Series And Parallel Combinations of Rubber Bands Testing Hooke’s Law
Sharma O and Sharma D
Published on: 20230531
Abstract
This experimental work focuses on a handson experiment done testing Hooke’s law of series and parallel combinations of spring to rubber bands. It is very exciting to see if rubber bands can follow the theory given by Hooke or not. Whether rubber bands can behave like springs when they are attached in series and parallel combinations. In this work, six rubber bands were taken and then each of them was measured for their spring constant individually. Then, series and parallel combinations of 2, 3, 4, 5, and 6 bands were made to test their spring constants. These spring constants were then compared with the single band’s spring constants. It is found that rubber bands follow Hooke’s Law with some uncertainties and sources of errors. When bands are attached in series combinations, they become more flexible than a single band, and when they are attached in parallel combinations, they become stiffer than a single band. Rubber bands are made of polymer that behave like a stretchy chainlike molecule, but their molecules are not made of springs hence there are high percentage errors found when equivalent spring constants were compared with single bands using Hooke’s Law.
Keywords
Rubber band; Springs; Stand; Physics; Hooke’s law; Polymer; Series; Parallel combinations; Logger pro; Data analysisIntroduction
Springs are made of coils and stretch when a force is applied to them. If the spring is flexible, it stretches more with less force; if the spring is stiff, it stretches less even though more force is applied. Stiffness is the measurement of the nature of the spring and can be used to find the spring constant. The flexible springs have lower spring constants and stiff springs have higher spring constants. Spring follows Hooke’s Law. When springs are connected to series and parallel combinations, the equivalent spring constant is different from the single springs and can be found using Hooke’s Law of Series and Parallel combinations.
In this experimental work, Hooke’s law of series and the parallel combination is tested with rubber bands. Since rubber bands do not have real coils but they stretch like spring, can Hooke’s law still be applied to them?
A rubber is a stretchy material made up of either natural polymer or synthetic polymer. Their molecules behave like springs within their material instead of having the shape of a coil. Natural rubber is a polymer that has a long chainlike molecule that contains repeated units or sets of molecules which is called a polymer. The term polymer comes from the Greek “poly” (many) and “mer” (parts). The chemical name for natural rubber is polyisoprene [1]. Rubbers can be natural or synthetic.
The main difference between natural and synthetic rubber is in their origin and how they are created. Natural rubber is produced from the latex obtained from a tree whereas synthetic rubber is an artificial polymer produced by using petroleum byproducts [2]. Both rubbers are polymers and have long chainlike structures that coil and uncoil.
Figure 1: Showing the difference between Natural and Synthetic Rubber (taken from the internet [3]).
Rubbers are used for several purposes in the real world where a large quantity of rubber is used to manufacture automobile tires [2]. Other uses may count as it is used in medical devices, surgical gloves, aircraft, automobiles, car tires, pacifiers, clothes, to grocerytotie tie bags, hair bands, wrist bands, or just rubber bands, etc. Natural rubber is obtained from latex, a milky liquid present in either the latex vessels (ducts) or in the cells of rubberproducing plants that is why it is stretchy like a spring [4].
The monomer of natural rubber is 2methyl1, 3butadiene (isoprene), CH2=C (CH3) – CH = CH2. The polymerization reaction can be written as [2]: nCH2 = C (CH3) – CH = CH2  [CH2C (CH3) = CHCH2] n–
Rubber bands that are used in the real world to tie objects or hair are Natural rubber [5]. Due to high demand in the application and easy access or availability, the natural rubber that is used as rubber bands in daily life is used for these experiments.
Some of the experiments were done with rubber bands in the past two years with my group and published some good results there about how rubber bands behave at room temperature or higher temperatures. It can be seen from these publications that a rubber band follows Hooke’s Law and Young’s Modulus [6]. A rubber band may show multiple spring constants [7]. The rubber band also shows the effect of temperature and pressure on its spring constant [8].
Following Hooke’s theory and laws that can be found on these websites [9,10], two to six rubber bands were used in series and parallel combinations to get their detailed results to see how they stretch when some weight is loaded on them.
