The purpose of this experiment was to investigate the spring constants of springs in parallel and series configurations. First, the spring constant for a single spring was investigated then spring constants for springs in parallel and series arrangements was also investigated and analyzed. The experiment was based on the Hooke’s Law which states that "the extension of a helical spring is directly proportional to the weight applied, provided the elastic limit of the spring is not exceeded." This shows that when load is applied to a spring, the spring undergoes extension to a certain point where it cannot return back to its initial status or it gets deformed. Before the elastic limit of a spring is not exceeded, there exist a spring constant which is the ratio between the load and change in spring extension before the elastic limit. The arrangement of springs also affects the level of extension of springs hence affecting the spring constant. Springs can either be arranged in series or in parallel as shown in the figure below.
From Hook’s Law: F = m*a = -kΔl = -k.x
The spring extension depends on the amount of force F, applied and the spring constant k. For two springs arranged in parallel as shown above, the extension would be half of one spring 0.5x because the weight is shared by the two springs. Since k = F/x, the spring constant for springs in parallel will be given by;
k = F/0.5x = 2k
For springs in series arrangement, the same amount of force acts on each spring therefore the extension on each spring is x and the total extension becomes 2x. Since k = F/x, the spring constant k will be given by 1/ks.
From the information above, it can be concluded that the general formula for spring constant for parallel arrangement, kp = ka + kb
and for series arrangement, 1/ks = 1/ka + 1/kb
Where ka and kb are the spring constants for the first spring and the second one.
The load and extension results were collected for a single spring, two springs in parallel and then in series. A graph of extension against force was plotted for each of the three and the gradient of the line of best fit taken as the spring constant.
Materials used
Assorted springs
Rulers (1m and 30 cm)
Clamp stand
Boss and stand
Sotted masses
Experimental Procedure
The clamp, stand and boss were set up firmly making sure that the stand was in balance
A spring of was suspended from the clamp and accurately measured the length before stretching
A mass of 10 g was applied on the spring and the extension recorded. This was repeated for three trials. Other 9 mass increments of the same mass were applied on the spring and each increment repeated thrice. Small masses of 10g were used to assess the change in extension over minimum load and the ten trials could give a good range for plotting the graph of force against extension to determine the spring constant.
The procedure above was repeated for two identical springs arranged in series.
The same procedure was also repeated for identical springs in parallel arrangement.
Laboratory health and safety
Some of the health and safety issues during the experiment was on the danger posed by the equipment and masses used. Every student had to attend the experiment in closed shoes to avoid any injuries.
Experimental set up
Data Collection and Analysis
Table 1: Results for single spring
Mass (kg)
Force (N)
Ext. 1 (cm)
Ext. 2 (cm)
Ext. 3 (cm)
Average (m)
0.10
1.00
3.80
3.70
3.60
0.0370
0.20
2.00
7.70
7.50
7.60
0.0760
0.30
3.00
11.40
11.50
11.60
0.1150
0.40
4.00
15.40
15.30
15.60
0.1543
0.50
5.00
19.20
19.30
19.40
0.1930
0.60
6.00
22.90
22.80
22.70
0.2280
0.70
7.00
23.80
24.00
23.90
0.2390
0.80
8.00
30.30
30.20
30.90
0.3030
0.90
9.00
34.20
34.30
34.20
0.3423
1.00
10.00
38.50
38.30
38.40
0.3840
The gravitational acceleration was taken to be 10 m/s2
Force = mass*gravity = 0.1*10 = 1.00 N
Length before extension, L = 2.2 cm
Sample calculation, = (3.80 + 3.70 +3.60)/3 cm = 3.70 cm = 0.0370 m.
Graph 1: Extension against force for single spring
From the linear equation, y = 0.0353x + 0.0078, gradient = 0.0353 = single spring constant
Table 2: Results for springs in series
Mass (kg)
Force (N)
Ext. 1 (cm)
Ext. 2 (cm)
Ext. 3 (cm)
Average (m)
0.10
1.00
6.80
6.90
7.00
0.0690
0.20
2.00
14.70
14.80
14.60
0.1470
0.30
3.00
21.20
21.10
21.40
0.2127
0.40
4.00
29.00
28.90
28.70
0.2887
0.50
5.00
36.50
36.60
36.70
0.3660
0.60
6.00
43.60
43.70
43.30
0.4350
0.70
7.00
50.80
50.80
50.70
0.5076
0.80
8.00
57.80
57.70
57.90
0.5780
0.90
9.00
64.70
64.60
64.70
0.6460
1.00
10.00
72.00
72.10
72.30
0.7213
Length before extension, L = 4.6 cm
Graph 2: Extension against force for springs in series
Gradient = 0.073 = spring constant for springs in series
Table 3: Results for springs in parallel
Mass (kg)
Force (N)
Ext. 1 (cm)
Ext. 2 (cm)
Ext. 3 (cm)
Average (m)
0.10
1.00
1.10
1.30
1.20
0.0120
0.20
2.00
3.60
3.40
3.50
0.0350
0.30
3.00
4.90
4.80
5.00
0.0470
0.40
4.00
6.70
6.50
6.60
0.0660
0.50
5.00
8.30
8.40
8.20
0.0830
0.60
6.00
10.30
10.40
10.60
0.1043
0.70
7.00
12.10
12.20
12.30
0.1220
0.80
8.00
14.10
13.90
14.30
0.1410
0.90
9.00
15.80
16.10
15.90
0.1593
1.00
10.00
17.50
17.60
17.60
0.1760
Length before extension, L = 2.2 cm
Graph 3: Extension against force for springs in parallel
Slope = 0.018 = spring constant for springs in parallel
Conclusion and Evaluation
The experiment was carried out well as per the procedures. The results for the force and extension for each spring setting was obtained and analyzed graphically and the gradient taken to be the spring constant. From the results, it was found out that the spring constant for series arrangement (0.073) was almost twice that of single spring (0.0353). The spring constant for the parallel setting (0.018) was also almost half of the single spring constant. The results obtained for the three spring arrangements fairly supported Hooke’s Law. The possible sources of errors for this experiment could be inaccurate measurement of spring extension and wear and tear aspect of the springs used. The range of values used and the number of repetitions were sufficient for data analysis.
References
https://socratic.org/questions/what-is-the-spring-constant-in-parallel-connection-and-series-connection
(Accessed June. 3, 2018)
http://www.cyberphysics.co.uk/topics/forces/springs_series_parallel.html (Accessed June. 3, 2018)
https://physics.info/elasticity/ (Accessed June. 3, 2018)
https://www.britannica.com/science/Hookes-law (Accessed June. 3, 2018)
