THE UNIVERSITY OF ZAMBIA
SCHOOL OF ENGINEERING
DEPARTMENT OF MECHANICAL ENGINEERING
VIBRATIONS AND CONTROL ENGINEERING I (MEC 4702)
LABORATORY REPORT
AUTHOR:
MWAPE CABINET
ID #:
2020003082
LAB #:
01
LAB TITLE:
VERIFICATION OF THE RELATION OF THE
SIMPLE PENDULUM
DATE OF THE LAB: 02ND APRIL, 2024.
DUE DATE:
09TH APRIL, 2024.
FOR THE ATTENTION OF: DR. CHILESHE.
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Table of Contents
1. TITLE
2. OBJECTIVE
3. INTRODUCTION
4. THEORY
5. EQUIPMENT
6. PROCEDURE
7. DATA COLLECTION
8. DATA ANALYSIS
9. DISCUSSION
10. CONCLUSION
11. DECLARATION
12. REFERENCE
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1. TITLE: TO VERIFY THE RELATION OF A SIMPLE PENDULUM
Variation of the period of the simple pendulum in response to the variations
(changes made to the length of the string
2. RATIONALE
By the end of this experiment, understanding of the theory stating that the
period of oscillation of the simple pendulum is only affected by the
acceleration due to gravity and the length of the string to which the bob is
attached as opposed to been affected by the size nor mass of the bob. This will
further lead to the verification of the relationship between the acceleration due
to gravity, length and period, hence frequency in cycles per second (Hertz) of
the simple pendulum.
3. INTRODUCTION
It is obvious that when an object of mass m hanging on a string of length l is
given an angular displacement of size Ө, and the let free, the body starts
swinging about the vertical axis and momentarily comes to rest of there are no
extra external forces acting on the mass. This phenomenon can be best termed
as a simple pendulum demonstration. The swinging of the mass to and from
either side of the vertical axis is referred to as oscillation. The time it takes for
the mass to move from the position defined by the angle Ө to the other side and
back to the same initial position is called a period of oscillation and the number
of oscillations made per unit of time, usually in seconds is known as the
frequency of swing and is measured in cycles per second, the unit called the
Hertz named in honour of the 19th-Century Germany physicist Heinrich Ruldof
Hertz. In this context, the term body of mass is used to refer to a round piece of
a steel ball hung on the thread vertically placed that is capable of swinging as
aforementioned.
Mainly, there are two factors that determine the magnitude of period and hence
frequency and these include;
• Magnitude of the acceleration due to gravity.
• Length of the thread om which the mass is hung.
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Having discussed the above idea, it then becomes an important aspect to get an
understanding of these quantities as they are of special property applications in
many engineering systems such as the mechanical clock.
4. THEORY:
With the above outline of the simple pendulum, the interest is then shifted to
wanting to know how actually do these [above] factors affect the system. To do
this, certain assumptions are made which include the uniformity of the
acceleration due to gravity, such that it remains constant because the equation
used in this verification is only valid for the parameters of constant measure per
test.
There is only one equation comprising of quantities of, the period, length and
the acceleration due to gravity that can be used to calculate the magnitude of the
period of the simple pendulum, given the length of the thread and the value of
the gravity in the particular area where the examination is conducted from.
Such that if the equation is as follows,
𝑇𝑇 = 2𝜋𝜋�
Where; T signifies the period in seconds
𝐿𝐿
𝑔𝑔
L stands for the length of the thread in centimetres
.g for the acceleration due to gravity in m/s2
Note; the above equation (a) only applies when quantities used are of fixed
magnitudes per test.
5. EQUIPMENT:
During this experiment, equipments used were the following
• A nylon thread of two metres (2m) long
• 2 steel balls of masses and respectively
• A steel one-meter (1m) ruler
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The figure 1.0 above shows the demonstration of the experimental setup during
the workshop test session.
6. PROCEDURE
As mentioned in the theory part of this report, the verification of the relation is
expected to be done on bodies of different masses, hence, this experiment was
done in two parts with a body of different mass per test. However, before all, the
average mass of the respective balls was taken from the digital scale and were
recorded.
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EXPERIMENT. A- testing with the ball of mass 113.5 g
One end of the nylon thread was tied to the ball and its (thread’s) length was
measured from the centre of mass of the ball to mark out the length of
105centimetres from which the string was hung on the support. After hanging,
the ball was left to be in vertical equilibrium position before giving it a
displacement as stated in the introductory part of this report. After this, the ball
was displaced from its equilibrium position by an angle approximately
equivalent to π/6 radians meanwhile setting the stopwatch in such a way that it
was started immediately the ball was released to oscillate. The stopwatch was
read carefully and noted down the time it took the ball to make 10 oscillations.
The just above procedure was repeated but with the thread length of 100
centimetres measured from the centre of mass of the ball. The result for the time
taken by the ball to make 10 oscillations as in the first experiment was read and
recorded.
The same was for the third time repeated but with the shortest thread length of
all tests measuring 50centimetres from the centre of mass of the ball.
The process of displacing the ball and reading the time it took the ball to make
10 oscillations was repeated 3 times for each of the three tests based on the
thread length and calculated for the average time in an effort to reduce errors.
