saif aldin ali madi سيف الدين علي ماضي s96aif@gmail.com In this laboratory practice the simulation of a simple pendulum was carried out with the objective of determining the acceleration of gravity and its uncertainty, through the data obtained in the simulation. In this one was made the assembly of a simple pendulum through a rope, a weight, a grader and the base for pendulums, which allowed us to obtain through the following instruments rule and timer, data as the time in which 20 oscillations are completed and the length of the pendulum, taking into account the uncertainties of each instrument, these data were organized into tables and then used in the realization of graphs expressing the time as a function of the length of the pendulum, in addition to calculating the acceleration of gravity and its uncertainty.

1. 1
[Vibrations Laboratory]
University of Baghdad
Name: Saif al-Din Ali Madi
Experiment Name: Simple Pendulum and Compound
Pendulum

2. 2
TABLE OF CONTENTS
ABSTRACT.........................................................................I
OBJECTIVE........................................................................II
INTRODUCTION..............................................................V
THEORY..........................................................................VI
APPARATUS&Procedure...............................................VII
Calculations ……………….................................................VIII
DISCUSSION ...............................................................VIIII

3. SAIF AL-DIN ALI MADI
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
- 3 -
Simple Pendulum and Compound Pendulum
1. ABSTRACT
In this laboratory practice the simulation of a simple pendulum was
carried out with the objective of determining the acceleration of gravity
and its uncertainty, through the data obtained in the simulation. In this
one was made the assembly of a simple pendulum through a rope, a
weight, a grader and the base for pendulums, which allowed us to
obtain through the following instruments rule and timer, data as the time
in which 20 oscillations are completed and the length of the pendulum,
taking into account the uncertainties of each instrument, these data
were organized into tables and then used in the realization of graphs
expressing the time as a function of the length of the pendulum, in
addition to calculating the acceleration of gravity and its uncertainty.
2. OBJECTIVE
The purposes of this experiment are:
1) Study the motion of a simple pendulum,
2) Study simple harmonic motion.
3) Learn the definitions of period, frequency, and amplitude
4) Learn the relationships between the period, frequency, amplitude
and length of a simple pendulum
5) Determine the acceleration due to gravity using the theory, results,
and analysis of this experiment.

4. SAIF AL-DIN ALI MADI
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
- 4 -
3. INTRODUCTION
Many things in nature wiggle in a periodic fashion. That is, they vibrate.
One such example is a simple pendulum. If we suspend a mass at the
end of a piece of string, we have a simple pendulum. Here, the to and
fro motion represents a periodic motion used in times The time that it
takes to make one complete oscillation is defined as the period T.
Another useful quantity used to describe periodic motion is the
frequency of oscillation. The frequency f of the oscillations is the
number of oscillations that occur per unit time and is the inverse of the
period, f= 1 / T. Similarly, the period is the inverse of the frequency.
T= 1 / t. The maximum distance that the mass is displaced from its
equilibrium position is defined as the amplitude of the oscillation
Another factor involved in the period of motion is, the acceleration due
gravity (g). Which on the earth is 9.8 m / s^2. It follows then that a long
pendulum has a greater period than a shorter pendulum.
The period of a simple pendulum can be found by;
𝜏 = 2𝜋√
𝑙
𝑔
Simple Pendulum:-
Is a ball suspended by a rope known length of the center is pulled to a
small angle and then leave is calculated 20 frequencies of a particular
lens and divide the number 20 on the time to calculate the pulse of one
move this oscillator frequency movement and as a derivation in the
theory of experience
Compound Pendulum:-
Is a rod of metal that calculates the center of the body and then takes a
certain measure of the center of the body to stabilize and the process of
recoiling and calculating the frequency and frequency of the process,
but after the process of rounding the place of fixation of the balance
center of the rod.

