Experiment: Finding the Spring Constant 'k' with Mass and Spring

This assignment presents a detailed physics lab experiment focused on determining the spring constant, 'k', using a mass and a spring. The experiment involves attaching varying masses to a spring, setting the system into oscillatory motion, and measuring the time period of the oscillations. Data is collected, including the time for multiple cycles, from which the time period for a single cycle (T) is calculated. The assignment then explores the relationship between the mass and the square of the time period (T^2), plotting a graph to determine the gradient and subsequently calculating the spring constant 'k'. The analysis includes calculations of 'k' using different methods, comparisons with a hypothesized value, and an evaluation of the accuracy and reliability of the results. It also discusses the identification of dependent and independent variables, and suggestions for improving the precision and reliability of the experimental process. The conclusion summarizes the findings, confirming the hypothesis and highlighting the advantages of using graphical methods for determining 'k' over simple averaging of values. Finally, the document provides recommendations for improvements in future experiments, focusing on repetition, control variables, and the use of precise measurement instruments.

Finding k with a Mass and a Spring 1
FINDING K WITH A MASS OF A SPRING
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Finding k with a Mass and a Spring 2
Finding k with a Mass and a Spring
Abstract
The primary aim of the experiment is to find the value of constant k using a mass and a
spring. It is a practical experiment involving the use of objects of same mass added gradually on
a string and set in motion. By calculation of the duration taken to compete one cycle and plotting
a graph of mass against time, it is possible to find the correct value of constant k.
Introduction
One of the common methods of determining spring constant is by using oscillating
springs and recording the measurements. The most effective method of analyzing this motion of
objects attached to a spring requires the incorporation of Newton’s second law of motion
(Rodriguez and Gesnouin 2007). When a mass is attached to a spring and set into an oscillatory
motion, the time it takes to complete a cycle is determined by its mass and the spring constant.
Since the aim of the experiment is to determine the spring constant, it uses this knowledge to
carry out procedures repeatedly and plot a graph to find the accurate value of the constant.
According to Hooke’s Law, different springs have varying degree of elasticity that determines
their extension constant (Cocco and Masin 2010). Comparatively, the springs used as shock
absorbers in vehicles are less elastic than those used in home appliances as return springs
(Michels et al. 2013). Therefore, their extension constants differ. Determining the spring
constants illustrates their application in varying circumstances. Thus, this experiment is
important since it educates the learners on practical approaches of Hooke’s Law and machines
using the principle.
Scientific Hypothesis

Finding k with a Mass and a Spring 3
Since the experiment determines the value of k, it hypothesizes its value by comparing
the measurement with a specified accepted value. Thus, the hypothesis for the experiment is:
k = 25
Since experimentations are prone to errors, there is an allowance for such mistakes. Therefore,
values close to 25 are acceptable. Hence, the most suitable hypothesis is:
k = 25 ± 1
Method
Equipment list
To carry out the experiment, the following items are need: a spring, 10 masses (50 grams
each), a thirty-centimetre ruler, clamp stand, boss, clamp, stop clock, and plasticine.
Task
 Fasten the clamp stand using a boss.
 Suspend the string whose constant is to be measured.
 Hang a 50g mass and displace it to set it into a simple harmonic motion.
 Measure the time taken for the pendulum to complete 10 cycles. Record this in a table
along with the time period for 1 cycle.
 Write down the precision of the stop clock.
 Repeat this experiment twice more and take an average for the time taken for 1 cycle, T.
 Include a column in your table of T2, where T is the time period of 1 cycle.
 Include a column labelled k.
 Plot a graph of time period squared, T2 (on the x-axis) against mass, m (on the y-axis).
 Plot a line of best fit using the values obtained from the experiment and calculate its
gradient.

