Hooke's Law Experiment: Investigating Spring Constant

This document details a physics experiment designed to investigate Hooke's Law, focusing on the relationship between the mass attached to a spring and the time it takes for the spring to complete oscillations. The experiment involves setting up a spring with a mass holder and slotted masses, measuring the time for 20 oscillations for different masses, and recording the data. The results are presented in tables and graphs, showing the relationship between mass and time period. Data analysis involves processing the data, squaring the periodic time, and plotting the results to find the spring constant. The document includes a discussion of the results, comparing experimental values with theoretical values, and provides conclusions and recommendations for improving the experiment's accuracy. The experiment uses concepts of simple harmonic motion and provides a practical application of Hooke's Law.

HOOK’S LAW EXPERIEMENT 1
Hooke’s law Experiment.
Student name
Professor name
University name
State name
Date.

HOOK’S LAW EXPERIEMENT 2
Objectives of the experiment
The objective of the experiment is to establish the relationship between mass at the end of
a swinging spring and the total amount of time that is taken for 20 complete oscillations.
List of materials
 Spring
 Retort stand
 Mass holder
 Clamp and stand
 Meter ruler
 Slotted mass
 Digital stopwatch
Theory of the experiment
The elasticity of a spring can be tested through accurate measurement of loads which are
added to the spring. The masses are changed and the varying readings recorded for each
cases (Dell'Isola et al 2010). When a given applied force stretches the spring, the production
of a restoring force is achieved. The restoring force therefore causes a simple harmonic
motion and a linear relationship is established between the restoring force and displacement
of the load from the equilibrium position. The relationship is generally represented as F=-kx
(Dell'Isola et al 2010). The applied force is usually in an opposite direction to the
displacement direction. The force is expressed in newton (French 2017). The force is
responsible for the extension that is caused on the spring utilize during the experiment.
Experimental set-up diagram.

HOOK’S LAW EXPERIEMENT 3
Methodology
1. An experimental diagram was set up as shown in in the diagram above. One spring end
was attached to the support stand
2. A slotted mass approximately 0.15kg was attached to the other end of the spring.
3. A ruler was the used to measure the spring extension after the mass was added.
4. The slotted mass holder was then stretched by 5cm and then released.
5. The spring was then left to oscillate for exactly 20 complete oscillations. The total time
taken for these 20 complete oscillations were then accurately recorded using a stopwatch
and tabulated.
6. The process was then repeated for 5 times.
7. The results for the five investigated were then recorded and tabulated. The average
reading was then recorded for the obtained values.
8. Steps 2 and 7 were then repeated for different mass slots of 0.25kg, 0.35kg, 0.45kg, and
0.55kg respectively.

HOOK’S LAW EXPERIEMENT 4
Results
Slotted (M) / kg
(±0.01kg)
Time for exactly
20 oscillations
(±0.01s)
Time taken for exactly
20 oscillations/(+-
0.01s)
Time taken for
one oscillation
(T) / s (±0.01s)
First try Second try Third try
0.15 5.7 5.6 5.8 5.7
0.25 7.3 7.5 7.4 7.4
0.35 9.4 9.5 9.6 9.5
0.45 11.4 11.5 11.2 11.37
0.55 12.1 12.0 12.3 12.13
A graphical plot of the data.
A graph can be plotted to show the relationship between the mass and the average time
taken for the 20 complete oscillations (French 2017). Different slot masses were used in this
investigations.
A graph of mass against periodic time.

HOOK’S LAW EXPERIEMENT 5
Time period T/s Mass m/Kg
5.7 0.15
7.4 0.25
9.5 0.35
11.37 0.45
12.13 0.55
Mass in kg
5 6 7 8 9 10 11 12 13
0
0.1
0.2
0.3
0.4
0.5
0.6
Chart Title
Column C Column D
Data analysis
A nonlinear relationship is established between the mass that is slotted and the total time
taken for one complete oscillations (Gati et al 2016). The is then processed in order to find a
relationship between these two variables (Dell'Isola et al 2010). The graph is parabolic. Hence
the T can be manipulated to become T2 so that the graph can be plotted with periodic values of
time.
Data processing
For effective data processing, the periodic time has been squared.
Slotted (M) / kg
(±0.01kg)
Time for one complete oscillation (T) / s
(±0.01s)
Time for one complete oscillation squared (T2)
s (±0.02s)
0.15 0.23 ± 4.5% 0.05 ± 0.36 x 10-2

