chapter 11 damped harmonic motion - oscillatory...
TRANSCRIPT
Chapter 11
Damped Harmonic Motion -Oscillatory Pendulum
11.1 Lecture - Simple Pendulum Motion
In this lesson, we will continue our study of simple harmonic motion. Instead of lookingat a linear oscillator, we will study an angular oscillator – the motion of a pendulum. Inour study of the linear motion of a spring-mass system, we observed that friction dissipatedenergy of the system after a large number of oscillatory periods, such that the system wouldeventually come to rest. We will see in this lesson that a similar phenomena takes placein angular motion as well. In lecture this week, we will solve two problems: Problem A -analysis of the motion of a simple pendulum using Newton’s Laws of motion in the absenceof friction and Problem B - analysis of the dissipation of the total mechanical energy of asimple pendulum using the work and energy theorem.
During lab we will be introduced to a new sensor - the angular encoder. The encoder isa device that can be a�xed to a rotating shaft to measure the angular position of the shaft.We will collect data for the angular position of a pendulum as a function of time and thensave this data to a text file.
During studio we will compare the laboratory data to the theoretical models developedduring lecture and continue our skills development with the MATLAB software environment.
11.1.1 Problem A. Formulate
State the Problem
We are given a mass m suspended from a rigid rod of length R. The rod is mounted on abearing assembly such that it is able to pivot about a point at the origin. In a system likethis, the mass at the end of the rod is often called the “bob.” The bob is initially manuallydisplaced through some angle ✓
0
and released from rest.Use Newton’s laws to develop an expression for the angular position of the bob as a
function of time.
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State the Known Information
m = Known [kg] � Mass of bob (Constant) (11.1)
R = Known [m] � Length of Pendulum (11.2)
✓0
= Known [radians] � Initial Displacement of bob (11.3)�!V
0
= 0 [m/s] � Initial Velocity of bob (11.4)
State the Desired Information
✓(t) = ? [radians] ↵ Displacement of m vs. time (11.5)
11.1.2 Problem A. Assume
mrod ⇡ 0 [N ] � Neglect mass of rod (11.6)
Wmi!f ⇡ 0 [J ] � Neglect frictional work (11.7)
Qmi!f ⇡ 0 [J ] � Neglect heat transfer (11.8)
11.1.3 Problem A. Chart
The schematic diagram and free body diagrams are shown below in Figure 11.1.
11.1.4 Problem A. Execute
Recall the governing equations:
If :X�!
F = 0 Then : �!a = 0 Newton’s 1st Law (11.9)
X�!F =
d(m�!V )
dtNewton’s 2nd Law (11.10)
�!F Action = ��!
F Reaction Newton’s 3rd Law (11.11)�!F g = g ·m # Newton’s Law of Gravity near Earth (11.12)
E2
� E1
= Q1!2
�W1!2
Work Energy Theorem (11.13)
From the schematic diagram and FBD we observe that there is a tension T in the rod oflength R. The tangential force due to gravity has a magnitude mg sin ✓, and it acts todecrease the magnitude of the angle |✓|. It is common nomenclature to describe the arclength with the symbol s measured from the lowest position of the pendulum. Using ourknowledge of trigonometry and geometry we can write s = R✓, , where ✓ is in radians. Noticefrom the FBD that the radial force due to gravity will be balanced by the tension in the
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Figure 11.1: Schematic diagram of pendulum experiment and free body diagram of bob atend of pundulum.
rod. The motion of the pendulum is forced by the tangential component of the force due togravity. From the FBD we can write Newton’s Second Law in the tangential direction as
XFtangential = �mg sin ✓ = m
d2s
dt2or (11.14)
d2s
dt2= �g sin ✓ or, since s = R✓ (11.15)
d2s
dt2= �g sin
s
R(11.16)
[m]
[s2]=
[m]
[s2][�] Units
From calculus, it is known that the sine of an angle (in radians) can be expressed by theseries:
sin ✓ = ✓ � ✓3
3!+
✓5
5!� ✓7
7!+ · · · (11.17)
When the angle ✓ is a small angle, then |✓| >> | ✓33!
| >> | ✓55!
| >> | ✓77!
| · · · , and we can use thesmall angle approximation that sin ✓ ⇡ ✓. Using the small angle approximation for the sinefunction in Equation 11.16 allows us to write:
d2s
dt2⇡ �g
s
R= � g
Rs = �!2s (11.18)
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where we have defined the natural frequency as !2 ⌘ g/R. The solution of the OrdinaryDi↵erential Equation (ODE) 11.18 will be studied in a later course on di↵erential equations.The solution is presented here, without derivation, as:
s = s0
cos(!t) (11.19)
where s0
= R✓0
is the initial (maximum) displacement measured along the arc of the circleshown in the schematic diagram. While the derivation of Equation 11.19 is beyond the scopeof this course, you can take the first and second derivative of Eq. 11.19 and substitute theresults into Equation 11.18 to show that it is indeed a solution of the ordinary di↵erentialequation. The period of the harmonic motion of the simple frictionless pendulum with smallinitial angle displacements is
T =2⇡
!=
2⇡pgR
(11.20)
[s] =[radians/period]
[radians/s]=
[radians/period]q[m]
[m/s2
Units
Notice that the period of the pendulum is independent of the mass of the bob! Also, theperiod of the pendulum depends only upon length of the rod and the acceleration of gravity.Thus, using a simple time piece, and carefully measuring the length of the rod providesus with a highly accurate method to experimentally estimate the acceleration of gravity.Earlier in this class, we used a high speed data acquisition system to measure the positionof an object falling through the air. Such elaborate equipment was not available in the timeof Newton. However, it was perfectly reasonable to estimate the period of oscillation of alarge pendulum and to measure its length. As an exercise, you may use your pendulumobservations as an alternative means of estimating the local acceleration of gravity at thesurface of the Earth.
