ABSTRACT
Referring from the resultant graph, we can
see that the obtained graph line is straight. This shows that both variables on
the graph are linearly related with each other. From the experiment, which to
determine the spring constant (k), the experimental value and theoretical value
is almost same. It been shown that the experimental value is 2.091 N/mm
theoretical is 1.71 N/mm with percentage error of 18.21%. The data collected
has told us that the value of extension will be rise if the amount of the load
is increase. In findings the frequency, we have got the answer with some
difference between the theoretical and experimental value. The relation between
frequencies with mass can be seen because weight of the mass can change the
value of frequency. The large amount of load given to the spring will deduct
the value of frequencies in vibration. During data collection, we encountered
some errors. This is maybe due to random error. As a conclusion, we managed to
obtain the spring constant, (k) value for the spring tested. We also managed to
find the spring oscillation’s natural frequency, (f). By obtaining the graphs,
we also succeeded in finding the relationship between the displacement, (x) and
the generated force of spring, (F). Through the graph, we also managed to
figure out the relations between the mass load of the spring, m and the
oscillation periodic time, (T). By reaching the experiment’s entire objective,
the experiment is a success.
OBJECTIVE
From this experiment, we able to find:
1. Determine the spring constant (k).
2. Determine the natural frequency (f).
INTRODUCTION
Vibration refers to mechanical oscillations
about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road. Vibration is
occasionally "desirable". For example the motion of a tuning fork, the
reed in a woodwind instrument or harmonica, or the cone of a loudspeaker is desirable vibration, necessary for the correct functioning of the
various devices. Free vibration occurs when a mechanical system is
set off with an initial input and then allowed to vibrate freely, and the
action of forces inherent in the system itself. The mechanical system will then
vibrate at one or more of its "natural
frequency"
and damp down to zero, which are properties of the dynamic system established
by its mass and stiffness distribution.
Procedure
Part 1 :
Determination the spring constant (k)
It is appropriate to plot the extension of the spring
with the recorder.
1.
The paper
was fit and the pencil is setup.
2.
Measure the
initial length of the spring.
3.
The adjuster
is used to set carriage such that stylus is 40mm line on chart paper
4.
Load spring
by placing weights on carriage
5.
The recorder
is start again after each weight is added
6.
The
extension values is measured manually.
Part 2 :
Determination the natural frequency (f)
1.
The graph
paper and pen is put in the mechanical recorder
2.
The adjuster
is adjust so that the gap between the carriage is constant and the pen is at
the centre of the graph paper.
3.
The carriage
is pull downward to give the initial displacement and let it oscillate freely
4.
Using the
mechanical recorder , record the mechanical oscillation of the system knowing
that the recorder velocity is 20mm/s .
5.
The
procedure is repeated by adding more masses.
Results
Determination
the spring constant (k)
Mass
(kg)
|
Total
mass (kg)
|
Force,F
(N)
|
Extension,x
(mm)
|
Spring
constant,k (Nmm)
|
0
|
1.25
|
12.2625
|
0
|
0
|
2
|
3.25
|
31.8825
|
10
|
3.1883
|
4
|
5.25
|
51.5025
|
20
|
2.5751
|
6
|
7.25
|
71.1225
|
30
|
2.3071
|
8
|
9.25
|
90.7425
|
40
|
2.2686
|
10
|
11.25
|
110.3625
|
50
|
2.2073
|
Determination
the natural frequency (f)
Mass (kg)
|
Total mass,m (kg)
|
Length in 3 period (mm)
|
Period,T (s)
|
Natural frequency,f (Hz)
|
0
|
1.25
|
10
|
3.33
|
0.300
|
2
|
3.25
|
18
|
6.00
|
0.167
|
4
|
5.25
|
22
|
7.33
|
0.136
|
6
|
7.25
|
25
|
8.33
|
0.120
|
8
|
9.25
|
29
|
9.67
|
0.103
|
10
|
11.25
|
32
|
10.67
|
0.094
|
4.1 RESULTS
1. Determining the
spring constant (k)
Mass, m (kg)
|
Spring Elongation, x (mm)
|
Force, F (N)
|
1.25
|
0
|
12.263
|
3.25
|
11.5
|
31.883
|
5.25
|
23
|
51.503
|
7.25
|
34.5
|
71.123
|
9.25
|
46
|
90.743
|
11.25
|
57.5
|
110.363
|
Table 4.1.1: Tabulated data to determine spring constant, k
k, N/mm
|
|
Experimental
|
Theoretical
|
Percentage error (%)
|
1.706
|
1.71
|
0.234
|
Table 4.1.2: Percentage error between experimental and
theoretical value
Sample of Calculation:
F=kx
F=ma
Therefore,
ma=kx
k= (m2-m1)a
(x2-x1)
k= (11.25-5.25)(9.81)
(57.5-23)
= 1.706N/mm
Percentage
error = 1.71-1.706 x 100%
1.71
= 0.234%
1.
