Investigations into the Spring Constant
Abstract
In this investigation we sought to investigate the spring constant by using a balanced
force equation for a vertically suspended spring mass system in equilibrium, and the natural
frequency of the spring as it oscillates harmonically. We will measure the distance the spring
is stretched by the weight of the mass, and when the system is in equilibrium, and use that
to derive a spring constant. We will then stretch the system a distance and measure the
period of oscillation, then use it to derive an experimental spring constant. Our null
hypothesis is that the forces will be the same, but we expect to reject it. The difference
between the two derived values differed vastly, as is consistent with modern physical
understanding.
Introduction
The use of springs in our modern world are vast, stretching from ball point pens to
scales to automobiles and guns. One of the earliest known uses of advanced springs was by
King Tutankhamen, who used leaf springs on his carriages to make his journeys more
smooth (1). Springs have been investigated by many people, including Leonardo Da Vinci,
who invented a small spring to be used in guns (1).
In classical physics, it is known that the restoring force on an object under pressure
is directly proportional to the amount it is stretched or compressed (2). This gives the
equation below;
F = -Kx
(1)
Where F equals the restoring force of the spring, x equals the amount the spring is
stretched or compressed, and K equals the spring constant of the spring in question. This
equation is known as Hooke’s Law, named after Robert Hooke (2). With Hooke’s law, there are
limitations, including that it is not valid beyond the elastic limit of the material. This means
that no material can be compressed or stretched beyond its elastic limit, and if it is, it will
be permanently deformed (2).
If one were to place a mass on a vertically suspended spring, as in the interest of this
investigation, the spring would stretch until equilibrium has been reached between the
restoring force of the spring and the force of gravity on the mass. Thus, one can write a
balanced force equation, as shown below.
kx = mg
(2)
This equation can be manipulated to isolate for the spring constant of the spring in
question, providing a theoretical value for the spring constant.
𝐾 =
𝑚𝑔
𝑥
(3)
This system would remain stationary unless acted upon by an external force, as
stated in Newton’s first law of motion (3). Indeed, if the spring mass system is stretched a
distance x, it will oscillate in a mostly constant way, and the motion of the system can be
described as harmonic.
It is known that the motion of a harmonic oscillator, such as the one described above,
will oscillate with a period T, and this period is described with the equation below (4).
𝑇 = 2π
𝑚
𝑘
(4)
Therefore, if one were to measure the period of oscillation of the mass spring system when
stretched a certain distance, one could derive the spring constant through manipulating the
equation above as such.
𝑇
2
𝑘 = 𝑚/( 2π )
(5)
Thus, there are multiple ways to derive a spring constant for a mass spring system, with the
first method calculating static stiffness and the second method calculating dynamic
stiffness. It is important to note that these values will not be the same, because in both
cases, the mass of the spring itself, as well as the mass of the string is ignored in calculation
(5).
Method
The materials to use for this investigation are a tape measure, a spring, a retort
stand, a stopwatch, a 200g mass, scissors, and a sufficient amount of string to attach both
the spring to the mass and the spring to the retort stand.
To set up this investigation, two lengths of string are cut. One length is used to tie the
spring to the retort stand, and the equilibrium length of the spring is measured with the
tape measure. The other length of string is used to tie the 200g mass to the other end of the
string, and the new, stretched length of the spring is measured. Using a balanced force
equation for the forces acting on the affixed mass, a theoretical spring constant is
determined.
Figure 1: Experimental Arrangement
Experimental Method:
1. Pull the 200g mass down, to extend the spring a distance of 0.01m
2. Release the mass, and after it has oscillated 3-4 times, begin the stopwatch, being
careful to begin the stopwatch at the moment when the mass is at its bottom
maximum amplitude.
3. Count 20 full bounces, and stop the stopwatch when 20 bounces have been
completed.
4. Record the time on the stopwatch, and divide that number by 20 to get the time for
the period of oscillation.
5. Repeat steps 1-4 for 30 trials.
Limiting factors for this experiment include that the stopwatch only measures to
centiseconds, so the degree of accuracy would be to centiseconds only. Another limiting
factor is that the tape measure only measures to millimetres, so the degree of accuracy
would be to millimetres only. Human error must also be considered, as it is possibly the
largest source of error for this investigation. This human error stems from the inability of
the human eye to recognize the exact moment the mass spring system completes its 20th
oscillation. In addition, there is a slight delay from the moment the 20th oscillation is
finished and the moment that the stopwatch is stopped, which will increase the
experimentally determined period.