A graphical representation of the harmonic motion of the ideal pendulum in the absenceof friction is shown in Figure 11.2.
11.1.5 Problem A. Test
The units of our analysis are correct. The period of a simple pendulum is independent ofthe mass of the pendulum bob. This can be confirmed experimentally. As the length, R, ofthe rod increases the period T of the pendulum increases. This makes intuitive sense. Ourconfidence is high.
11.1.6 Problem B. Formulate
State the Problem
Building upon the analysis presented in Part A, we now wish to incorporate an analysis ofthe e↵ects of friction on simple harmonic motion. Both the pendulum of the current lessonand the spring-mass system of the previous lesson exhibit characteristics of harmonic motion.
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Figure 11.2: Angular position of a simple pendulum as a function of time in the absence offriction.
Given a mechanical system operating in harmonic motion, estimate the energy dissipationdue to an average friction force which acts in opposition to and is directly proportional tothe velocity of motion. Use the work and energy theorem to develop an expression for thedecay in the total mechanical energy of a system.
State the Known Information
We start with the knowledge that the motion of an ideal pendulum in the absence of frictionmay be described using the results from Part A:
s(t) = s0
cos(!t) [m] � Arc Displacement (No friction) (11.21)
✓(t) = ✓0
cos(!t) [radians] � Angular Displacement (No friction) (11.22)
We know that the pendulum is released from rest from an initial angle ✓0
as was givenpreviously in Equation 11.3. Since the initial speed of the pendulum is zero, the total initialmechanical energy present in the system at time zero is simply the gravitational potentialenergy of the bob:
E0
= PE0
= mgR(1� cos ✓0
) [J ] � Initial Energy (11.23)
where the datum for elevation, z = 0, is chosen to be at the bottom of the pendulum arc asillustrated in the schematic diagram of Figure 11.1.
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State the Desired Information
We expect friction to do work always against the motion of the system. Thus, the mechanicalenergy stored in the pendulum system is always working to overcome the frictional resistancedue to the bearings and air resistance. We expect this friction to dissipate eventually all ofthe mechanical energy in the system. We are to develop an expression for predicting thetotal mechanical energy E(t) in the system as a function of time:
E(t) = ? [J ] ↵ Total Energy vs. Time (11.24)
11.1.7 Problem B. Assume
We make an assumption that the friction force always acts in opposition to the motion ofthe pendulum, and is directly proportional to the magnitude of the velocity:
�!F friction = �b
�!V [N ] � Friction Force (11.25)
Qi!f ⇡ 0 [J ] � Neglect Heat Transfer (11.26)
Equation 11.25 is a common engineering model for frictional damping in mechanical systems.While the actual instantaneous friction is dependent the velocity, it is also common to reportan average friction force over a single period of oscillation of a harmonic system. Thecoe�cient b describes the amount of friction in a system. When b ! 0 there is no frictionin the system. As b ! 1 then the frictional resistance to motion is infinite and the system’seizes’, or ’locks up’. Equation 11.26 states that there is no heat transfer to or from thesystem during the time interval of interest. This is consistent with the analysis of Part A.
For Part B, we will use the average friction force acting during one period of oscillation toestimate the work done by the system to overcome friction. This analysis is mathematicallynot quite perfect, but it is certainly su�cient to provide us with an engineering understandingabout the influence of frictional damping on simple harmonic motion.
11.1.8 Problem B. Chart
The schematic diagram and free body diagrams were already presented for Problem A.
11.1.9 Problem B. Execute
Recall the governing equations:
If :X�!
F = 0 Then : �!a = 0 Newton’s 1st Law (11.27)
X�!F =
d(m�!V )
dtNewton’s 2nd Law (11.28)
�!F Action = ��!
F Reaction Newton’s 3rd Law (11.29)
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�!F g = g ·m # Newton’s Law of Gravity near Earth (11.30)
�E = E2
� E1
= Q1!2
�W1!2
Work Energy Theorem (11.31)
We used Newton’s laws in Part A to model the motion of the pendulum in the absence offriction. Now, we will use the work and energy theorem to evaluate the influence of frictionupon the system. The total mechanical energy E of the system at any instant of time t isgiven by:
Total Ez}|{E(t) =
Kinetic Ez }| {KE(t) +
Grav. Pot. Ez }| {PE(t) +
Elastic Pot. Ez }| {SE(t) [J ] Total Energy (11.32)
For the pendulum, we neglect the elastic potential energy since there are no springs presentin the system. If we were to analyze the friction present in a spring-mass system similarto that studied in the previous lesson, then clearly the elastic potential energy would besignificant. For the pendulum, we have
Total Ez}|{E(t) =
Kinetic Ez }| {KE(t) +
Grav. Pot. Ez }| {PE(t) [J ] Total Energy (11.33)
At time t = 0 we know that the system starts from rest at an initial angular displacement of✓0
from the vertical as illustrated in schematic. The total mechanical energy in the systemat time t = 0 is:
E0
= E(t = 0) = PE(t = 0) = mgR(1� cos ✓0
) [J ] Initial Energy (11.34)
The total mechanical energy in the system at any instant of time t is:
E(t) =mV 2
2+mgR(1� cos ✓) [J ] Total Energy (11.35)
From our earlier work with the spring-mass system, we observed that the mechanical energyin the system is continuously traded back and forth. For the spring/mass system, we had toconsider SE, KE, and PE. For the simple pendulum, we only need to consider only KEand PE. The average value of kinetic energy and potential energy during a single period ofthe pendulum motion is one half the total energy at the beginning of each period:
