Pohl's Pendulum Lab Report: Resonance & Oscillations

People’s Democratic Republic of Algeria
Ministry of Higher Education and Scientific Research
National Polytechnic School of Constantine
Preparatory Class- First Semester (2nd year)
Lab Report: Physics III
Pohl’s Pendulum
Prepared by:
Djeffal Selman
Email: djeff[email protected]
Or
Professional Email : selman.djeff[email protected]tine.dz
Academic Year:
2023/2024

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Lab report
Pohl’s Pendulum
Equipment and objectives
1.1
1.1.1 Definition
APohl pendulum, also known as Pohl’s pendulum or resonator, is a physics experiment
that demonstrates the concept of resonance. It consists of a heavy mass suspended by a
spring. When the pendulum is subjected to periodic vibrations at its natural frequency,
it exhibits resonance, with its oscillations greatly amplified. This experiment is used for
educational purposes to illustrate resonance phenomena.
Figure 1.1: Experimental Setup for Pohl Pendulum Dynamics: Components

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1.1.2 Objective
The objective of studying a Pohl pendulum is to investigate and understand the mechanical
oscillation of a system that is free, damped, and forced. The Pohl pendulum provides
a practical way to observe and analyze these different aspects of mechanical oscillations
in a controlled experimental setup. Here are a few reasons why we study Pohl’s
pendulum:
1. Understanding Simple Harmonic Motion (SHM): grasping concepts such as
amplitude, frequency, and period in the context of oscillatory motion.
2. Damping Effects: The experiment also allows for the study of damping effects in
oscillatory systems. Damping is the process by which energy is gradually removed from
a vibrating system, and Pohl’s pendulum can demonstrate how damping influences the
behavior of the system.
3. Energy Transfer: Pohl’s pendulum provides insights into how energy is conserved
and transferred in oscillatory systems.
4. Practical Application: The principles learned from Pohl’s pendulum have appli-
cations in various fields, including physics, engineering, and even in understanding
phenomena such as vibrations in mechanical systems.
5. Experimental Techniques: Performing experiments like Pohl’s pendulum helps de-
velop skills in experimental design, data collection, and analysis. These skills are
valuable in scientific research and various technical fields.
In essence, studying Pohl’s pendulum is not just about understanding a specific exper-
iment; it’s about gaining a deeper comprehension of fundamental principles in physics and
applying that knowledge to real-world situations.
Theoretical Foundations
1.2
In the following sections, we will explore the governing equations, solutions, and key concepts
associated with each scenario, shedding light on the captivating world of Pohl pendulum
dynamics. Through these investigations, we gain a deeper understanding of mechanical
oscillations, their underlying principles, and their real-world applications. Let us embark on
this journey through the oscillatory realm of the Pohl pendulum [1-4].
1.2.1 Free Oscillation
In the case of free oscillation, there is no external force acting on the pendulum, and the
damping is either negligible or absent. The equation governing this motion is a simple
second-order differential equation for angular displacement (θ) with respect to time (t):
Id2θ
dt2+Cθ = 0 (1.1)

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This is a linear homogeneous differential equation, and its solution represents the natural
oscillations of the Pohl pendulum. The solution for free oscillation is given by:
θ(t) = θ0cos(ω0t+φ)
Here: - θ(t)represents the angular displacement of the pendulum at time t. - θ0is the
initial amplitude of the oscillation. - ω0is the natural angular frequency (pulsation propre)
of the pendulum when it’s in free oscillation and not damped. - φis the phase angle, which
determines the starting position of the oscillation.
1.2.2 Damped Oscillation
When damping is introduced, you can modify the equation to include the damping force (f)
term:
Id2θ
dt2+ 2λdθ dt +ω2
0θ= 0 (1.2)
Here, λrepresents the damping coefficient, and it is defined as λ=f
2I. The equation
describes how damping affects the oscillations of the pendulum. In the case where λ<ω0,
the solution of this equation is:
θ(t) = Ae−λt cos(ωt +φ)
Where: - Ais the initial amplitude of the damped oscillation. - ωis given by ω=
pω2
0−λ2. - φis the phase angle, which determines the starting position of the oscillation.
This solution describes how the amplitude of the oscillation decreases over time due to
damping (λ).
1.2.3 Forced Oscillation
In the presence of an external driving force (M·cos(ωt)), the equation becomes:
Id2θ
dt2+fdθ dt +Cθ =M·cos(ωt)(1.3)
In this case, ’M’ represents the magnitude of the external force, and ’ω’ is its angular
frequency. This equation describes how the Pohl pendulum responds to the applied force.
The amplitude of the pendulum’s motion in this forced oscillation scenario can be affected
by damping. The logarithmic decrement (δ) is a measure of how quickly the amplitude
decreases with time. It is given by:
δ=1
nln θ(t) θ(nt +T)=1 nλT (1.4)
Where: - δis the logarithmic decrement. - nis the number of complete oscillations. -
Tis the period of the forced oscillation, defined as T=2π
ω. - λis the damping coefficient.
-Iis the moment of inertia of the system. The logarithmic decrement (δ) quantifies how
rapidly the amplitude decreases over time. A smaller δindicates slower decay, while a larger
δindicates faster decay. It is an important parameter in analyzing the behavior of the
pendulum in response to the applied force.
The solutions to these differential equations provide insights into the behavior of the
Pohl pendulum under different conditions: free oscillation, damped oscillation, and forced

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oscillation. The specific solutions depend on the initial conditions and the parameters of the
system (I,f,C,M,ω).
Experiment
1.3
1.3.1 Exploration of natural oscillations and lightly damped oscil-
latory phenomena
We proceed as follows: first, we initiate the experiment by positioning the pointer at 19 and
then releasing it. Next, we observe the reactions after five back-and-forth movements. At
this point, we record the graduation corresponding to the pointer’s position. We then repeat
this experiment by performing ten back-and-forth movements, followed by fifteen, and so
on. To record and document our results, we use the table 1.1.
Table 1.1: Experimental Data 1
n5 10 15 20 25
θ(t)
t(s)
θ(t+nT )
(δ)
T=t
n(Period)
T0=P25
i=5 Ti(average)
Question 1:
By carefully analyzing the data collected in Table 1.1, this approach allows us to discern
the nature and characteristics of the motion involved in the studied system. Based on the
results obtained in Table 1.1, what is the type of motion with justification?
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Question 2:
Determine the damping factor of attenuation (damping) λusing equation 1.4?
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