HOOK’S LAW EXPERIEMENT 6
0.25 0.32 ± 3.2% 0.10 ± 0.64 x 10-2
0.35 0.44 ± 2.3% 0.19 ± 0.83 x 10-2
0.45 0.51 ± 2.0% 0.26 ± 1.04 x 10-2
0.55 0.56 ± 1.8% 0.31 ± 1.15 x 10-2
Analysis and presentation of the processed data
A graphical representation of the relationship between slotted mass and the square of the
periodic time denoted as shown in the graph.
Analysis of the graph and discussions
Gradient of best fit line = 0.5/0.32 = 1.56 kgs-2
The second plotted graph directly shows that slotted mass is directly proportional to the
square of the periodic time obtained from the experiment.
Expressed mathematically as m α T2
The general relating formulae for the relationship between the mass and the squared time
was obtained as shown below (Lee et al 2013).

HOOK’S LAW EXPERIEMENT 7
T = 2π/k =
Where k is the constant to be investigated in the experiment.
Therefore,
T = 2π
T2 =
Comparing the above equation to the general linear relation = mx + c
y =m,
m = 4π2/k
x = T2.
The line gradient is therefore equals 4π2/k
The above equation can then be use to find out the spring constant (Kuhn and Vogt 2012).
Using the line of best fit.
1.57 = 1/
= 1/1.57
k = 63.69 Nm-1

HOOK’S LAW EXPERIEMENT 8
The experimental value of the spring constant was therefore found as 63.69 Nm-1
The supporting equation is thus generally represented as. T = 2π (Kuhn and Vogt 2012).
T2 =
Therefore, T2 α m
The extension was 1.6cm.
The spring constant as determined from Hooke’s Law (Gati et al 2016).
F = kx, k = 1/0.016
= 62.5 Nm-1
The deviation is thus 63.69-62.5= 1.19
Percentage deviation is thus (1.19/63.69) x100%=1.87%
Conclusions and recommendations
The experiment was used to investigate Hook’s law through the evaluation of the
relationship between slotted mass and the time taken for each complete oscillations (Lee et al
2013). Periodic times for oscillations. When the squared values of periodic time are plotted
against the slotted mass, a linear relationship is now obtained between these experimental values
and data.
The following modifications can be effected to ensure more precise experimental results.
1. During the recording of the experimental results, another ruler is used in pointing on to
the fast ruler. This will ensure precise and more accurate readings hence avoiding
parallax error.

HOOK’S LAW EXPERIEMENT 9
2. Ultra-sonic motion detector that is positioned below oscillating spring to ensure more
accurate time readings during the data recording during the experiment. The use of
ultrasonic motion detectors will help in minimizing the extent of human interactions thus
minimizing any form of human error during the experiment.
3. Recommendations is also made on the the number of reading taken during the
experiment. More experimental readings are likely to reduce the extent of errors on the
experimental data.
References

HOOK’S LAW EXPERIEMENT 10
Lee, C.K., Tan, S.C., Wu, F.F., Hui, S.Y.R. and Chaudhuri, B., 2013, September. Use of Hooke's
law for stabilizing future smart grid—The electric spring concept. In 2013 IEEE Energy
Conversion Congress and Exposition (pp. 5253-5257). IEEE.
Kuhn, J. and Vogt, P., 2012. Analyzing spring pendulum phenomena with a smart-phone
acceleration sensor. The Physics Teacher, 50(8), pp.504-505.
French, A.P., 2017. Vibrations and waves. CRC press.
Dell'Isola, F., Sciarra, G. and Vidoli, S., 2010. Generalized Hooke's law for isotropic second
gradient materials. Proceedings of the Royal Society A: Mathematical, Physical and
Engineering Sciences, 465(2107), pp.2177-2196.
Gati, E., Garst, M., Manna, R.S., Tutsch, U., Wolf, B., Bartosch, L., Schubert, H., Sasaki, T.,
Schlueter, J.A. and Lang, M., 2016. Breakdown of Hooke’s law of elasticity at the Mott
critical endpoint in an organic conductor. Science advances, 2(12), p.e1601646.