(KE)ave = (PE)ave =E
2[J ] Average KE and PE (11.36)
We can use the average kinetic energy of the pendulum during one period of oscillation toestimate the average speed of the pendulum during a period (the instantaneous speed variesfrom zero at the peak of the motion to a maximum value at the bottom of the arc):
(KE)ave =1
2m(V 2)ave [J ] Average KE (11.37)
(V 2)ave =2
m(KE)ave =
2
m
E
2=
E
m[J ] Average Speed (11.38)
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We introduced the assumption that the instantaneous friction force is directly proportionalto and in the opposite direction of the velocity, as given by Equation 11.25. Since work isthe dot product of force with displacement, we can write the instantaneous work done byfriction as:
Wf =�!F friction ·�!x [J ] Instantaneous Work Done by Friction (11.39)
The RATE at which friction does work Wf is called the frictional power dissipation, and itmay be approximated by
Wf =�!F friction ·
d�!xdt
=�!F friction ·
�!V
[J ]
[s]Instantaneous Power Dissipation by Friction
(11.40)
Our assumption for the friction force, Equation 11.25, may be used in Equation 11.40 toyield
Wf = �b�!V ·�!V = �bV 2
[J ]
[s]Instantaneous Power Dissipation by Friction (11.41)
The average frictional power dissipation rate during one period of oscillation may be approx-imated by:
W frictionave = �b(V 2)ave
[J ]
[s]Average Power Dissipation by friction (11.42)
The work done by the PENDULUM is the opposite sign of the work done by the FRICTION:
W Pendulumave = +b(V 2)ave
[J ]
[s]Average Power Expended by Pendulum to Overcome Friction
(11.43)
Now, using our assumption of negligible heat transfer (Equation 11.26) in the work energytheorem of Equation 11.31 allows us to write:
�E = �W1!2
Simplified Work Energy Theorem (11.44)
If we divide both sides of Equation 11.44 by �t and using Equations 11.38 and 11.43 we get:
�E
�t= �W
1!2
�t= �W Pendulum
ave = �b(V 2)ave = � b
mE Rate Form of Work & Energy
(11.45)
If we take the limit as �t ! 0, then the ordinary di↵erential equation describing the energyof a simple harmonic oscillator working to overcome friction is thus given by
dE
dt= � b
mE Simple Oscillator with Friction ODE for Energy (11.46)
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The solution of Equation 11.46 is
E(t) = E0
e�(b/m)t Energy vs. Time (11.47)
where E0
= (PE)0
= mgR(1� cos ✓0
). The solution of Equation 11.46 is beyond the scopeof this course, but will be studied in your later course on di↵erential equations. You cantake the first derivative of Equation 11.47 and substitute into Eq. 11.46 to demonstrate thatwe do indeed have a valid solution.
We know from the schematic diagram that the PE of the pendulum is proportional tothe maximum angle of displacement. If the PE decays exponentially, then it follows thatthe maximum angle will also decay exponentially. Thus, we hypothesize that the maximumangular displacement of a pendulum moving in the presence of friction should obey:
✓max(t) = ✓0
e�(b/m)t Maximum Angle vs. time (11.48)
A graphical representation of Equation 11.48 is shown in Figure 11.3. As time goes to infinity,
Figure 11.3: Decay in maximum angle of deflection due to the energy dissipated in overcom-ing friction.
we realize that the motion of the pendulum will cease and come to rest at ✓ = 0, which isthe minimum potential energy condition. In fact, this idea of minimum potential energy isa very powerful driver in nature. This natural tendency towards minimum energy systemsforms the foundation of an entire branch of mathematics known as “variational calculus”and is used by engineers to solve problems in solid body mechanics, heat transfer, energysystems, bio-mechanical systems and a host of other application areas.
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11.1.10 Problem B. Test
The units of our analysis are correct. As the amount of friction increases, the pendulummotion will dampen out more rapidly. The coe�cient b must be estimated experimentally.
11.1.11 Combining Harmonic Motion with Frictional Damping
In Problem A, we developed Equation 11.20 to describe the undamped harmonic motion ofan ideal pendulum in the absence of friction. In Problem B, we developed Equation 11.48 todescribe the decay in the maximum angle of displacement as a function of time, due to thework done by the pendulum in overcoming the friction forces in the bearing and atmosphere.
Since the energy is dissipated continuously during motion, as the pendulum works toovercome friction, the amplitude of the oscillation will decay over time. We can combine theangular decay term describing the amplitude of the pendulum oscillation with the harmonicresponse term. The combined response of undamped oscillations being dampened by theenergy decay term is illustrated in Figure 11.4.
Figure 11.4: Combination of angular decay and undamped harmonic motion to predict thedamped harmonic motion of a pendulum in the presence of friction.
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The angular position of the pendulum will obey the expression
✓(t) = ✓0
e�(b/m)t cos(!t) (11.49)
where the natural frequency is given by ! =p
g/R and the damping constant b mustbe determined experimentally. Derivation of the solution shown in Equation 11.49 will bestudied in your later class on ordinary di↵erence equations. The coe�cient in the exponentialterm is called the damping ratio or coe�cient, and it is defined as ⇠ ⌘ b/m. The symbol ⇠ isthe lower case Greek letter Xi. The damped harmonic motion of a pendulum is illustratedin Figure 11.5.
Figure 11.5: Damped harmonic motion of a pendulum in the presence of friction.
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11.2 Lab - Simple Pendulum Motion
11.2.1 Scope
This week you will investigate the transient response of a simple pendulum with variablemass and variable length when subjected to an initial disturbance. The resulting motion isa perfect example of damped harmonic motion.