5. SAIF AL-DIN ALI MADI
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
- 5 -
4. THEORY
A-Simple pendulum
∑𝑭 = 𝒎𝒙̈
−𝐰 𝐬𝐢𝐧 𝛉 = 𝐦𝐱̈
𝒎𝒙̈ + 𝒘 𝐬𝐢𝐧 𝜽 = 𝟎
𝒘 = 𝒎𝒈
𝐬𝐢𝐧 𝜽 ≈ 𝜽
𝒎𝒙̈ + 𝒎𝒈𝜽 = 𝟎
𝒎𝜽̈ 𝒍 + 𝒎𝒈𝜽 = 𝟎
𝜽̈ +
𝒈
𝒍
𝜽 = 𝟎
𝜽 = 𝑨 𝐬𝐢𝐧 𝒘𝒕
𝜽̈ = −𝒘 𝟐
𝑨 𝐬𝐢𝐧 𝒘𝒕
−𝒘 𝟐
𝑨 𝐬𝐢𝐧 𝒘𝒕 +
𝒈
𝒍
𝑨 𝐬𝐢𝐧 𝒘𝒕 = 0
𝑨 𝐬𝐢𝐧 𝒘𝒕 [−𝒘 𝟐
+
𝒈
𝒍
] = 𝟎
𝒎𝒙̈ + 𝒌𝒙 = 𝟎
𝒙̈ +
𝒌
𝒎
𝒙 = 0
𝐰 𝐧
𝟐
=
𝐤
𝐦
𝐰 𝐧 = √
𝐤
𝐦
𝐰 𝟐
=
𝐠
𝐋
→ 𝐰 = √
𝐠
𝐋
𝐰 =
𝟐𝛑
𝛕
→ 𝜏 =
𝟐𝝅
𝒘
𝜏 𝟐
=
𝟒𝝅 𝟐
𝑳
𝒈

6. SAIF AL-DIN ALI MADI
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
- 6 -
B. Composite pendulum
∑𝐌 𝐨 = 𝑱𝜽̈
−𝐰 𝐬𝐢𝐧 𝛉 = 𝑱𝜽̈
𝑱𝜽̈ + 𝒉 𝐬𝐢𝐧 𝜽 = 𝟎
𝜽̈ +
𝒉𝒘
𝑱
𝐬𝐢𝐧𝜽 = 𝟎
𝐬𝐢𝐧 𝜽 ≈ 𝜽
𝜽̈ +
𝒉𝒘
𝑱
𝐬𝐢𝜽 == 𝟎
−𝒘 𝟐
𝑨 𝐬𝐢𝐧 𝒘𝒕 +
𝒉𝒘
𝑱
𝑨 𝐬𝐢𝐧 𝒘𝒕 = 𝟎
𝑨 𝐬𝐢𝐧 𝒘𝒕 [−𝒘 𝟐
+
𝒉𝒘
𝑱
] = 𝟎
𝐰 𝟐
=
𝒉𝒘
𝑱
𝐰 𝟐
=
𝒉𝒎𝒈
𝒎𝐤 𝟐 + 𝒎𝐡 𝟐
𝟒𝝅 𝟐
𝜏 𝟐 =
𝒉𝒈
𝐤 𝟐 + 𝐡 𝟐
𝐰 = 𝒎𝒈
𝑱 = 𝐈 𝐨+𝐦𝐡 𝟐
= 𝒎𝐤 𝟐
+ 𝒎𝐡 𝟐
𝐰 =
𝟐𝛑
𝛕
𝜏 𝟐
𝒉 =
𝟒𝝅 𝟐
𝒌 𝟐
𝒈
+
𝟒𝝅 𝟐
𝒉 𝟐
𝒈
𝟒𝝅 𝟐
𝒌 𝟐
𝒈
= 𝒄
Theory
𝒌 𝟐
=
𝒍 𝟐
𝟑

7. SAIF AL-DIN ALI MADI
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
- 7 -
5. APPARATUS
1. Sub frame (crossbeam) B1
2. Small wooden ball B2
3. Small steel ball B3
4. Inextensible flexible cord
5. Stopwatch or clock
6. Meter nule
Procedure
1. The simple pendulum is composed of a small spherical ball
suspended by a long, light string which is attached to support
stand by a string clamp, The string should be approximately
125 cm long and should be clamped by the string clamp
between the two flat pieces of metal so that the string always
pivots about the same point,
2. Use a meter to measure the length of the string from hanging
point to the center of the ball,
3. Displace the pendulum about 10 ° from its equilibrium
position and let it swing back and forth, Measure the total time
that it takes to make 20 complete oscillations and record that
time in your spreadsheet
4. Increase the length of the pendulum and repeat the
measurements made in the previous steps for multi string
lengths.
5. Record the measurement for steel and wood spherical
6. Calculate the period of the oscillations for each length by
dividing the total time by the number of oscillations, 20, Record
the values in the appropriate column of your data table
7. Examine your graph of 𝜏2 versus L and check to see if there
is a linear relationship between 𝜏2and L so that the data points
lie along a line
frame Metal rod
steel
ball
wooden
ball

