Testing Real Springs
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Spring Force Lab Report
Joseph W.
Philopateer Christian College
SPH4UA, Physics 12
Ms. Kallini
18/10/2023
Testing Real Springs
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Introduction
The purpose of this lab experiment is to test real springs and be able to determine under
what conditions, if any, they obey Hooke’s law. This lab experiment also explores what occurs
within the spring constant of 2 or more springs that are linked together. The main forces that will
be focused on are the spring force and gravity.
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Spring Force Lab Report
Hypothesis/Prediction
As the mass is attached to the spring from below, it displaces the string by
stretching it from its “relaxed” state by a fixed amount. The mass causes the string to be
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stretched downward, therefore, Δ𝑥 the vertical stretch that the spring experiences is in
the downward direction meaning that the spring force, FX, is in the upward direction.
This is shown by observing the FBD of the forces acting on the mass as shown in the
diagram. By observing how the spring force is opposite to gravity, and the mass does
not accelerate in any direction, the net force acting on the mass is 0. This also concludes
the fact that the spring force is equal to gravity, or to be specific, mass * gravity. This is
how the magnitude of the spring force can be calculated using the FBD.
Keeping in mind that the equation of the spring force is also
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FX = -kΔ𝑥, knowing how the stretching of the string is in the
opposite direction of the spring force, the graph of FX versus x will
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result in a linear graph with a negative slope since Δ𝑥 must be
negative since it is in the opposite direction of FX. The resulting
slope of the graph will be equal to k, the spring constant. This graph
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on the right illustrates the linear relationship between FX vs Δ𝑥 with FX on the vertical axis and
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Δ𝑥 the horizontal.
By using the same reasoning, the spring constant of 2 springs hung linearly would greatly
differ from the individual spring constants. This is because the KT relies on both the K1 and K2
both both strings combined in some form. Since KT is the resultant slope of both K1 and K2, the
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equation of the total spring constant should have some form of addition or even subtraction. In
other words, KT = K1 + K2. or KT = K2 - K1, or finally, KT = K1 - K2 These equations should result
in being able to solve for the total spring constant in some way.
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Analysis
Spring 1
The first spring that was selected to be suspended from the clamp and measured the
length after it stretched due to various masses hung from it was incredibly stiff. This resulted in
very little stretching or even no stretching at all depending on the mass hung from it. The initial
length of the spring while at rest was 4cm. The table below describes the mass (in grams) of the
mass attached to it compared to the new length of the spring. The table also includes the distance
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that the spring had stretched from rest, Δ𝑥, and finally, the spring force acting on the mass and
spring.
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Mass (g)
Spring Length (cm)
FX (N)
30
4
0
0.29
40
4.2
0.2
0.39
100
6.3
2.3
0.98
200
10
6
1.96
Δ𝑥 (cm)
The lighter masses had very little effect on the spring due to its stiffness, therefore it had
barely stretched by less than a centimeter. As the mass gets heavier, the spring stretches much
further. This is due to the equation from the FBD, FX = mg, and the equation of the spring force,
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FX = -kΔ𝑥. combining the 2 equations results in -kΔ𝑥 = mg. The direction of the Δ𝑥 is opposite to
the spring force. Therefore, while keeping the direction of the spring force to be positive, the
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magnitude Δ𝑥 will become negative resulting in the spring constant and Δ𝑥 both becoming
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positive as kΔ𝑥 = mg. Finally, after isolating for Δ𝑥, the final equation becomes Δ𝑥 = (mg)/k.
This final equation can be seen throughout all the springs and shows that as the mass increases,
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the magnitude Δ𝑥 will also increase since the numerator will also increase. The following graph
displays the relationship of the spring force FX, vs the distance
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stretched, Δ𝑥. Δ𝑥 is considered to be negative since it is in the
opposite direction of the spring constant. It is mainly a linear
relationship, with slight deviation that is caused due to
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measurement errors for the spring length and calculating the Δ𝑥.
The negative slope on the graph represents the spring constant K, or
in this case -K. After calculating the slope of the graph using the
first and last points, the resulting slope is -0.27. This means that the spring constant for the first
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spring is 0.27 N/m. To calculate the slope, however, the units Δ𝑥 needed to be converted from cm
to m to respect the proper SI units.
Spring 2
The next spring to be tested was considerably less stiff than the previous string.
Therefore, the masses had a larger effect on the spring and were able to be stretched both more
easily and further. The initial length of this spring was estimated to be about 9.6 cm. After
repeating the same steps as with the previous spring, the results were as shown:
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Mass (g)
Spring Length (cm)
30
13.4
3.8
0.29
40
15.6
6
0.39
Δ𝑥 (cm)
FX (N)
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100
24.2
14.6
0.98
200
37.3
27.7
1.96
This table shows the results of the second experiment. The spring force remains constant
since the spring constant was calculated using the equation FX = mg. The stiffness of this spring
drastically changes the results from the previous experiment. Although this spring was longer
than the previous spring, it can be noted that it stretched much further than the first spring. This
shows how the stiffness of a spring can greatly affect the amount it can stretch. The relationship
between the spring force and the distance stretched by spring 2 once
again results in a negative linear graph, however, one key difference
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is that the slope is not as steep due to the major increase of the Δ𝑥.
This means that the spring constant of spring 2 should be smaller
than the spring constant of spring 1. The following graph proves this
relationship as the slope of the graph once again appears to be
decreasing at a constant, linear rate, with some slight deviation due
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to some form of error. As the Δ𝑥 increases, the magnitude of FX
decreases. Although the slopes of both graphs appear to be similar, this graph has a much flatter
slope by looking at the increments along the horizontal axis. Once again, after calculating the
slope using the beginning and final points, the result is K = 0.07 N/m. This is true due to the fact
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the fact that by re-arranging the previous equation, Δ𝑥 = (mg)/k, to isolate for k, the equation
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becomes K = (mg)/ Δ𝑥. As the spring stretches further, the Δ𝑥 increases result in the spring
constant, K, decreasing in magnitude.