EXPERIMENT. B- testing with the ball of mass 83.2 g
Exactly the same as in the preceding outline of experiment A was done except
the ball was of different mass.
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7. DATA COLLECTION
For experiment A
The table 1.0 below shows the data entry for the tests using the ball of mass
113.50 gm
Trial
Length L
number n of
time t for n
Average
Period T
Period T
number
of thread
oscillations
oscillations
time t (s)
(actual) (s)
(theoretical)
(cm)
n
(s)
1
2
150
10
24.75
24.39/10
24.39
2.46 s
3
24.06
=2.44
1
19.75
19.87/10
2
100
10
19.73
19.87
2.01 s
3
20.12
=1.99
1
13.83
13.94/10
2
3
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24.36
50
10
13.89
14.11
13.94
1.42 s
=1.39
For experiment B
The table 2.0 below shows the data entry for the tests using the ball of mass
83.20 gm
Trial
Length L
number n of
time t for n
Average
Period T
Period T
number
of thread
oscillations
oscillations
time t
(actual) (s)
(theoretical)
(cm)
n
(s)
(s)
1
2
23.63
150
10
23.87
23.81/10
23.81
2.46 s
3
23.93
=2.38
1
19.85
19.59/10
2
100
10
19.78
19.59
2.01 s
3
19.15
=1.96
1
14.41
14.35/10
2
50
3
10
14.31
14.33
14.35
1.42 s
=1.44
8. DATA ANALYSIS
the actual period of oscillations T were calculated simply by dividing the total
average time taken for a set of counted oscillation (t measured in seconds s) by
the number of counted oscillations n.
Stated as; T = t/n
From collected data, the calculated values were as shown in the tables
For Experiment A
using the formula stated in the theory part of this report and the calculated
average times as from the data collection table, periods of oscillation was
calculated as follows;
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for: l = 150cm = 1.5m
g = 9.81m/s2
𝐿𝐿
𝑇𝑇 = 2𝜋𝜋�𝑔𝑔
T = 2π√(1.50/9.81) = 2.456919879 = 2.46s
for: l = 100cm = 1m
g = 9.81m/s2
𝐿𝐿
𝑇𝑇 = 2𝜋𝜋�𝑔𝑔
T = 2π√(1.00/9.81) =2.006066681 =2.01s
for: l = 50cm = 0.5m
g = 9.81m/s2
𝐿𝐿
𝑇𝑇 = 2𝜋𝜋�𝑔𝑔
T = 2π√(0.5/9.81) =1.418503353 =1.42s
For Experiment B
By employing the same approach stated in the theory part of this report and the
calculated average times as from the data collection table, periods of oscillation
were calculated and found to be just the same as those in found in experiment
A above. This is so simply because the parameters for the length and that of
the acceleration due to gravity remained unchanged as in experiment A
9. DISCUSSION:
From the results shown above, it can vividly be seen that when a mass is hung
on a string and given an angular displacement then let free, the mass swings
about its vertical equilibrium axis as stated in the introduction part if this report.
The period of oscillations made lie directly proportional to the length of the
thread on which it (mass) is attached such that the increment in the thread length
results in the increased period of swing measure. From the experiments
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performed, it can be stated clearly that the mass of the ball does not affect the
period of oscillation because its (period) calculations from experiment B
performed using a ball of different mass still shows the same values of the period
at different lengths of the thread corresponding to those (lengths) in experiment
A.
In comparison of the actual to theoretical values for the period of oscillations
found in this experiment, there exists infinitesimal deviations of the theoretical
values from actual ones, citing the human error which potentially resulted from
delays in either releasing the ball or starting the stopwatch. However, this error
can be cleared off if the same test is repeated more than the three times
performed in this experiment and get the average which reveals no noticeable
difference from the values read per trial.
In addition, the lab was so essential in that it imparted knowledge to the
experimenters which is part of the things not known previously and also to
broaden the understanding of how important this principle of simple pendulum
is and how essentially it can be applied in many mechanical systems to alleviate
the societal challenges.
Furthermore, the errors encountered during this workshop session could also
have been propelled by inadequate preparation for the work performed as the
instruction manual was received only few minutes before the actual session
could start, leaving students with literally no time to read, comprehend and
hence, appreciate the outlined procedures.
10. CONCLUSION:
The data analysed in this report which was collected in the experiment
conducted compared to the actual data, it suffices to conclude that was the
workshop experiment was a success as verification of the relations of the simple
pendulum was achieved as per rationale in spite of encountering minimal
deviations between the actual results and the theoretical ones. Eradicating these
errors may be done by taking the tests repeatedly, a suggestion already submitted
in the discussion section above.
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11. DECLARATION
I MWAPE CABINENT of Computer Number 2020003082 do declare
that I conducted this laboratory experiment on Tuesday the 2nd day of
April in 2024 and that this report is accurate to the best of my
knowledge and is a true representation of my laboratory results.
Signature:
.
12. REFERENCES
•
computer controlled torsional vibration unit (ATE-344) instruction manual
•
https://labs.phys.utk.edu/mbreining/phys221 core/m11 /The pendulum.
•
http://www2.tntech.edu
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