Experiment 6: Spring constants

Error Analysis: Force constant in springs
Maria Loraine Menorca
National Institute of Physics, University of the Philippines Diliman, Quezon City [e-mail] Abstract
This experiment aims to fully understand the analysis of possible errors in the measurement of the force constants in different spring systems. Springs can be utilized to store and release energy. According to Hooke’s Law, the response of a spring to external forces is characterized by the equation F = -kΔx. In this experiment, two springs were combined in series and in parallel. In a series spring system, the two springs are hooked together end-to-end while for the parallel spring system, the two springs were hooked to the iron stand parallel to each other. Both of the spring systems were connected to five different masses as its trials. The force constants in a single spring system was determined to be 17 ± 20 and 19 ± 20 for the .06 and .065-meter springs, respectively. In the series spring system, the experimental k was found to be 8 ± 7 and a theoretical value of 9 ± 30. Comparing the two results, the calculated uncertainty value was 87.5% with a percent deviation of 11% ± 4000%. For the parallel spring system, the experimental k was measured to be 48 ± 40 and was again compared to its theoretical value of 36 ± 20. The relative uncertainty and percent deviation was found to be 83.3% and 30% ± 200%, respectively. Having relatively greater uncertainty values than deviation values, the experiment was rendered a success. For further improvements, it is advised to use other methods of finding the force constant in springs and then compare the error analysis of these two parallel experiments.

1. Introduction

Springs are devices that can be utilized to store and release energy. It can be combined in series and parallel. The response of each of the spring system to external forces can be characterized by the use of Hooke’s Law where F is the force exerted on the external agent by the spring called restoring force and k, which is often referred to as the spring constant, is the proportionality constant that characterizes the “stiffness” of the spring when connected in series or in parallel. [2] F = -kΔx (1) In a single spring system, x is the natural equilibrium length of the vertical spring. Having a restoring force of the same magnitude with the weight of the mass connected to it, the force (mg) is then just equal to the spring constant k multiplied to the total displacement of the system. With this relationship, k can be experimentally determined as the slope of the equation obtained from the force vs. displacement plot.

(2) k = (3) For a spring connected in series, the two springs hooked together forms a single and longer spring. In this system, the forces of the two springs can be rendered equal in magnitude and thus forming the following relationship: k1x1 = k2x2 = mg (4) xtotal = x1 + x2 (5)

Using these equations, the equation for the effective force constant can be obtained.

x2 = mg/ k2 (6) ks (x1 + x2) = mg (7)

ks (mg/ k1+ mg/ k2) = mg (8)

(9)

For a spring connected in parallel, the two springs are hooked parallel to each other to act as one unit. Here, the net force of the spring system is the addition of the forces exerted by the two springs. Forming this relationship, the total effective force constant can be obtained through the equation below. [1]

k1x + k2x = mg = kpx (10)

In this experiment, the analysis of the possible errors in the measurement of the force constants, as external forces are applied in different spring systems, will be investigated fully. It also aims to differentiate systematic and random errors while the possible sources of errors are identified and taken into account. Specifically, this experiment sought answers to the following questions:
1.) What is the force constant of a single spring?
2.) What is the force constant of the two springs in series?
3.) What is the force constant of the two springs in parallel?
4.) What are the methods used to measure the force constants and how will it be rendered true?

This experiment only focuses on the error analysis of the measured force constants of springs in series and parallel. The force constant in single spring system was used as the basis for the theoretical results.

2. Methodology

The materials used in this experiment were an iron stand, ruler, weighing scale, two springs of different lengths and five different masses. In the determination of the force constants of the two springs, the length of the springs before and after it is connected to a mass, was measured separately. The data obtained was then used to calculate for their force constants using equation 3. For the determination of the force constant of springs in series, the two springs are hooked together end-to-end and was connected to five different masses as its trials. With this condition, the equation 9 was used to calculate for the force constant. In the determination of the force constant of springs in parallel, the two springs was hooked to the iron stand side by side then was connected to five different masses as its trials. The relationship of these variables was then used to calculate for the force constant using equation 10. The forces in each of the spring system was determined to be the weight of each of the masses which is mass multiplied to the acceleration due to gravity.