According to Hooke’s law, the equivalent spring constant of parallel and series combination can be given respectively as k = k1 + k2 and 1/k = 1/k1 + 1/k2 [9,10].
Figure 2: Showing parallel (left) and series (right) combinations of springs (taken from the internet [9,10]).
In this research, the following goals were tested with natural rubber bands for series and parallel combinations or not.
Hypothesis
 Rubber bands will stretch more in series combinations and be more flexible.
 Rubber bands will stretch less in parallel combinations and be stiffer.
 Series and parallel combinations of rubber bands will follow Hooke’s law.
Variables
Independent: Mass or Force applied
Dependent: Extension in the rubber band
Constant: Gravity
Control: Single rubber band, first rubber band R1.
Research Questions:
 What is the role of the number of rubber bands in a series combination for spring constant?
 What is the role of the number of rubber bands in parallel combination for spring constant?
Will rubber bands follow Hooke’s law for series and parallel combinations?
Figure 3: Six Rubber Bands used.
Materials and Method
Six small rubber bands of the same type can be seen in Figure 3, used for experiments. Wooden stand, hanger, pan, masses/weight, hook, ruler, calculator, laptop, MS Office.
 Take six identical, same type but different in colors rubber bands.
 Find a wooden stand and hang one rubber band on it. Hand a hanger or pan of 50 g on it. Measure its initial length as Lo.
 Now hang another 50g mass and measure its new length, L.
 Increase mass in steps of 100 g up to 600 g from now and keep measuring new length L.
 Calculate weight from mass, W = M*g in Newton and change in length dL = L – Lo, in meters.
 Now plot the graph between weight and change in length and find a linear fit for every six bands individually and see their spring constant.
 Now hang two bands in parallel combination on the wooden stand and hang the hanger of 50 g. Measure the initial length, Lo.
 Now add another 50 g to it and measure the new length, L.
 Now keep increasing mass in the step of 100 g until 1000 g and note down the new length, L. Calculate the weight hanging and change in length.
 Plot the graph for 2 bands in parallel combination and find the spring constant.
 Now repeat steps 7^{th} to 11^{th} for 3, 4, 5, and 6 bands in parallel combinations.
 Now repeat steps 7^{th} to 12^{th} for a series combination of rubber bands. See Figure 4.
 Now make a summary table for all spring constants found by graphs for single bands, and parallel and series combinations of bands.
 Compare experimental equivalent spring constants of bands with theoretical equivalent spring constants of bands.
Calculate the percentage error between theoretical and experimental equivalent spring constants. Find sources of error if the percentage error is higher.
Theory
Hooke’s Law of Spring: F =  k*dL    1
Where F is the force applied in Newton, dL = L Lo = change in length in meters, and k is the spring constant in N/m.
Hooke’s Law for parallel combination of springs: k eq = k1 + k2 + …    2
Where k_eq is the equivalent spring constant of the parallel combination of the springs, k1, k2 are the respective spring constants of each spring.
Hooke’s Law for series combination of springs: 1/k_eq = 1/k1 + 1/k2 + …    3
Where k_eq is the equivalent spring constant of the series combination of springs, k1, k2 are the respective spring constants of each spring.
Percent error between theoretical and experimental values of spring constant:
% Error = (k_th – k_exp)*100/k_th    4
Where k_th, k_exp are the theoretical and experimental values of spring constants.
Hooke’s Law Graph
Figure 5: Hooke’s Law Graph plotted dL vs F for a spring. Taken from Internet sites [11].
Figure 6: Hooke’s Law Graph plotted F vs dL for a spring. Taken from Internet sites [12].
Figure 5 shows how Hooke’s law works when force is applied, and spring extends and follows elasticity and spring constant limit, but the linear slope will give the reverse of spring constant whereas Figure 6 shows if the graph is plotted reversely as F vs dL then the linearity slope will give spring constant directly. Raw data details can be seen in Tables 13.
Data Collection
Table 1: Data Details for Single Bands.