Determine the natural frequency (f)
Mass, m (kg)
|
5 oscillations, a (mm)
|
Time for 5 oscillations,T5 (s)
|
Time for 1 oscillation,T1 (s)
|
Natural frequency, f (Hz)
|
1.25
|
27
|
1.35
|
0.27
|
3.704
|
3.25
|
35
|
1.75
|
0.35
|
2.857
|
5.25
|
37
|
1.85
|
0.37
|
2.703
|
7.25
|
41
|
2.05
|
0.41
|
2.439
|
9.25
|
47.5
|
2.38
|
0.48
|
2.105
|
11.25
|
53
|
2.65
|
0.53
|
1.887
|
Table 4.1.3: Tabulated data to determine the natural frequency,
f
|
Natural frequency,f (Hz)
|
|
Mass, m (kg)
|
Experimental
|
Theoretical
|
Percentage error (%)
|
1.25
|
3.704
|
5.887
|
37.08
|
3.25
|
2.857
|
3.651
|
21.74
|
5.25
|
2.703
|
2.872
|
5.91
|
7.25
|
2.439
|
2.444
|
0.21
|
9.25
|
2.105
|
2.164
|
2.71
|
11.25
|
1.887
|
1.962
|
3.84
|
Table 4.1.4: Percentage error between experimental and
theoretical value
Sample of Calculation:
T5 =
a
v
= 47.5mm
20mm/s
= 2.38s
T1 = T5
5
= 2.38
5
= 0.48
fexp = 1
T1
= 2.105 Hz
f
theo =
=
= 2.164 Hz
Percentage error = 2.164-2.105 X 100%
2.164
= 2.71%
5.1
DISCUSSION
Based on the experiment conducted, the
value of spring constant, k and natural frequency, f is determined. Hooke's Law states that the
restoring force of a spring is directly proportional to a small displacement.
In equation form, F = -kx where x is the displacement of the
spring. As the displacement is acting downwards, it is considered that it
acting in negative direction. Therefore, F = -k (-x) is also F= kx. The
proportionality constant k is specific for each spring. For this
experiment, the theoretical value of k of the spring is 1.71Nmm-1.
Even though initially there is no additional mass attached to the spring, the
mass of the carriage is taken into account, which is 1.25kg. Therefore for each
additional mass, their values are added with 1.25kg and the initial mass
considered as 1.25kg instead of 0kg. From
Figure 4.1.1, the relation between force, F and spring elongation, x is
proportional. As the force acting on the spring increases, the elongation of
the spring also increase. The value of experimental spring constant, k is
obtained from the gradient of the graph. It is observed that the value of
experimental and theoretical spring constant, k is slightly different that
carried 0.234% of error. The different between the values is very small. The
error can therefore be assumed to be ignored. Hence, the values of experimental
and theoretical spring constant, k are approximately equal.
To determine the natural frequency, each mass on
the spring is allowed to vibrate to obtain a sinusoidal graph. The length of
five oscillations is recorded to obtain the time of five oscillations by
dividing the length with the velocity of the mechanical recorder. Then, the
time for one oscillation is obtained by dividing the time for five oscillations
with five. The experimental natural frequency, f is obtained as the reciprocal
of time for one oscillation. Each value of natural frequency, f is different
for each mass attached to the spring. The theoretical natural frequency, f of
the spring is calculated using the formula given. It is a function of spring
constant, k and mass, m. For the mass 1.25kg and 3.25kg, the percentage errors
between the experimental and theoretical values of natural frequency, f of the
spring are differed immensely. The percentage errors are more than 20% and
should be considered and analysed. This may caused by several factors and
errors. Disturbances during the experiment may also contribute to the vastly
percentage errors. For the other additional mass, the percentage errors between
the experimental and theoretical values of natural frequency, f of the spring
are only minor and can be considered as insignificant difference.
6.1 CONCLUSIONS
In a nutshell, Hooke's Law states that the restoring force of a
spring is directly proportional to a small displacement. In equation form, F =
-kx where x is the displacement of the spring. As the displacement
is acting downwards, it is considered that it acting in negative direction.
Therefore, F = -k (-x) is also F= kx. The force, F is proportional to the
spring elongation, x and the spring constant is the slope of the graph of force
versus spring elongation. The value of spring constant, k
obtained from the experiment is approximately equal to the theoretical value of
the spring constant, k. However the values of experimental natural frequency, f
of the spring for mass of 1.25kg and 3.25kg are significantly differ from their
theoretical values. This
may caused by several factors and errors. Disturbances during the experiment
may also contribute to the vastly percentage errors. The natural frequency, f
depends on the spring constant, k and mass attached to the spring, m. The
percentage errors between the experimental and theoretical values for natural
frequency, f of other additional mass are very small and considered to be
insignificant.
REFERENCES
APA FORMATTING AND STYLE GUIDE (2011, August
17). (6th ed.) University of Malaya Library (UML).
All about
Hooke’s Law (n.d.). Retrieved December 9, 2011 from: http://asms.k12.ar.us/classes/physics/GENERAL/KENNETH/HOOKE.HTM
Simple
Harmonic Motion (n.d.). In UCLA.
Retrived December 9, 2011 from: http://www.physics.ucla.edu/demoweb/demomanual/harmonic_motion_and_waves/simple_harmonic_motion/simple_harmonic_motion.html
A.R Zamri (n.d.). Vector Dynamics and Vibration. University of Technology MARA, UiTM.
Hibbeler,
R.C. (2007). Engineering Mechanics
Dynamics 11th Edition in SI Units: Vibration (pg 620 to 622).
Pearson Education, Inc. In Jurong, Singapore.