To reduce limiting factors, the same person operated the stopwatch each time, and
counted the oscillations. Also, the stopwatch was started after 3-4 oscillations of the mass, to
produce the most accuracy possible for the investigation.
Analysis
When we performed this experiment, we expected that the theoretically derived
spring constant would be different from the experimentally derived spring constant. We
knew this because in a theoretical world, the spring constants would be the same, but in our
real world, the mass of the spring and the mass of the string must be accounted for.
We compared a theoretically derived value for the spring constant of a spring and the
experimentally derived value for spring constant. We tested this by using a balanced force
equation to derive a theoretical value, then measured the period of oscillation of the spring
mass system when stretched a distance of 1 cm, and derived a spring constant with
calculations. Our assumption held, as the theoretically derived value differed vastly from the
experimentally derived value. Our investigation did not yield outliers. The level of
significance was to 0.01 seconds. Because the only value we measured was time, our
experimental uncertainty is half of the smallest time increment. The stopwatch measured to
the centisecond (0.01s), so the experimental uncertainty is 0.005 seconds.
Our key result was that two values for the spring constant of the spring were
drastically different. The mean period of oscillation was 0.322 seconds, and standard
deviation in our data set was 0.005. Using a formula that relates period to spring constant,
we determined the experimental spring constant to be 76.246N/m, in stark contrast to the
theoretically determined spring constant of 122.625 N/m. Our Z score test yielded a Z value of
65.5, with a probability (p value) of 0. This means that we reject our null hypothesis, meaning
that the experimentally derived value and the theoretically derived value for spring constant
differed.
Figure 2: A graph of Trial # vs Period of Oscillation
Discussion
In our investigation, we wanted to show that the spring constant we calculated from
experimentally gathered data was different from the spring constant we calculated from our
balanced force equation of the affixed mass when in equilibrium. The reason our results
differed so much was because of the difference between static and dynamic coefficients, and
we expected this result given that it aligns with modern physical theory. Because our mass
was relatively small, the effect of neglecting the mass of the spring itself, and the mass of
the string were very great, which also skewed our results.
In this investigation there were multiple causes of error growth. One of which was
that the spring used in the investigation was most likely used before, so using Hooke’s Law
to calculate the spring coefficient would yield a less accurate result than if a new spring was
used. In addition, for our experimental calculations, we used the mean period of oscillation
to calculate the spring coefficient, when it would have been more precise to calculate the
spring coefficient with each period from each of the 30 trials, and then took the mean spring
coefficient. Another big cause of error growth was human error, which made a big impact on
the results. The human error stemmed from the inability of the human eye to recognize the
exact moment the mass spring system completed its 20th oscillation. In addition, there was
a slight delay from the moment the 20th oscillation was finished and the moment that the
stopwatch was stopped, causing the period to increase slightly.
The results of this experiment could be extended into all sorts of fields, including the
automobile industry, most all types of industrial machinery, furniture, even in the crafting of
pens. In the furniture industry, there is a direct correlation, as springs are used to hold up
cushions on a couch for example, and when someone sits on said couch, the spring will
compress, and there will be a restoring force. To improve this experiment, one could use a
photogate, which is a much more precise way to time oscillations and therefore period, than
using a stopwatch and the naked eye. In addition, one could do more calculations to factor in
the weight of the spring, and/or use a bigger mass, however that would cause the spring to
degrade more quickly, and thus change its spring constant.
Appendix
(1) Data
Citations:
1.
Hagens Spring Group. (2019). The history of the spring. Hagens Fjedre. Retrieved
January 27, 2023, from
https://www.hagens.com/en/about-the-hagens-spring-group/the-history-of-the-spri
ng.aspx
2. Bellis, Mary. (2020, August 27). Biography of Robert Hooke. Retrieved January 27, 2023
from https://www.thoughtco.com/spring-coils-physics-and-workings-4075522
3. Newton, I. (1833). Philosophiae naturalis principia mathematica (Vol. 1). G. Brookman.
4. University of Brimingham. (2022). Physics - simple harmonic motion. University of
Birmingham. Retrieved January 27, 2023, from
https://www.birmingham.ac.uk/teachers/study-resources/stem/Physics/harmonic-m
otion.aspx
5. Garrett, S. (2020). The Simple Harmonic Oscillator. In Understanding Acoustics (pp.
59–131). essay, Graduate texts in Physics .