Single Total Weight (N) 
R1 ΔL1 (m) 
R2 ΔL2 (m) 
R3 ΔL3 (m) 
R4 ΔL4 (m) 
R5 ΔL5 (m) 
R6 ΔL6 (m) 
4.91E01 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
9.81E01 
2.00E03 
2.00E03 
2.00E03 
2.00E03 
2.00E03 
2.00E03 
1.96E+00 
5.00E03 
1.20E02 
4.00E03 
6.00E03 
4.00E03 
4.00E03 
2.94E+00 
1.40E02 
4.30E02 
8.00E03 
4.10E02 
2.60E02 
1.50E02 
3.92E+00 
3.80E02 
7.60E02 
3.00E02 
6.90E02 
5.60E02 
4.10E02 
4.91E+00 
6.30E02 
9.00E02 
5.20E02 
8.70E02 
8.00E02 
6.20E02 
Table 2: Data Details for Parallel Combination of Bands.
Parallel W (N) 
R12 dL (m) 
R123 dL (m) 
R1234 dL (m) 
R12345 dL (m) 
R123456 dL (m) 
0.4905 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
0.981 
1.00E03 
0.00E+00 
0.00E+00 
0.00E+00 
0.00E+00 
1.962 
2.00E03 
5.00E04 
5.00E04 
5.00E04 
0.00E+00 
2.943 
3.00E03 
1.00E03 
1.00E03 
1.00E03 
0.00E+00 
3.924 
8.00E03 
1.50E03 
1.00E03 
1.00E03 
5.00E04 
4.905 
1.30E02 
2.00E03 
3.00E03 
1.10E02 
1.00E03 
5.886 
2.80E02 
4.00E03 
4.00E03 
1.50E03 
1.00E03 
6.867 
1.20E02 
5.00E03 
2.00E03 
1.50E03 

7.848 
1.80E02 
7.00E03 
3.00E03 
2.00E03 

8.829 
2.80E02 
1.40E02 
5.50E03 
4.00E03 

9.81 
5.00E03 
Table 3: Data Details for Series Combination of Bands.
Series W (N) 
R12 dL (m) 
R123 dL (m) 
R1234 dL (m) 
R12345 dL (m) 
R123456 dL (m) 
0.4905 
0 
0 
0 
0 
0 
0.981 
0.004 
0.004 
0.004 
0.005 
0.008 
1.4715 
0.008 
0.008 
0.014 
0.017 
0.021 
1.962 
0.019 
0.022 
0.04 
0.061 
0.072 
2.4525 
0.04 
0.05 
0.089 
0.125 
0.151 
2.943 
0.07 
0.085 
0.148 
0.201 
0.232 
3.4335 
0.102 
0.135 
0.199 
0.266 
0.308 
3.924 
0.122 
Results
A graph is plotted between the change in length, dL, and weight hung, W for a rubber band, R1, as shown in Figure 7 below.
Hooke’s Law Graph for Rubber Band, R1
Figure 7: Graph for R1, one single rubber band, dL vs W, in Excel.
This figure shows that R1 follows two trends of linearity not one following Hooke’s Law. The linear slope of the graph will not directly give the spring constant, but it will be the reverse of the spring constant.
Graph for parallel combination of two rubber bands, dL vs W
Figure 8: Graph for R1 and R2 in parallel combination showing dL vs W.
Figure 8 shows that R1 and R2 are connected in parallel combination and follow two trends of linearity, not one using Hooke’s Law. The linear slope of the graph will not directly give the spring constant, but it will be the reverse of the spring constant.
Graph for series combination of two rubber bands, dL vs W
Figure 9: Graph for R1 and R2 in series combination showing dL vs W.
Figure 9 shows that R1 and R2 are connected in series combination and follow two trends of linearity, not one using Hooke’s Law. The linear slope of the graph will not directly give the spring constant, but it will be the reverse of the spring constant.
Now graphs are plotted following Hooke’s Graph that is shown In Figure 6 for all bands so that linearity slope can give spring constant directly. Differences can be seen directly when the graph is plotted dL vs F or F vs dL for Rubber band R1 in Figure 10 and Figure 11.