3. Results and Discussions

Table 1. Single Spring system with .06m spring

F(N) x(m) k
M1 (0.046kg)
0.4547
.02
22.753
M2 (0.097kg)
0.9506
.045
21.124
M3 (0.194kg)
0.1902
.095
2.002
M4 (0.240kg)
2.3569
.12
19.640
M5 (0.291kg)
2.8528
.145
19.674
kave= 17. 035 k1 = 17 ± 20

Table 2. Single spring system with .065m spring

F(N) x(m) k
M1 (0.046kg)
0.4547
.02
22.75
M2 (0.097kg)
0.9506
.042
22.63
M3 (0.194kg)
0.1902
.085
2.24
M4 (0.240kg)
2.3569
.105
22.45
M5 (0.291kg)
2.8528
.125
22.82
kave = 18.5744 k2 = 19 ± 20

The tables above show the calculations for the force constant of a .06 and .065-meter spring. The total displacement x(m) of each spring system were obtained by subtracting the final length (xfinal) of the spring to the initial length of the spring (xinitial). The data was then used to calculate for force constant k in each of the five different masses. Following equation 3 we get,

k = = = 22.753 = k1(1) k = = = 22.630 = k2(2)

For each of the spring systems, the values of k obtained was then averaged to get their total effective Force constant. The uncertainties of the value was also considered.

Table 3. Force constant of springs in series

F(N) x(m) k
M1 (0.046kg)
0.4547
0.045
10.104
M2 (0.097kg)
0.9506
0.095
10.006
M3 (0.194kg)
0.1902
0.19
1.001
M4 (0.240kg)
2.3569
0.245
9.620
M5 (0.291kg)
2.8528
0.28
10.189
kave = 8.184 ks(expt) = 8 ± 7 The table above shows the measured force constant in series. The experimental force constant for each trial was calculated using equation 3 and was then averaged to get the total effective force constant. The theoretical force constant was determined using equation 9. For deviations and possible errors of the value obtained, the experimental k value was compared with the calculated theoretical k value.

ks(theo) = = = 8.97 ± 29.96 = 9 ± 30

Relative uncertainty = q(%) = (q ÷ < q >) × 100 = (7 ÷ 8) × 100 = 87.5%

Relative deviation = = = 0.1 ± 4.0
Absolute deviation = = |(9 ± 30)-(8 ± 7)| = 1 ± 37
Percent deviation = = = 11 % ± 4000%

Table 4. Force constant of springs in parallel kave = 47.582 kp(expt) = 48 ± 40

The table above shows the measurement of the force constant of springs in parallel. The experimental force constant for each trial was calculated using equation 3 and was then averaged to get the total effective force constant. The theoretical force constant was determined using equation 10. For deviations and possible errors of the value obtained, the experimental k value was compared with the calculated theoretical k value.

kp(theo) = k1 + k2 = (17 ± 20) + (19 ± 20) = 36 ± 20

Relative uncertainty = q(%) = (q ÷ < q >) × 100 = (40 ÷ 48) × 100 = 83.3%

Relative deviation = = = 0.3 ± 0.7
Absolute deviation = = |(36 ± 20)-(48 ± 40)| = 12 ± 20
Percent deviation = = = 30% ± 200%

4. Conclusion and Recommendation

The force constants in a single spring system was determined to be 17 ± 20 and 19 ± 20 for the .06 and .065-meter springs, respectively. In the series spring system, the experimental k was found to be 8 ± 7 and was compared to the theoretical value of 9 ± 30. The calculated uncertainty value was 87.5% and the percent deviation was 11% ± 4000%. For the parallel spring system, the experimental k was measured to be 48 ± 40 and was again compared to its theoretical value of 36 ± 20. For possible errors, their uncertainty and deviation values were calculated. The relative uncertainty and percent deviation was found to be 83.3% and 30% ± 200%, respectively. Having the values yielded from the experiment result to small deviations from their theoretical values and relatively higher uncertainty values, it can be concluded that the values are acceptable and the experiment was a success. However, the experiment’s accuracy could be increased by using other methods of finding the force constant of a spring, such as the determination of the slope in the force vs. displacement plot of each spring system. Through this, computational errors can be minimized and thus, a more effective result. With this, the error analysis of the two parallel experiments can be compared.

References

[1] “Experiment 6: Springs in series and parallel” Mechanics Laboratory AM 317. Web. 18 Apr. 2015. <http://www.csun.edu/~ehrgott/EXP6.>
[2] Young, Hugh D., Roger A. Freedman, and A. Lewis Ford. “Work done on a spring scale” University Physics. 13th Edition ed. Pearson Education, 2012. 188-189. Print.

References: [1] “Experiment 6: Springs in series and parallel” Mechanics Laboratory AM 317. Web. 18 Apr. 2015. <http://www.csun.edu/~ehrgott/EXP6.> [2] Young, Hugh D., Roger A. Freedman, and A. Lewis Ford. “Work done on a spring scale” University Physics. 13th Edition ed. Pearson Education, 2012. 188-189. Print.