 ```Slinky Stretch:
Effect of Spring Resting Length on Maximum Stretch of
Spring When Allowed to Decompress Under Force of
Gravity
Ryan LaPointe
Section B
December 18, 2015
Lab Partners: Patrick Lei, Dany Alkurdi
Introduction
The purpose of this experiment was to demonstrate the law of
conservation of energy using a spring, namely a slinky children’s toy. How
does increasing the resting length of a spring affect the maximum
stretched length of the spring that occurs when the spring is allowed to
decompress vertically under the force of gravity? If the resting length of
the spring is increased, it is hypothesized that the maximum stretched
length of the spring when allowed to decompress vertically under the force
of gravity will increase linearly, because the resting length of a spring is
inversely proportional to the spring constant.
Methodology
The slinky was held against the bottom of a horizontal surface. A number
of turns of the slinky were then held in place as the rest of the slinky was
allowed to fall under the force of gravity. Using a camera, the maximum
stretched length of the spring was determined through video analysis.
This procedure was repeated with various numbers of turns of the slinky
held in place. Both before and after this experimentation, force
measurements were taken from the slinky in order to determine the
coefficient of friction of the entire slinky. The slinky was placed horizontally
on a table and one end was attached to a force measuring device. The
slinky was then stretched to the lengths of 0.00, 0.10, 0.20, 0.30, and 0.50
meters and the force exerted by the slinky on the force sensor was
recorded. The slinky was also massed.
Diagram
𝐶𝑀
𝑍0
𝐿0 𝐿𝑡𝑜𝑡𝑎𝑙
𝑍
𝐿
𝐶𝑀
Constants and Equations
Equations
Constants
𝑀 = 64.2 g
𝐿𝑡𝑜𝑡𝑎𝑙 = 16.4 cm
𝑔 = 9.8 m/s 2
4
𝑚𝑔
3
4
𝑚𝑔
3
∆𝐿
𝑘
or ∆𝐿 =
𝑀
(𝐿 )
𝑚=
𝐿𝑡𝑜𝑡𝑎𝑙 0
2
∆𝑍 = ∆𝐿
3
𝑘=
(See Appendix A for derivations)
Summarized Data
Setting
AVG
L
(m)
STDEV
%RSD
Ltheo
%err
k
ktheo
%err
L0
m
Z
ΣEi
ΣEf
of L
of L
(m)
of L
(N/m)
(N/m)
of k
(m)
(kg)
(m)
(J)
(J)
%
change
of ΣE
0.33
0.14
0.008
1.149
0.220
58.80
3.49
1.26
64.02
0.05
0.0212
0.16
0.034
0.034
0.000
0.50
0.30
0.007
0.769
0.506
67.89
2.44
0.83
65.96
0.08
0.0321
0.35
0.111
0.111
0.000
0.67
0.52
0.010
0.918
0.909
75.98
1.91
0.62
67.53
0.11
0.0430
0.60
0.254
0.254
0.000
0.83
0.78
0.007
0.499
1.394
78.01
1.56
0.50
67.90
0.14
0.0533
0.91
0.477
0.477
0.000
0.91
0.92
0.011
0.729
1.676
82.15
1.45
0.46
68.63
0.15
0.0584
1.07
0.615
0.615
0.000
Avg.
0.813
Avg.
72.57
Avg.
66.81
Graphs
L vs Amount of Slinky Used
L (m)
Measured
Theoretical
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
L = 2.521s - 0.6925
R&sup2; = 0.9837
L = 1.3595s - 0.3489
R&sup2; = 0.9869
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Spring Constant vs Amount of Slinky Used
Calculated
Theoretical
Spring Constant, k (N/m)
4.0
3.5
k[s] = 1.3345s-0.868
R&sup2; = 0.9997
3.0
2.5
2.0
k[s] = 0.4145s-1
R&sup2; = 1
1.5
1.0
0.5
0.0
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Analysis
The average percent relative standard deviation (%RSD) of the collected
data was 0.813%, indicating high precision in the data collection. All
theoretical and calculated models have a high coefficient of determination
(r2), denoting that the models fit the data with high fidelity. The models are
only valid in the first quadrant, because a spring of negative or zero length
cannot exist and neither 𝑘 nor ∆𝐿 can be negative. The theoretical and
calculated models strongly differ. Regarding ∆𝐿 vs amount of slinky used,
the average error of the theoretical model with respect to the calculated
model was 72.57%; for 𝑘 vs amount of slinky used, the average error was
66.81%. A possible source of error in this experiment is the friction
between the slinky and the table when the force of the spring was being
measured in order to generate the theoretical 𝑘 model. This source of
error would influence the theoretical 𝑘 model in a direction consistent with
that present in the graph.
Conclusion
The results of this experiment do support the hypothesis that the maximum
stretched length of a spring when allowed to decompress vertically under
the force of gravity will increase linearly as the resting length of the spring
is increased, because a linear trend line fits a graph of ∆𝐿 vs amount of
slinky used with an r2 of 0.99. Further research should be conducted to
develop a method for determining the coefficient of the function without
using approximations.