Graph Plotted for R1 in Logger Pro with Linear Fit
Figure 10: Graph for R1 as dL vs W, in Logger Pro with a linear fit. The slope is the reverse of the spring constant.
Reverse Graph for R1 in Logger Pro with Linear Fit
Figure 11: Graph for R1 as W vs dL, in Logger Pro with a linear fit. The slope gives spring constant.
Reverse Graph for all six bands as single bands in Logger Pro with Linear Fit
Figure 12: Graph for all single bands from R1 to R6 as W vs dL, in Logger Pro with a linear fit. The slope gives the spring constant.
From this Figure 12, it is clearly seen that all rubber bands show almost identical values of spring constants, and their data points are overlapping on each other. All of six bands show two linearity fits with two linear slopes showing they have two linear trends of extension and follow Hooke’s Law in two ways with two spring constants as k and K’. All six bands have the same k and k’ to each other. The first spring constant of all six bands is k = 296.4 N/m and the second spring constant for all bands is k’ = 40.2 N/m.
Now graphs are shown for parallel and series combinations of bands as 2, 3, 4, 5, and 6 bands and plotted following Hooke’s Figure 4 to get spring constant directly from the slope. All of them show two linear trends as single bands and follow Hooke’s law. Details can be seen in the Figure from Figure 1326. Comparative results of all series and parallel combinations of all bands are shown in Figures 2326. The calculated data details are shown in Tables 46.
Two Bands in Parallel
Figure 13: The graph for R1 and R2 in parallel combination has two linear trends.
Three Bands in Parallel
Figure 14: Graph for R1, R2, and R3 in parallel combination has two linear trends.
Four Bands in Parallel
Figure 15: Graph for R1, R2, R3, and R4 in parallel combination has two linear trends.
Five Bands in Parallel
Figure 16: Graph for R1, R2, R3, R4, and R5 in parallel combination has two linear trends.
Six Bands in Parallel
Figure 17: Graph for R1, R2, R3, R4, R5, and R6 in parallel combination has two linear trends.
Two Bands in Series
Figure 18: Graph for R1 and R2 in series combination has two linear trends.
Three Bands in Series
Figure 19: Graph for R1, R2, and R3 in series combination has two linear trends.
Four Bands in Series
Figure 20: Graph for R1, R2, R3 and R4 in series combination has two linear trends.
Five Bands in Series
Figure 21: Graph for R1, R2, R3, R4, and R5 in series combination has two linear trends.
Six Bands in Series
Figure 22: Graph for R1, R2, R3, R4, R5 and R6 in series combination has two linear trends.
Discussion
Comparative Graph of all 26 bands in Parallel
Figure 23: Graphs for all bands from R1R6 in parallel combination show a comparison between them.
Comparative Graph of all 26 bands in Series
Figure 24: Graphs for all bands from R1R6 in series combination show a comparison between them.
Comparative Graph for 2 bands in Series and Parallel
Figure 25: Graph for two bands only for series and parallel combination to show a comparison of extension in the same range of X axis.
Linear fit to Figure 25 for 2 bands in Series and Parallel
Figure 26: The best linear fit to the most linear data points of two bands in series and parallel from Figure 25. The slope of 2 bands in series and parallel match with the values found in Figures 13 and 18.
Table 4: Data of Spring Constant of Single Bands.
Spring Constants for Single Bands 

Rubber Bands 
k (N/m) 
k'(N/m) 
R1 
296.4 
40.2 
R2 
296.4 
40.2 
R3 
296.4 
40.2 
R4 
296.4 
40.2 
R5 
296.4 
40.2 
R6 
296.4 
40.2 
Table 5: Data of Spring Constant of Bands in Parallel.
Spring Constants for Parallel Combination of Bands 

# Bands 
k (N/m) 
k'(N/m) 
k_th (N/m) 
k_th'(N/m) 
% Error in k 
% Error in k' 
2 
832.6 
112.3 
592.8 
80.4 
28.8 
28.41 
3 
1346 
124.6 
889.2 
120.6 
33.94 
3.21 
4 
1391 
257.3 
1185.6 
160.8 
14.77 
37.5 
5 
3249 
528.2 
1482 
201 
54.39 
61.95 
6 
3308 
630.6 
1778.4 
241.2 
46.24 
61.75 
Table 6: Data of Spring Constant of Bands in Series.