11.2.2 Goal
The goals of this laboratory experiment are to
1. validate the work - energy theorem,
2. understand the oscillatory response of a harmonic system in the presence of frictionaldamping, and
3. begin to understand the physics of angular motion.
11.2.3 Units of Measurement to Use
All reports shall be presented in the SI system of units. Raw data may be collected in avariety of units.
Table 11.1: Units of measurement to be used for damped oscillatory pendulum system.Quantity Basic units Derived units
Length [m] [m]Mass [kg] [kg]Time [s] [s]
Velocity [m/s] [m/s]Force [kgm/s
2] [N ]Energy [kgm2
/s
2] [J ] or [N ][m]Angle [radians] [radians]
Frequency [radians/s] [radians/s]Period [s] [s]Work [kgm2
/s
2] [J ] or [N ][m]
11.2.4 Reference Documents
Review the lab videos for this week, and pay close attention to the proper installation anduse of the rotary encoder. Also, you may find the tutorial on rotary encoders on Wikipediato be of use, at http://en.wikipedia.org/wiki/Rotaryencoder.
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11.2.5 Terminology
The following terms must be fully understood in order to achieve the educational objectivesof this laboratory experiment.
Energy Displacement ForceKinetic Energy Velocity WorkGravitational Potential Energy Speed PendulumArc Angle FrequencyPeriod Damping FrictionBearing Decay Super-positionEncoder Radians Degrees
11.2.6 Summary of Test Method
On the myCourses site for this course you will find links to one or more videos on YouTubefor this week?s exercise. Watch all of the available videos, and complete the online lab quizfor the week. The videos are your best reference for the specific tasks and procedures tofollow for completing the laboratory exercise.
11.2.7 Calibration and Standardization
By now in this course, students should be in a position to conduct independent calibrationsof hardware, and properly configure the use of all hardware, without having detailed instruc-tions. Note that you will treat the USB encoder device as a primary instrument, and thuswill not calibrate it. The only requirement is to use the interface software to set the zeroangle of the encoder at the position when the pendulum bob hangs at rest.
11.2.8 Apparatus
All required apparatus and equipment components are described and demonstrated in theinstructional videos for this exercise, or will be familiar from common or previous use.
11.2.9 Measurement Uncertainty
The encoder device in use for this investigation is capable of representing each full revolutionof its shaft with 2048 digital pulses that are captured by the control electronics and thesoftware program that interfaces to the system. Therefore, knowing that each revolution is360 [degrees], and treating the device as a primary instrument, the instrument least countof the rotational position measurement (the encoder) becomes 360 [degrees]/2048 = 0.176[degrees]. In turn, the uncertainty of the encoder as a primary instrument is then taken tobe:
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✏encoder = ±1
2ILCencoder ⇡ ±1
20.176[degrees] ⇡ ±0.089[degrees] (11.50)
11.2.10 Sampling, Test Specimens
The transient response of the pendulum will depend upon the applied mass, its location onthe shaft, and the initial angle from which the mass is released to initiate motion. Everygroup member should run and record his/her own set of trials, for at least two di↵erentmasses and locations on the shaft, following the overall guidelines noted in the videos formaximum angle and so forth. It is always recommended that every student run several trialsto ensure that at least one good data set will result for each setup.
11.2.11 Preparation of Apparatus
All required equipment for conducting the laboratory exercise is made available either withinone or both of the drawers attached to the lab bench, or available from the laboratory instruc-tor. You are expected to bring all other necessary materials, particularly your logbook and aflash drive for storing electronic data as appropriate. You are to follow the general specifica-tions for team roles within the lab. Although there are specific, individual expectations foreach role, you are each responsible overall to ensure that the objectives and requirements ofthe laboratory exercise are met, and that all rules and procedures are followed at all times,especially any that are related to safety in the lab. When finished, all equipment is to bereturned to the proper location, in proper working order.
subsectionProcedure - Lab Portion
Record all observations and notes about your lab experiment inyour logbook.
The instructional videos for this exercise cover the specific procedures to follow as you setup the apparatus to make measurements, and for actually collecting data with the variousdevices and software interfaces. More generally, you should always observe the followinggeneral procedures as you conduct any of the exercises in this laboratory.
1. Come prepared to lab, having watched the videos in detail, then completing the asso-ciated lab quiz and preparing your logbook before you arrive to class.
2. Follow the basic outline of elements to include in your logbook related to headers,footer, and signatures.
3. As you conduct the exercise, please pay attention to the following safety concerns:
• Watch for tripping hazards, due to cables and moving elements.
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• Watch for pinch points, during assembling and disassembly.
• Be careful of shock hazards while connecting and operating electrical components
4. Every week, for every exercise, your logbook will minimally contain background notesand information that you collect before the lab, at least one schematic of the apparatus,various standard tables for recording the organization of your roles and equipmentused, the actual data collected and/or notes related to the data collected (if doneelectronically for instance), and any other information relevant to the reporting andanalysis of the data and understanding of the exercise itself.
5. All students should create and complete a table indicating the sta�ng plan for theweek (that is, the roles assumed by each group member), as shown in Table 1.2.
6. All students should create and complete a table listing all equipment used for the exer-cise, the location (from where was it obtained: top drawer, bottom drawer, instructor?)and all identifying information that is readily available. If the manufacturer and se-rial number are available, then record both (this would be an ideal scenario). If not,record whatever you can about the component. In some, cases, there will be no specificidentifying information whatsoever either because of the simplicity of the component,or because of its origin. In these cases, just identify the component as best you can,perhaps as “Manufactured by RITME.” The point here is to give as much informationas possible in case someone was to try to reproduce or verify what you did. Refer toTable 1.3.
