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Prepared By: Bukhary Sulayman F Sheikh: University of Zakho College of Engineering Mechanical Department

This document analyzes the simple pendulum through mathematical equations. It begins with an introduction to pendulums and their history. It then discusses assumptions made in the model, key variables like mass and length, and derives the equation for the period of a simple pendulum based on these variables and Newton's Second Law. Applications of pendulums are also mentioned. The document is a university-level analysis of using mathematical equations to model simple pendulum motion.

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Ahmed Amir
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53 views12 pages

Prepared By: Bukhary Sulayman F Sheikh: University of Zakho College of Engineering Mechanical Department

This document analyzes the simple pendulum through mathematical equations. It begins with an introduction to pendulums and their history. It then discusses assumptions made in the model, key variables like mass and length, and derives the equation for the period of a simple pendulum based on these variables and Newton's Second Law. Applications of pendulums are also mentioned. The document is a university-level analysis of using mathematical equations to model simple pendulum motion.

Uploaded by

Ahmed Amir
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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UNIVERSITY OF ZAKHO

COLLEGE OF ENGINEERING
MECHANICAL DEPARTMENT

ENGINEERING ANALYSIS
(APPLICATION OF SIMPLE PINDULUM)

Prepared by:
Bukhary Sulayman F sheikh

SUPERVISED BY:
Dr.HUSSIEN
OUTLINE:
1-introduction.
2-Application.
3- Assumptions
4-Variable.
5-Equation.
6-Reference.
Introduction
History and definition
1.1

“Pendulum” from the Latin“pendulus”meanin“hanging”

Galileo Galilei began experimenting with pendulums in


1602. Galilei first became interested as a university
student when Galilei was watching a lamp swinging in a
cathedral in Pisa, Italy. Galilei discovered that the period
almost entirely upon length. Galilei theorized that a clock
could be made using a pendulum.
Christiaan Huygens was the first person to use this idea
when Huygens con- structed a clock using a pendulum in
1665. For the day it was very accurate, only losing one
minute per day. Huygens later improved this to a loss of
ten seconds per day.
A simple pendulum may be described ideally as a point mass suspended by
a massless string from some point about which it is allowed to swing back
and forth in a place. A simple pendulum can be approximated by a small
metal sphere which has a small radius and a large mass when compared
relatively to the length and mass of the light string from which it is
suspended. If a pendulum is set in motion so that is swings back and forth,
its motion will be periodic. 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 = l/f. The maximum distance that the mass is displaced
from its equilibrium position is defined as the amplitude of the oscillation.
When a simple pendulum is displaced from its equilibrium position, there
will be a restoring force that moves the pendulum back towards its
equilibrium position. As the motion of the pendulum carries it past the
equilibrium position, the restoring force changes its direction so that it is
still directed towards the equilibrium position. If the restoring force F G
is opposite and directly proportional to the displacement x from the
equilibrium position, so that it satisfies the relationship.
𝐹 = −𝑘𝑥
then the motion of the pendulum will be simple harmonic motion and its
period can be calculated using the equation for the period of simple
harmonic motion
𝑚
𝑇 = 2𝜋√
𝑘
It can be shown that if the amplitude of the motion is kept small, Equation
(2) will be satisfied and the motion of a simple pendulum will be simple
harmonic motion, and Equation (2) can be used.

The restoring force for a simple pendulum is supplied by the vector sum of
the gravitational force on the mass. mg, and the tension in the string, T. The
magnitude of the restoring force depends on the gravitational force and the
displacement of the mass from the equilibrium position. Consider Figure 1
where a mass m is suspended by a string of length l and is displaced from
its equilibrium position by an angle θ and a distance x along the arc through
which the mass moves. The gravitational force can be resolved into two
components, one along the radial direction, away from the point of
suspension, and one along the arc in the direction that the mass moves. The
component of the gravitational force along the arc provides the restoring
force F and is given by.
𝐹 = −𝑚𝑔 sin 𝜃
where g is the acceleration of gravity, θ is the angle the pendulum is
displaced, and the minus sign indicates that the force is opposite to the
displacement. For small amplitudes where θ is small, sinθ can be
approximated by θ measured in radians so that Equation (3) can be written
as.
𝐹 = −𝑚𝑔𝜃
The angle θ in radians is x/l, the arc length divided by the length of the
pendulum or the radius of the circle in which the mass moves. The
restoring force is then given by.
𝑥
𝐹 = −𝑚𝑔
𝑙
and is directly proportional to the displacement x and is in the form of
Equation (1) where mg k = l . Substituting this value of k into Equation (2),
the period of a simple pendulum can be found by
Therefore, for small amplitudes the period of a simple pendulum depends
only on its length and the value of the acceleration due to gravity.

1.1 Applications
Pendulums have many applications and were utilized often before the
digital age. They are used in clocks and metronomes due to the
regularity of their period, in wrecking balls and playground
swings,due to their simple way of building up and keeping energy.
They are even found in various scientific instruments, from
seismographs to early torpedo guidance systems, due to their
sensitivity to disturbance. A predecessor to the seismograph was
based on an inverted pendulum, Chang Heng’s Dragon Jar invented
at around 123 A.D.

2 Assumptions
All models are full of assumptions. Some of these
assumptions are very accurate, such as the pendulum is
unaffected by the day of the week. Some of these
assumptions are less accurate but we are still going to
make them, friction does not effect the system. Here is a
list of some of the more notable assumptions of this model
of a pendulum.

 Friction from both air resistance and the system is


negligible.
 The pendulum swings in a perfect plane.
 The arm of the pendulum cannot bend or stretch/compress.
 The arm is massless.
 Gravity is a constant 9.8 meter/second2.
Variables
m = mass at the swinging end of the pendulum (kilograms)

g = acceleration due to gravity (meter/second2)

L = length from the swivel point to the center of mass (meters)

θ = angle between the string position to the string position at

rest (radians) t = time (seconds)

T = period of the pendulum (time for one complete cycle) (seconds)


1 Equations
We will now derive the simple harmonic motion equation
of a pendulum from Newton’s second Law.
F = ma
Now take the square root of both sides while ignoring the negative
because we are solving for time and either time will be the same
distance from our t0. It is just a matter of forwards or backwards in
time. On the left hand of our equation lies the rate of change of the
angle with respect to time, but we are going to solve for the period,
so we need the time with respect to the angle, because of this we
are going to inverse the entire equation and integrate from 0 to θ0.
We will now multiply the whole thing by four to get the period. The
change in time to get from 0 to θ0 is only one forth of the entire
cycle of the pendulum. This gives us our new equation of
Sample problem:
REFERENCE:
1-Differential Equations with Boundary Value Problems Polking, Boggess, Arnold
2-http://calculuslab.deltacollege.edu/ODE/7-A-2/7-A-2-h.html
3-https://www.acs.psu.edu/drussell/Demos/Pendulum/Pendula.html
4-https://www.academia.edu/31390648/LAP_REPORT_THE_SIMPLE_PENDULUM

5-
http://www.webassign.net/question_assets/tamucolphysmechl1/lab_1/manu
al.html
6- ADVANCED ENGINEERING MATHEMATIC, Peter V,O Neil. Seven
edition,

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