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Spring 3
The third and final spring to be tested was considered to be the loosest of all the springs
tested. The masses caused the spring to easily stretch much further than the other springs. The
initial length of the final spring at rest was estimated to be around 9.5 cm. Although it is shorter
than spring 2, due to the large difference in flexibility between both springs, spring 3 was able to
produce much different results during the experiment as shown:
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Mass (g)
Spring Length (cm)
30
20.3
10.8
0.29
40
24.4
14.9
0.39
100
49.6
37.4
0.98
200
79.4
69.9
1.96
Δ𝑥 (cm)
FX (N)
This table shows how the stiffness of each spring can drastically affect the length of the
stretched spring. Spring 3 was shorter than spring 2, however judging by the graph, the flexibility
of the spring allowed for the spring to be able to stretch much
further than any of the previous springs. The larger stretching
also concludes with the idea that the spring constant for spring
3 must be the least out of all 3 springs. This correlation brings
the idea that the stiffer a spring is, the higher the spring
constant it will have. The graph displaying the relation for the
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spring force vs the Δ𝑥 of spring 3 shows this point of the
slope is the shallowest of all of all the 3 slopes. The spring
constant of spring 3 is K = 0.03 N/m
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One thing to note out of all the graphs and springs is that since they follow a linear path,
the graph can be extended in any direction. This means that even if the graph is extended to
where the spring force is equal to 50 N, the spring constant will remain the same in each graph,
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and this Δ𝑥 can be calculated using the equation FX = -kΔ𝑥.
Springs 1+2
For this final experiment, spring 1 was attached to the clamp while spring 2 was attached
to the other end of spring 1. The different masses were attached to the bottom end of spring 2.
This final experiment was done to see how the spring constants of each spring would affect each
other. The total length of both springs attached was about 13.6 cm. Keeping in mind the different
stiffnesses of each spring, these were the results of the final experiment:
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Mass (g)
Spring Length (cm)
30
22.6
9
0.29
40
24.3
10.7
0.39
100
36
22.4
0.98
200
52.6
39
1.96
Δ𝑥 (cm)
The results of the experiment show that by hanging both
springs off each other linearly, the springs can stretch further
than if they were separate. This shows that the spring constants
can combine in a way that results in a smaller total spring
constant. This exact point is proven when looking at the graph
of both springs combined. The graph shows that stretching is
indeed much further than the stretching of the individual
FX (N)
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springs. The slope of the graph is much flatter with the total spring constant being equal to
KT = 0.06 N/m. While looking at each spring constant for the individual springs, K1 = 0.27, and
K2 = 0.07, there is a slight connection to combining the 2 constants to calculate the total spring
constant. When taking the reciprocal of each constant, the result is that the reciprocal of K1 + the
reciprocal of K2 = the reciprocal of KT. (1/0.27) + (1/0.07) = 1/0.06. This in turn means that
1/K1 + 1/K2 = 1/KT. This is similar to calculating the total resistance in a parallel circuit where
the equation is equal to 1/RT = 1/R1 + 1/R2 + … + 1/Rn.
Additional Observations
After conducting the experiments and discovering the graphs of the relationships in all
the individual springs and having 2 springs attached, the original hypothesis and prediction are
found to be somewhat accurate to the results. The original hypothesis would be that the graphs
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between the spring force vs the Δ𝑥 would follow a perfectly linear path. This was not fully true
since there were some forms of error while conducting the experiments. While attempting to
measure the length of the springs, they may be slightly bent therefore the measurement was the
real total length of the spring. At other times, some springs were too flexible and weak to the
point where adding slightly too much mass at the end resulted in the spring breaking. In the end,
the results of the graph did indeed show that linear relationship with a negative slope
representing the spring constant, -K. Another part of the prediction was that the spring constants
of springs 1 and 2 would combine in a way whether by addition or subtraction to get the total
spring constant when both springs were attached. This prediction was once again slightly correct
since the way to calculate the total spring constant is by adding the reciprocal of the individual
spring constants and getting the reciprocal of the final answer, 1/KT = 1/K1 + 1/K2 + … + 1/Kn. In
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conclusion, the hypothesis and prediction were pretty accurate due to not accounting for the
possibility of some sort of error.
Conclusion
In conclusion, the purpose of this lab was to test real springs and be able to determine
under what conditions, if any, they obey Hooke’s law. Hook’s law states that the amount of force
exerted by a spring is directly proportional to the displacement of the spring, or in other words,
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FX = Δ𝑥. However, since the direction of the displacement is opposite to that of the spring force,
the equation is in reality, FX = -Δ𝑥. This is especially true in an ideal spring. An ideal spring is
considered to be any spring that obeys Hook’s law which does not experience any form of
friction. It never loses the elastic energy and will continue to be in simple harmonic motion
forever unless it is disturbed. This is not possible in reality because some mechanical energy will
always be lost from the system as a different form of energy such as thermal, sound, or other
forms of energy. A real spring such as in the experiments, experiences damped harmonic motion
due to friction. Real springs are usually considered to be underdamped as they continue to
oscillate between stretching and compression, however, eventually, they will reach equilibrium
with time.
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References
DiGiuseppe, Maurice, and Christopher T. Howes. “Newton’s Laws of Motion.” Physics 12,
Nelson Education, Toronto, Ont., 2011