7. For the Lab Manager only: create a key sign-out/sign-in table for obtaining thekey to the equipment drawers, as shown in Table 1.4.
8. All students should create a table or series of tables as appropriate to collect his/herown data for the exercise, as well as any specific notes related to the data collectionactivities. In those cases where data collection is done electronically, there may not beany data tables required.
9. Many of the laboratory exercises will require the use of a specific software interfacefor measurements and/or control. In all cases, these will be made available on themyCourses site unless stated otherwise.
10. The Scribe (or a designated alternative) should take a photo of each group memberperforming some aspect of the laboratory exercise for inclusion in the lab reportthat will be generated during the studio session. Refer to the example lab report formore details.
11. Record all relevant data and observations in your logbook, even those that may nothave been explicitly requested or indicated by the textbook or videos. If in doubtabout any measurements, it is better to make the measurement rather than not.
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12. When you are finished with all lab activities, make sure that all equipment has beenreturned to the proper place. Log out of the computer, and straighten up everythingon the lab bench as you found it. Put the lab stools back under the bench and out ofthe way.
13. Prepare for the upcoming studio session for the week by carefully read and understandsub-Section 3 of the textbook, and complete the Studio pre-work prior to your arrivalat Studio.
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11.3 Studio - Matlab Simulation of Damped HarmonicMotion
This week in Studio you will complete an analysis of damped pendulum motion. You willcreate two plots for angular displacement: one theoretical and one experimental. The ex-perimental plot will be derived directly from data obtained in lab and read from a file intoyour MATLAB code. The theoretical curve will use an experimental estimate for dampingratio and the experimentally observed initial displacement along with the theoretical valuenatural frequency. Ideally, the experiment plot should lay on top of the theory, but therewill be some di↵erences that accumulate over time. You will investigate the e↵ect of thesedi↵erences and discuss them as part of your analysis.
Now that you have been introduced to the MATLAB environment, you are ready to takethe next step, and write some code on your own. The commands will be very similar tothose employed in last weeks Studio. It is convenient to store code and re-use it time andtime again. As part of your studio exercise, we will ask you to vary some of your inputparameters to predict results of future experiments.
11.3.1 Calculation and Interpretation of Results
Our theoretical expression for damping says that the maximum angular displacement is givenby
✓max(t) = ✓0
e�⇠t which can provide (11.51)
✓2
= ✓1
e�⇠(t2
�t1
) or (11.52)
⇠ =ln(✓
2
/✓1
)
t1
� t2
(11.53)
The term ⇠ is the damping ratio, and you will need this value for your studio exercise.Other equations needed for Studio this week were derived in the lecture portion of the text.You will need equation 11.49, which is provided here for reference
✓(t) = ✓0
e�⇠t cos(!t) equation 11.49
11.3.2 Procedure - Studio Portion
Your pre-work for Studio this week includes two portions: tasks that you will complete inyour logbook to prepare for Studio, and a MATLAB activity that will help you spend yourtime more productively upon arrival in Studio.
Studio Pre-work (Logbook portion)
Prior to arriving at Studio, you should have acquired the necessary data in lab, recordeddata in your logbook and stored your experimental data on a thumb drive. You should also
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have a corresponding schematic drawing in your logbook that clearly identifies where eachmeasurement was made in symbolic notation. After Lab and before Studio, pleasecomplete the following tasks.
1. Refer to the measurements that you made during the Lab experiment and record thependulum rod length R in your logbook.
2. Refer to the measurements that you made during the Lab experiment and record thependulum bob mass m in your logbook.
3. Visually inspect your raw data to observe one peak value of ✓ near the beginning of theoscillatory motion. We refer to this as “Peak 1.” Determine the value of the angulardisplacement at Peak 1 and record the value as ✓
1
in your logbook. Determine thetime at which Peak 1 occurs and record this value as t
1
in your logbook.
4. Visually inspect your raw data to observe the next successive peak value of ✓ followingPeak 1. We refer to this as “Peak 2.” Determine the value of the angular displacementat Peak 2 and record the value as ✓
2
in your logbook. Determine the time at whichPeak 2 occurs and record this value as t
2
in your logbook.
5. Use Equation 11.53 to estimate the value of the damping ratio ⇠ and record this valuein your logbook.
6. Finally, inspect your data set to see what the final time is and record this value in yourlogbook. We will refer to this value as tf .
7. Be sure to save the data file as a Tab Delimited format; this will enable MATLAB toread the data file.
Studio Pre-work (MATLAB Portion)
After Lab and before Studio, please complete the following MATLAB exercise.Please upload the MATLAB portion of studio pre-work to your individual drop-box for thecorresponding week, using a filename of the format Lastname Studio11Prework.m. Re-member that MATLAB script files cannot have spaces or dashes in their filenames. You will receive a quiz grade based on the completeness of your submission.
Write a MATLAB script to do the following tasks. Be sure to include title block andcomments in your script file indicate the purpose of each line and section you write.
1. Create a first array using the variable name x, fill it with whole number values from1 to 15, and display the array results to the screen. We will use a new function inMATLAB this week to do this. Type: linspace(1, 15, 20);. This creates an arraystarting at one, ending at 15, and having 20 elements. This is a very convenient wayto define “vector” arrays (i.e., 1-D matrices). Note that this will be a row vector(i.e., 15 columns wide x 1 row long). Typically, when data is read from files it will
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be in columnar format, and when manipulating array variables in MATLAB like inequations or plots, it is necessary to have the arrays in the same format. Therefore, wewill transpose the x array by typing the following on the next line: x = x’;. This isanother example of how handling matrix or array type variables is extremely convenientin MATLAB.