Appendix A: Equation Derivations
Calculating 𝑘 of the entire spring:
Hooke’s law: 𝐹 = 𝑘∆𝑥
𝐹
𝑘=
∆𝑥
Obtaining theoretical 𝑘 for each setting using the 𝑘 of the entire spring:
𝑠 is the amount of spring used (between 0 and 1).
1
𝑃𝐸𝑠 = 𝑘 (∆𝑥 )2
2
First, we know the following:
1
1
2
𝑃𝐸𝑠𝑠𝑚𝑎𝑙𝑙 = 𝑘𝑠𝑚𝑎𝑙𝑙 (∆𝑥 )2𝑠𝑚𝑎𝑙𝑙 = 𝑘𝑠𝑚𝑎𝑙𝑙 (𝑠∆𝑥𝑙𝑎𝑟𝑔𝑒 )
2
2
𝑃𝐸𝑠𝑠𝑚𝑎𝑙𝑙 = (𝑠)𝑃𝐸𝑠𝑙𝑎𝑟𝑔𝑒
Combining these two equations, we get:
1
2
𝑘𝑠𝑚𝑎𝑙𝑙 (𝑠∆𝑥𝑙𝑎𝑟𝑔𝑒 ) = (𝑠)𝑃𝐸𝑠𝑙𝑎𝑟𝑔𝑒
2
1
1
2
2
𝑘𝑠𝑚𝑎𝑙𝑙 (𝑠∆𝑥𝑙𝑎𝑟𝑔𝑒 ) = (𝑠)𝑘𝑙𝑎𝑟𝑔𝑒 (∆𝑥𝑙𝑎𝑟𝑔𝑒 )
2
2
1
1
2
2
𝑘𝑠𝑚𝑎𝑙𝑙 (𝑠 2 )(∆𝑥𝑙𝑎𝑟𝑔𝑒 ) = (𝑠)𝑘𝑙𝑎𝑟𝑔𝑒 (∆𝑥𝑙𝑎𝑟𝑔𝑒 )
2
2
2
𝑘𝑠𝑚𝑎𝑙𝑙 (𝑠 ) = (𝑠)𝑘𝑙𝑎𝑟𝑔𝑒
𝑘𝑠𝑚𝑎𝑙𝑙 (𝑠) = 𝑘𝑙𝑎𝑟𝑔𝑒
Calculating ∆𝑍 for each setting:
(Derivation from Finding the Center of Mass of a Soft Spring by Juan D.
Serna)
𝑁 is the amount of the slinky measured in number of turns.
𝜃 is the amount of the slinky measured as an angle.
𝑐 ≡ 𝐿0 /2𝜋𝑁
𝑚𝑔(2𝐿0 − 𝑐𝜃 )𝑐𝜃
∆𝑧 =
2𝑘𝐿20
2𝜋𝑁
1
𝑚𝑔(2𝐿0 − 𝑐𝜃 )𝑐𝜃
𝐿0 𝑚𝑔
∫(𝑧)𝑑𝑚
𝑍=
=
∫
[𝑐𝜃 +
]
𝑑𝜃
=
+
𝑚
2𝜋𝑁 0
2
3𝑘
2𝑘𝐿20
𝐿0 2
+ ∆𝐿
2 3
2
∆𝑍 = ∆𝐿
3
𝑍=
Calculating 𝑘 for each setting:
∆𝑃𝐸𝑔 = ∆𝑃𝐸𝑠
1
𝑚𝑔∆𝑍 = 𝑘 (∆𝐿)2
2
2𝑚𝑔∆𝑍
𝑘=
(∆𝐿)2
2
2𝑚𝑔 ( ∆𝐿)
3
𝑘=
(∆𝐿)2
4
𝑚𝑔
3
𝑘=
∆𝐿
Calculating theoretical ∆𝐿 for each setting:
4
𝑚𝑔
3
∆𝐿 =
𝑘
Appendix B: Collected Data
Before
Distance
(m)
0.00
0.10
0.20
0.30
0.50
Force
(N)
-0.01548
0.04260
0.10900
0.14050
0.30600
k
0.000
0.426
0.545
0.468
0.612
Average
0.410 Average
Average
Setting
Trial
0.33
0.50
0.67
0.83
0.91
1
2
3
0.700
0.870
1.067
1.348
1.472
0.704
0.877
1.088
1.357
1.493
0.722
0.877
1.086
1.350
1.481
After
Force
(N)
0.01233
0.04859
0.11320
0.14860
0.27300
k
0.000
0.486
0.566
0.495
0.546
0.419
0.414
Lowest position of bottom of slinky (m)
4
5
6
7
0.708
0.870
1.099
1.358
1.494
0.716
0.880
1.092
1.363
1.504
0.714
0.879
1.093
1.360
1.496
0.710
0.881
1.092
1.364
1.504
8
9
10
0.724
0.882
1.093
1.358
1.502
0.717
0.863
1.101
1.354
1.503
0.723
0.885
1.102
1.371
1.503
```