Finding k with a Mass and a Spring 4
Experiment Results
Mass (Kg) Time for 10
cycles, t (s)
Time for 1
cycle, T (s)
T2 4π2 k
0.05 2.992 0.2992 0.0895 39.4784 22.055
0.10 3.884 0.3884 0.1509 39.4784 26.162
0.15 4.997 0.4997 0.2497 39.4784 23.716
0.20 5.433 0.5433 0.2952 39.4784 26.747
0.25 6.270 0.6270 0.3931 39.4784 25.107
0.30 7.202 0.7202 0.5187 39.4784 22.833
0.35 7.912 0.7912 0.6260 39.4784 22.073
0.40 8.337 0.8337 0.6951 39.4784 22.718
0.45 8.500 0.8500 0.7225 39.4784 24.589
0.50 9.656 0.9656 0.9224 39.4784 21.400
Obtaining mean values of T, for masses 100. 150, 200, 250, 300, 350, 400, 450 and 500g:
Mean = 1/9(0.3884 + 0.4997 + 0.5433 + 0.6270 + 0.7202 + 0.7912 + 0.8337 + 0.8500 + 0.9656)
= 0.6910

Finding k with a Mass and a Spring 5
A graph of time period, T2 in seconds against mass in kilograms.
Calculating the gradient:
Gradient = ∆y/∆x
= 0.35 – 0.10
0.77 – 0.37
= 0.6250

Finding k with a Mass and a Spring 6
After the Experiment
a) Finding the values of k from m and T2:
To find k from m and T2 we use the formula:
Hence, k = (4π2 * m)/T2
Thus from m = 0.05, and T2 = 0.0895
k = (4π2 * 0.05)/0.0895
= 22.055
From m = 0.10, and T2 = 0.1509
k = (4π2 * 0.10)/0.1509
= 26.162
From m = 0.15, and T2 = 0.2497
k = (4π2 * 0.15)/0.2497
= 23.716
From m = 0.20, and T2 = 0.2952
k = (4π2 * 0.20)/0.2952
= 26.747
From m = 0.25, and T2 = 0.3931
k = (4π2 * 0.25)/0.3931
= 25.107
From m = 0.30, and T2 = 0.5187
k = (4π2 * 0.20)/0.5187
= 22.833
From m = 0.35, and T2 = 0.6260
k = (4π2 * 0.35)/0.6260
= 22.073
From m = 0.40, and T2 = 0.6951
k = (4π2 * 0.40)/0.6951
= 22.718

Finding k with a Mass and a Spring 7
From m = 0.45, and T2 = 0.7225
k = (4π2 * 0.45)/0.7225
= 24.589
From m = 0.50, and T2 = 0.9224
k = (4π2 * 0.50)/0.9224
= 21.400
b) Calculating the mean value of k:
The mean of k = 1/10(22.055 + 26.162 + 23.716 + 26.767 + 25.107 + 22.833 + 22.073 + 22.718
+ 24.589 + 21.400)
= 23.442
c) Difference between mean value from table and the accepted value:
From table = 23.445, Accepted value = 25.000
Difference = 25.000 – 23.445
= 1.555
d) Percentage difference:
= 1.555 * 100
25
= 6.22%
e) Calculating k from the gradient
Equation of the line of best fit is given by:
k = 4π2 * m/T2
= 4π2 * Gradient
= 4π2 * 0.625
= 24.674
f) Difference between the value of k calculated from the gradient and the accepted value
Difference = 25.000 - 24.674
= 0.326
g) Percentage difference

Finding k with a Mass and a Spring 8
= 0.326 * 100
25
= 1.304%
h) The most accurate value of k:
The most accurate value of k is that calculated from the gradient since it involves the
consideration of all the values during the process of plotting the graph and choosing those falling
within the required limit. It is for this reason that it has the smallest percentage of error which
illustrates that it is more accurate than the value of k found by averaging.
i) Dependent and independent variables from experiment part 2:
The independent variable is the mass of the 10 Masses. The masses are set in motion while the
time is recorded to determine the time taken for them to complete 10 oscillations. By adding
more masses, it is possible to increase efficiency by trying the experiment severally to reduce
errors.
The dependent variable is T2. The time measured depends on the cycles made by the masses set
in motion. Therefore, the results of T2 depends on the oscillations completed by different masses
displaced in a harmonic motion.
j) Reliability of the results:
Reliability is determined by the variation between the values collected during experimentation.
The results obtained from the graph is reliable since the percentage error of finding k from the
graph is smaller than that calculated by averaging the values. Therefore, there is small variation
between the accepted value and the that found after experimentation. Additionally, the drawing
of the best line of fit validates the inclusion of values that are more accurate and ignoring the
outliers to promote reliability.

