Value (N/m)
.15 +/- .02
.29 +/- .02
.33 +/- .02
.42 +/- .02
.6 +/- .02
.72 +/- .02
.8 +/- .02
9.81 +/- .2
5.07 +/- .2
4.46 +/- .2
3.55 +/- .2
2.45 +/- .2
2.04 +/- .2
1.84 +/- .2
Figure 4: Table showing raw data found by maintaining the same mass and changing the equilibrium
length to see the length’s effect on the k value.
As the equilibrium length increased, the k value decreased. Additionally, the displacement of the
mass is found to be about equal to the equilibrium length within a 2 cm range. The data in Figure
4 is then used to plot a graph of k vs. equilibrium length.
K vs. Equillibrium Length
12
Figure 5: Graph of the k value
calculated using equation 3 versus
equilibrium length. Regression
analysis finds a 1.3% error in the
slope.
K Value (N/m)
10
y = 1.5408x-0.915
8
6
4
2
0
0
0.2
0.4
0.6
Equillibrium Length (m)
0.8
1
The graph shows a nonlinear relationship between the k value and equilibrium length described
by the following equation when the mass is 150 grams:
k=1.5408l-.915 [4].
Where l is the equilibrium length.
Discussion:
Our results give strong evidence suggesting a bungee cord does not behave like an ideal
spring as described by Hooke’s Law (Equation 1). An ideal spring has a k value which remains
constant despite the weight hanging on the spring. Therefore, a linear relationship is observed
between weight (Fspring) and displacement. Our plot of weight vs. displacement did result in a
linear graph. However, the y-intercept is .35 meaning that there is no displacement with .35 N of
force acting on the cord. An ideal spring would exhibit a y-intercept of 0 since with no observed
displacement there must be no weight forcing the spring down. Additionally, the slope of our
line, 2.73, represents the theoretical k constant our cord possesses. Individual calculations using
Equation 3 show the k value is not equal to 2.73 for any of the masses used and, more
importantly, the k value was not constant when different masses are used (Figure 2). This means
the k value is not an inherent characteristic of the bungee cord as it is in an ideal spring. These
results make sense because an ideal spring is massless and does not experience any damping
effects which together ensure the k value remains constant. The bungee cord is neither massless
nor in a vacuum so its mass and air drag experienced will cause deviations from ideal spring
behavior. Both air drag and the cord’s mass will increase its k value compared to the cord’s
theoretical ideal spring constant. This explains why at .35 N of weight, no displacement of the
mass occurs.
The k value is then expressed as a function of equilibrium length at a constant mass of
150 grams to come up with Equation 4. The slope of the function, 1.54, is a property of the cord
at this constant mass. A longer equilibrium length results in a smaller k meaning that under the
same force, a longer cord will stretch further than a shorter cord. However, a major source of
uncertainty when modeling the bungee cord’s properties is the assumption the cord has never
been overstretched. If the bungee cord has experienced a force that exceeds its elastic limit only
a single time then the cord will never be able to restore its original elasticity properties. We did
not have control over these cords prior to experimentation and therefore the cords’ histories are
unknown. If this elastic limit was breached at any point then the experimental displacement and
k values would be larger than they should be as the cord would be less stiff and thus require less
force to cause a change in displacement. New, unused bungee cords would have to be used to
ensure optimal conditions for achieving reliable data.
An additional point of error is these experiments are not done in a vacuum. Therefore, air
drag is a factor which is not taken into account using Hooke’s Law. Because of air resistance the
acceleration due to gravity is not 9.81 m/s2. Rather, gravity has a smaller effective magnitude
impacting the calculated weights used to determine the cord’s k values. Our experimental k
values, then, are greater than they would be in a vacuum. With a smaller effective weight the
same displacement is measured signifying the cord is less stiff than what our calculations
determined.
Displacement measurements are also a subject of uncertainty. The measuring tape could
not be put directly up to the bungee cord as this would cause the hanging mass to sway and
possibly change how much the cord stretches. We use a pencil to approximate a straight line
from the knot holding the hanging mass to the measuring tape. If the measuring tape and pencil
are not exactly perpendicular to each other then the displacement measured can be too large or
too small depending on the angle relative to the horizontal. This in turn impacts the k values we
calculated since displacement is inversely proportional to k. In addition, the smallest possible
knots were made to tie the bungee cord to the metal stand and to the hanging mass. However,
these small knots cause slight, unmeasurable changes in the cord’s stretch. The cord’s overall
extension will be less than it would be under ideal conditions due to the stretch of the attached
loop. Applying a strong adhesive to one end of the cord to bind it to the stand is one possible
method of eliminating this error.
Conclusion:
Our experiments illustrate the bungee cord does not behave like an ideal spring. The cord
does not have a characteristic k value which, in an ideal spring, remains constant despite the
amount of force the cord experiences. Knowing this we find the relationship between the k value
and the cord’s equilibrium length with mass staying constant. This generated Equation 4 which
can be used to ultimately begin modeling an egg’s descent from the fourth floor of the Great
Hall. Knowing the k value and the mass of the egg we can calculate an equilibrium length that
will allow the egg to have a displacement where it barely reaches the ground.