2. Add some lines to your existing script to accomplish the following task. Create asecond array using the variable name y and fill it with zeros to initialize it. Refer backto last week’s script for details on how to do this. Be sure to define the dimensions ofy as 15 rows by 1 column to match that of x. Run the script to make sure it is workingproperly.
3. Add some lines to your existing script to accomplish the following task. Create a scalarvariable with the name k and assign a numerical value of k = 3.5. Echo the result tothe screen. Recall that any line in a script file that does not end with a semicolon willecho its result to the screen. Run the script to make sure it is working properly. Then,edit the script so that the intermediate results are no longer echoed to the screen. Runthe script to make sure it is working properly. Demonstrate that you understand howto enable and disable screen echo for any portion of a script that you desire.
4. Add some lines to your existing script to accomplish the following task. Now that youhave initialized x, y, and k, write a MATLAB expression to fill the array of y valueswith a computed result y = k ln(x). Run the script to make sure it is working properly.
5. Add some lines to your existing script to accomplish the following task. Create an xyplot with your script file, showing that you know how to plot an array of dependentvariables (y) as a function of an array of independent variables (x). Use your data fory = k ln(x) to plot the result over the interval 1 x 15. Recall from last week thatyou need to have the figure function followed by the plot(x,y) function. Run thescript to make sure it is working properly.
6. Now, edit your script, to change the value of k in whole numbers, to create a plot fory = k ln(x) where 1 x 15, and 1 k 8. Run the script at least eight times(changing the value of k and running it once for each value of k) to make sure it isworking properly. This is a tedious process, but will begin to demonstrate the value ofa scripting language to conduct “parametric studies.” A parametric study is when weperform a series of simulations of a physical phenomena to better understand how anengineering system will behave over a range of operating conditions.
7. You can appreciate that manually re-running the simulation as a parametric study canbecome rather tedious, particularly if each simulation takes quite a while to execute.We can use the “for loop” structure in MATLAB to help us do these repetitive tasks.Now, edit your script, to change the value of 1 k 8 in whole numbers using a “forloop” around the main body of your script, so that you create a series of eight plots
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with a single run of your simulation. That is, write your “for loop” such that the scriptdoes the entire parametric study. You need to include the calculation of y and plottingof xvs.y in the loop. Be sure you have x defined and have the ”figure” command priorto the for loop. Run the script to make sure it is working properly. You will only seethe final plot value when you run this because we redraw the plot every time it cyclesthrough the loop. To prevent the plot from redrawing itself, we will use the ”hold”function. In your script, type a line that is hold on right after the figure line. Thiswill tell MATLAB to preserve the plot figure while adding new plots to it. Be sure totype hold off after the loop. Run the script again, and now you should see all eightplots on the graph.
Videos
You may wish to review the video “Getting Started with MATLAB” found in the Week 10Content page in myCourses. Recall that this video demonstrates how to use the MATLABengineering analysis tool in “interactive mode.” You can see examples of how to do manyof the tasks requested in the Studio pre-work. Also, you may want to review the examplescript that was provided for Studio last week. If you get stuck on the “for loop” task, thenuse the MATLAB on-line resources or the MATLAB You Tube channel to find an exampleof how to write a “for loop” in an m-file script. If you find videos that your believe areparticularly useful, please suggest them to your Studio instructor, and we will add a list ofvideo resources to this section over time.
11.3.3 Steps to Complete Studio Analysis
1. CREATE A SCRIPT FILE: from within the MATLAB environment, use the pull-downcommand to execute “File - New - Script.” After the script editor window opens, usethe pull down command to execute “File - Save As” and save this file to your thumbdrive, in a folder named studio11 and a file named Lastname damped pendulum.m.Note, you can also use the script file from last week as a starting point to save a littletime. Just be sure to save it as a new file for this week right after opening it, and payspecial attention to changing all details for this week’s script.
2. CREATE A TITLE BLOCK: In the script editor window, create a title block for yourcomputer simulation program. In the ME department, your title block should alwaysinclude the name of the author, the academic term, the name of the course, and adescriptive title. You may add comments to explain the use and limitations of yourscripts. Over your career, as you develop a library of scripts, this will be an importantway for you to build upon previous knowledge. In MATLAB, the special character% is used to indicate that any text on the current line to the right of the characteris considered a comment, and is not considered a command to MATLAB. It is goodpractice to start each script with commands to clear all variables from workspacememory, and to clear the contents of the command window.
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% Author: R.N. Carter, Term 2155% Example program for MECE-102% This program is not intended to be copied electronically% Students should manually re-type this script to help learn MATLAB% Any text (like this) following a percent sign is a comment
clear % clear up the workspaceclc % clear command window
3. CREATE A LIST OF KNOWN INFORMATION: In the script editor window, createa number of scalar variables that contain known information for numerical simulation.These expressions should look similar to those shown below, but should be replacedwith numerical values appropriate for your experiment. By using the same table ofconstants as your experiment, you will be able to compare your simulated oscillationplots with your experimental measurements.
The semicolon at the end of each line inhibits MATLAB from printing intermediateresults to the screen. It is good engineering practice to include a comment on eachline, indicating the engineering units associated with each assignment statement. Notethat it is your responsibility, as the engineer, to verify that the units of every equationand constant are correct, since the simulation tools (both Excel and MATLAB) andthe program have no concept of units associated with the mathematical expressionsand assignment statements. Note that all the items shown as “XXX” representones that you need to insert the numbers that you measured from your labdata.