Finding k with a Mass and a Spring 9
k) Increasing reliability of results:
Repeated experimentation can help to facilitate reliability. The more the experiment is done, the
closer the values get closer to the accurate measurement (Kimberlin and Winterstein 2008).
Thus, repetition of the process ensures reliability. Furthermore, fixing control variables promotes
reliability (Fragassa, Pavlovic and Massimo 2014). For instance, in the experiment above, the
control has been fixed at 25. Hence, finding a measurement closer to it shows reliability of the
calculated value.
l) Increasing precision in the results:
Scientific experiments must yield precise measurements. Precision refers to the degree of
closeness of different measurements from an experimentation (Raudenbush, Martinez and
Spybrook 2007). For the experiment above, it is possible to increase precision by using new
stopwatches whose conditions are good for easy commencement of measuring and stopping
without lag.
m) Percentage uncertainty in the largest measurement:
The largest time measurement = 9.718 for 10 cycles, which is equivalent to, T = 0.9718s
Percentage uncertainty = (|the measurement - average measurement|) * 100
Average measurement
= (|0.9718 – 0.9656|) * 100
0.9656
= 10.423%
Corresponding value of T2 = 0.9444
Percentage measurement = (|the measurement - average measurement|) * 100
Average measurement
= (|0.9444 – 0.9224|) * 100
0.9224

Finding k with a Mass and a Spring 10
= 2.385%
Conclusion
As seen from the graph and calculations, the value of constant from the experiment is
24.674. Henceforth, the experiment confirms the hypothesis that k = 25 ± 1. It is also worth
noting that the value obtained from plotting a graph of line of best fit yields a better result than
finding averages of values obtained from the experimentation. This is because some outcomes of
the experiments have flaws which cause errors on inclusion for calculations. On the other hand,
on plotting the graph, the line of best fit cuts out these values and only includes those suitable for
the experiment. Therefore, the value of k obtained from the gradient of the graph is closest to the
hypothesized value.
Evaluations and Methods of Improvement
For the sake of improving subsequent experiments, it is important to consider reliability
and precision of results. Therefore, the researcher should do the experiment repeatedly to get
values close to the hypothesized value. Essentially, eliminating values that are too far from the
required measurement should be eliminated. Finally, the precision of results can be improved by
using reliable instruments and using many decimal places to minimize errors in averaging.

Finding k with a Mass and a Spring 11
References
Cocco, A & Masin, SC 2010, “The Law of Elasticity,’ Psicologica, vol. 31, pp. 647-657,
Available at: https://www.uv.es/revispi/articulos3FM.10/13Cocco.pdf [Accessed 26
January 2018]
Fragassa, C, Pavlovic, A, & Massimo, S 2014, “Using a Total Quality Strategy in a New
Practical Approach for Improving the Product Reliability in Automotive Industry,”
International Journal of Quality Research, vol.8, no. 3, pp. 297-310, Available at:
http://www.ijqr.net/journal/v8-n3/1.pdf [Accessed 26 January 2018]
Kimberlin, CL & Winterstein, AG 2008, “Validity and Reliability of Measurement Instruments
Used in Research,” Research Fundamentals, vol. 65, pp. 2276-2284, Available at:
http://www.ajhepworth.yolasite.com/resources/9817-Reliability%20and%20validity.pdf
[Accessed 26 January 2018]
Michels, LB, Gruber, V, Schaffer, L, Marcelino, R, Casagrande, LC, & Guerra, SR 2013,
“Didactic Press for Remote Experimentation Applied in Spring to Study Hooke’s Law,”
iJOE, vol. 9, no. 8, pp. 13-15, Available at:
http:online-journal.org/index.php/i-joe/article/viewFile/3325/2905 [Accessed 26 January
2018]
Raudenbush, SW, Martinez, A, & Spybrook, J 2008, “Strategies for Improving Precision in
Group-Randomized Experiments,” Educational Evaluation and Policy Analysis, vol. 29,
no. 1, pp. 5-29, Available at:
http://home.uchicago.edu/sraudenb/files/StratImprovingPrecisionEEPA07.pdf [Accessed
26 January 2018]

Finding k with a Mass and a Spring 12
Rodriguez, EE & Gesnouin, GA 2007, “Effective Mass of an Oscillating Spring,” The Physics
Teacher, vol. 45, pp. 100-103, Available at:
http://users.df.uba.ar/cobelli/LaboratoriesBasicos/EffectiveMassOfSpring.pdf [Accessed
26 January 2018]