Spring Constants for Series Combinations of Bands 

# Bands 
k (N/m) 
k'(N/m) 
k_th (N/m) 
k_th'(N/m) 
% Error in k 
% Error in k' 
2 
74.52 
17.49 
148.2 
20.1 
49.72 
12.99 
3 
62.43 
15.51 
98.8 
13.4 
36.81 
13.6 
4 
32.83 
9.14 
74.1 
10.05 
55.7 
9.05 
5 
20.68 
7.09 
59.28 
8.04 
65.11 
11.82 
6 
17.89 
6.22 
49.4 
6.7 
63.79 
7.16 
Data Tables 1 to 3 show the raw data collected for single, parallel, and series combinations of the rubber bands for weight and change in length. Following these tables, when graphs were plotted in Logger Pro for all bands as single, parallel, and series combinations, spring constants of single bands and equivalent spring constants of parallel and series combinations are found those are shown in Tables 4, 5, and 6.
It can be seen from Table 4 that all single bands are identical and have the same spring constants with two linearities. It can be seen in Figure 12 that is plotted for all 6 bands individually as a single band and then linear fit was taken to see their spring constant.
It can be seen from Table 5 that the equivalent experimental spring constants of parallel combinations of bands are higher than single bands and follow Hooke’s law of parallel combinations. When theoretical equivalent values of spring constant are compared with experimental values, some percent error is found in parallel combination, and it indicates that rubber bands are not truly springing with coils, but they are stretchy material made with polymers that behave like springs, but they are not truly spring.
Similarly, it can be seen from Table 6 that the equivalent experimental spring constant of series combinations of the bands is smaller than single bands and follow Hooke’s law. When theoretical values of equivalent spring constant are compared with experimental values, some percentage errors are found. It indicates that Rubber bands follow Hooke’s law as they showed a lower equivalent spring constant for series than single bands and a higher equivalent spring constant for parallel combinations. Rubber bands behave like a spring, but they are not a true spring. Rubber bands are made of Polymers, long chainlike molecules that behave like a spring but are not exactly a spring, and show high percent error from theoretical values.
Conclusion
In this research work, series, and parallel combinations of six rubber bands were tested to see if they follow Hooke’s Law or not. First experiments were performed for six single bands to find their spring constants. All the six bands follow Hooke’s law and gave two spring constants showing two linearities when weight was increased. The first spring constant follows lowers weights whereas the second spring constant follows moderate to higher weights.
All bands follow Hooke’s Law for Series and Parallel combinations of 26 bands. In parallel combination, the equivalent spring constant came larger than single bands whereas in series combination, the equivalent spring constant came smaller than single bands. This indicates that rubber bands follow Hooke’s Law in Series and Parallel combinations. The rubber bands become flexible in series and stiff in parallel combinations. The rubber bands stretch more in series and less in parallel combinations. So, all hypotheses came true.
Some unique behavior was seen with rubber bands that cannot be seen in springs. Two major discoveries were made in this research work; 1) Rubber bands show two linearities and show two spring constants for lower and higher weights. 2) Rubber bands follow Hooke’s Law as single band or band in combinations in series and parallel, but the experimental values of the spring constants for series and parallel are very different than their theoretical values. It indicates that Rubber bands do not behave like real springs, but they can be compared with springs. Rubber bands do not have coils like springs but their molecular structure that is polymer stretches in two different modes when they are stretched. First, they stretch slowly and then fast and show two linearities and hence they have higher percentage errors from theoretical values as theoretical values are calculated from Hooke’s law that is made for spring. It means that rubber bands behave like springs, but they are not truly springs. Their longchain molecule stretches like a spring for some time and then behaves like another spring for the rest of the time within the elastic limit as shown in Hooke’s Graphs.
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