% Set the known information and simulation parameters% for the simulationtheta_0 = XXX ; % [degrees] (value initial peak theta in lab data)t_0 = XXX ; % [s] (time of initial peak theta in lab data)t_f = XXX ; % [s] (time of last data point in lab data)m = XXX ; % [kg] (measured in lab)R = XXX ; % [m] (measured in lab)g = 9.81 ; % [m/s^2]xi = XXX ; % [1/s] (determined from data analysis)% Compute the natural frequency based on theoryOmega = sqrt ( g / R );
4. CREATE A LIST OF CALCULATION PARAMETERS: This week, we are using ananalytical solution for the pendulum system, so we are using MATLAB to perform acalculation rather than a simulation like we did last week with the spring-mass systemusing Euler’s method. We still need to define the time variable this week. To do this,
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we need to define the start, end, and increment times. We have already define thevalues for t
0
and tf . For this calculation a time increment of 0.01 seconds will besu�cient. Like in the prework, we will using the “linspace” function to define the timevector and then transpose it to switch it from a row-vector to a column-vector.
% Determine Calcualtion parametersdt = 0.01; ; % [s] time incrementnt = round((t_f-t_0) / dt) ; % [-] Use round function to ensure nt
% is an integert = linspace(t_0,t_f,nt); % defines a row vector for t that has nt
% values from t_0 to t_ft = t’; % transpose t so it is a column vector
5. CREATE A LIST OF DESIRED INFORMATION: In the script editor window, createa number of array variables that will be used to store your simulation results. We firstwill compute how many array elements will be needed to store the answer for a givensimulation duration and time increment. Then, we will create an array for each set ofanswers, with that many elements. It is good programming practice to initialize all ofthese variables in each element of the array to a value of zero, as shown below.
% Determine how many time steps are neededNTime = End_Time / Delta_Time ; % [-]% Initialize the vectors for storing data% Filling each vector with zeros is good programming practiceTime_Array = zeros ( NTime, 1 ) ; % NTime rows and 1 columntheta_Array = zeros ( NTime, 1 ) ;E_Array = zeros ( NTime, 1 ) ;
6. CALCULATE ANGLE: We now will calculate the angle of the pendulum with respectto time. Note, we could set up a for loop to do this and calculate a value of angle foreach time as we cycle through the loop using code something like this:
for i=1:nttheta(nt) = theta_0*exp(-xi*t(nt))*cos(omega*(t(nt)-t_0));
end
Note that we have modified the equation slightly from that shown in Equation 11.49by subtracting t
0
from t within the cosine function. This is done to account for thefact that the data acquisition of your experimental results does not typically start rightat a peak angle. By subtracting t
0
the calculated results will align properly with theexperimental ones. This code is the typical way in which arrays are calculated in mostprogramming languages. Again, since MATLAB is specifically designed to work witharray information, this calculation can actually be done in one line of code:
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theta = theta_0*exp(-xi*t).*cos(omega*(t-t_0));
This will create a ✓ vector with the same dimensions as the t vector using this equation.Note the period right before the * symbol before the cosine function. With-out this, the code will not run because MATLAB will assume we are trying to performa matrix operation (like a cross product) in the calculation. By adding the period priorto the multiplier, we are telling MATLAB that we want it to do the calculation witheach value in the t vector. So, all that is needed to do this week’s MATLAB calculationof ✓ is this single line of code!
7. READ YOUR EXPERIMENTAL DATA: We will use a MATLAB provided functionto read your experimental data from your data file. Use code similar to that shownbelow in your script:
% Next section obtains the data from the lab% Please include the text field ’TimeData’ in Row 1 Column A% Please include the text field ’AngleData’ in Row 2 Column Btdfread;
The MATLAB intrinsic function tdfread will cause the script to open a dialog windowasking the user to point to the data file that is to be opened. Verify that your filecontains the correct columnar data and is in a tab-delimited format. The commanddetermines the number of rows and columns in the data file and assigns each columnto a unique variable with a name corresponding to the first row text values. Foranyone with coding experience, this step may seem like a bit of magic, because onewould typically have to write many lines of code to accomplish this step in otherprogramming languages. Note, your can include the path and file name in the code tohave MATLAB automatically open the file without bringing up the dialog box. Thiswould look something like:
tdfread(’c:\temp\PendulumData.dat’);
The path and filename must be stated exactly correctly or the code will not run. Ifyou are having trouble getting these correct, go to the command window and type theclear command. Then type [fn, pn] = uigetfile(’*.*’). Leave the semicolon o↵the command line. This will bring up the file open dialog. Find your data file andopen it. The filename and pathname strings will be echoed to the workspace. You canthen copy these into your script.
8. PLOT THE COMPUTED DISPLACEMENT DATA: In the script editor window, wewill next enter commands to create a plot of ✓[deg] on the vertical axis vs. Time, t [s]on the horizontal axis. We will present the simulation data contained in theta Arrayand the experimental data contained in AngleData. The next set of commands enables
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the grid to be displayed, and creates appropriate labels for the figure title, y axis, andlegend.
% create a single figure called "Results" with one plot in it% Set the fonts for all labels to 18 pt and use hold on function to% maintain this property while adding various plot elementsResults = figureset(gca,’fontsize’,18); % sets all fonts in plot to 18 pthold on;% The next two lines set the plot position in inches. These can be% adjusted as needed by the user.Results.Units = ’inches’;Results.Position = [3 3 12 6];% Create a plot with both Calculated and Experimental angle resultsplot(t,theta*180/pi,TimeData,AngleData);grid on;xlabel(’Time [s]’);ylabel(’Angle [deg]’);legend(’Calculation’,’Experiment’,’Location’,’NorthEast’);title(’Damped Pendulum Motion’);hold off ; % It’s important to turn the hold function off or the plot
% will not update when the code is run again.