8. SAIF AL-DIN ALI MADI
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
- 8 -
6. Calculations
STEEL
NO L(M) τ(𝑆𝐸𝐶) τ(20 𝑂𝑆𝐶) 𝜏2
1 0.56 29.42 1.471 2.16384
2 0.54 29.20 1.46 2.1316
3 0.5 28.32 1.416 2.005
4 0.45 27.3 1.365 1.8632
5 0.4 25.9 1.295 1.677
WOOD
1 0.56 29.7 1.485 2.205
2 0.54 27.7 1.385 1.918
3 0.5 26.86 1343 1.804
4 0.45 25.63 1282 1.644
5 0.4 24.27 1.214 1.474
𝟒𝝅 𝟐
𝒌 𝟐
𝒈
= 𝒄
𝟎. 𝟐𝟕 =
𝟒𝝅 𝟐
𝒌 𝟐
𝟗. 𝟖𝟏
→ 𝒌 = 𝟎. 𝟐𝟓𝟗
Theory
𝒌 𝟐
=
𝟎. 𝟗𝟏𝟓 𝟐
𝟏𝟐
→ 𝒌 = 𝟎. 𝟐𝟔𝟒𝟏
𝐸 = |
𝟎. 𝟐𝟓𝟗 − 𝟎. 𝟐𝟔𝟒𝟏
𝟎. 𝟐𝟓𝟗
| ∗ 100 = 1.969%

9. SAIF AL-DIN ALI MADI
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
- 9 -
7. DISCUSSION
1) What type of graph you got and what is the relationship
that related between 𝝉 𝟐
and L
The drawing is linear so the relationship is positive i.e. that when
the first increase is increased. This relationship was forged after
manipulation of the equation to get to this drawing
0
0.5
1
1.5
2
2.5
0 0.1 0.2 0.3 0.4 0.5 0.6
𝜏^2
L(M)
𝝉 =
𝟏.𝟗−𝟏.𝟓𝟗
𝟎.𝟒𝟖𝟕−𝟎.𝟒𝟏
= 4.0259
𝒈 =
𝟒𝝅 𝟐
4.0259 = 9.808

10. SAIF AL-DIN ALI MADI
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
- 10 -
2) What effect does the mass of the ball have on the period of
a simple pendulum? What would be the effect of replacing the
steel ball with a wooden ball?
The mass on a pendulum does not affect the swing because force and mass are
proportional and when the mass increases so does the force. As the force
increases so does the acceleration and along with gravity are the factors that
affect the pendulum swing.
Mass does not affect the period of the pendulum only the length of the string
and the angle of amplitude of the pendulum. Only force will be affected by the
mass and this supported by newton' s second law of motion F=ma
Mass does not affect the period of the pendulum. When looking at the formula
for a period of a pendulum, it is " T = 2 (pi) Sqrt (l/g) ". In the formula, "l" is
the length and "g" is the force of gravity. Therefore, the mass does not affect
the period of the pendulum
The amount of mass at the end of a simple pendulum has no effect on the
period of the swing. There would be no change at all, assuming the center of
mass of the wooden ball was the same distance from the pendulum pivot as
was the center of mass of the steel ball.
3) What are the error that affect the calculation of the earth
gravity, and how can prevent these error
In the simple pendulum errors are in the measurement of the tendon and determine the
center of the ball and the amount of angle from which the body starts because of the
hypothesis that the angle is equal to the sin calculation of the frequency of the pendulum
and the time calculation of frequency
In the compound pendulum determine the center of the body and calculate the distance
between the center and the distance imposed and the operation of the weight position in
the state of equilibrium at the height and also the starting angle must be small and calculate
the time to frequency

11. SAIF AL-DIN ALI MADI
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
- 11 -
4) Pendulum application
Musical Instruments
Sound itself is produced from oscillations of the air. In a string instrument such as a
violin or a guitar, bowing or plucking the string provides the force needed to make
the string oscillate and produce sound. In a
wind instrument like a trumpet, the
vibrations are caused by the player's lips
while the sound is caused by exciting the air
molecules by blowing across the opening in a
flute. In a percussion instrument like the
triangle, the vibrations occur when the
instrument is struck.The vibration produced
in the string, column of air or body of the
instrument causes standing waves to be formed, which produces sound
5) The reason for the emergence of a graph similar to the sine
function in the pendulum
Because of the oscillation of the pendulum back and forth with a
periodic movement when Newton's law was applied, as we did in
the theory of experience,
( )