9. SAVE AND EXECUTE THE SCRIPT: Periodically save your work to your USB drive,so that you have a convenient recovery point in the event of a significant error. Aftersaving your completed script, run it by clicking the green arrow with ”Run” under it inthe Editor ribbon tools. Select ”Add to Path” if you get a pop up when first runningthe script.
It is highly unlikely that your script file will run properly the first time you make anattempt. In fact, debugging your script is an essential engineering skill. You will needto hone this skill continually throughout your academic studies and professional career.You may expect this process to be quite frustrating. However, when you finally getthe bugs out of your script and it runs successfully, you will also experience a strongsense of accomplishment.
One of the most important debugging skills that you can develop is an ability toclosely read and understand the error messages that are provided by your programmingsystem. MATLAB provides detailed error messages, but they can be quite confusing tonovice programmers. If you do not understand what an error message means, then youmay find it useful to search for help, using that precise error message as your searchstring. This will often help you locate example scripts or even video examples thatmay help you track down your bug.
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DO NOT DESPAIR! It is natural to get frustrated during debugging. Try to fosterthe mindset that debugging your program is analogous to figuring out a subtle puzzleor problem solution in a video game or homework assignment. Do not try to hurrythrough the process of debugging in order to just “get the script done.” If your classmate corrects a bug in your script for you, and you do not understand what theycorrected and why it worked, then you are depriving yourself of learning the central,most valuable, skill of this entire lesson. The debugging process is by far the mostimportant aspect of this portion of the weekly lesson. Please do not short circuit thedebugging process to save time. You will find that nurturing your debugging skill willsave you countless hours in the future.
After some bug tracking, you will finally get your script to run successfully. This isexcellent! Just as you were able to enter single commands interactively in the previousLab, you can run your entire script as easily as entering a single command! When youexecute the script, you should see a window pop-up, containing your plot, that willlook something like the one shown in Figure 11.6.
Figure 11.6: Results graph of the damped pendulum and calculation. Note the commentsin the plot illustrating how the calculation could be fine-tuned by adjusting the rod lengthand damping coe�cients to obtain better agreement between the two sets of data.
10. FINE-TUNE THE CALCULATION PARAMETERS: Note that the agreement be-tween the calculated and experimental results is not perfect. The peaks for the cal-culation are space slightly further apart than the experimental ones, and the peaksare diminishing in magnitude over time slightly more quickly than the experimentalones. You can fix these things by adjusting the value of the rod length (e.g.,one woulddecrease R to decrease the peak spacing for the case shown in Figure 11.6) and thevalue of the damping coe�cient (e.g. one would decrease ⇠ to correct the behavior seen
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in Figure 11.6). Make small adjustments in these two parameters for you script andrerun it to optimize the values of R and ⇠.
11. OBSERVATIONS AND ANALYSIS: Write responses to the following questions in yourlogbook. Be sure to include a justification for your answer by referring to the data,plots, and derivations that are contained within your logbook. You may want to cross-reference equations from Sections 11.1, 11.2.9 and 11.3.1 in your work.
(a) In your logbook, document your iterative problem-solving and trouble shootingmethod.
(b) Briefly describe how you ultimately found the errors in your simulation script,and resolved them.
(c) Explain not only how you found the correct results, but why you believe yourcalculation results to be accurate.
(d) Compare your predictive simulation results with your experimental results fromlast week. Fully explain the similarities and di↵erences between your experimen-tal observations and your theoretical predictions.
(e) Now you can uses your code to predict future experiments! If a pendulum has alength of 50 meters, and friction coe�cient b = 0.05, and bob mass = 5 kg, predictthe motion of the pendulum. Print out your graphs and paste them in your logbook. Be sure to clearly identify the simulations conditions of these graphs sothey are not confused with your experimental and simulation comparison graphs.
12. SUBMIT YOUR WORK: Remember to remove your USB drive from the computer,and take it with you when you leave the Studio. Save your MATLAB m-file to theUSB drive. You may want this file in the future! Please be sure to sign and date yourengineering logbook before you leave the studio and to submit your MATLAB m-fileto your individual Week 11 Dropbox on myCourses before leaving the room or within24 hrs.
13. CONGRATULATIONS! You have just completed the Studio portion for week 11.
14. WRITE THE REPORT: Please refer to section 11.3.4 Report on details for the reportsubmission. Before leaving Studio, decide on a date and time to meet with your teammates to prepare the report.
11.3.4 Report
Please use the same task distribution for writing the report that was outlined in Week1. This week we have added a theory section, which should be completed
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by the Team Manager. The scribe is responsible for compiling the report, howeverall team members are responsible for ensuring that the report is uploaded correctlyand on time.
Prepare a report to include only the following components:
• TITLE PAGE: Include the title of your experiment, “Damped Pendulum Mo-tion”, Team Number, date, authors, with the scribe first, the team member’srole for the week, and a photograph of each person beginning to initiate theirtrial, with a label below each photo providing team member’s name.
• PAGE 1: The heading should read Theory. In no more than one page, brieflydescribe the theory related to the experiment and simulation. Include importantequations relevant to the lab. Be sure to define every variable in the equations,and include units. The equations should be formatted correctly, with equationnumbers labeling each on the right side of the page. Format the equations in thesame structure as this book.
• PAGE 2: The heading on this page should read Experimental Set-up. Createa diagram of the experimental set-up. We will include only the diagram andits caption. Thus, is it important that your diagram clearly communicate theset-up, including each key component and where measurements were taken. Theimportant information to communicate are the variable names, distances, axisand datums that relate to your measurements and results. It is a good practiceto add a legend that defines any variables or components of the schematic thatare not obvious. At the bottom of the figure include a figure caption, for exampleFigure 1. A brief figure caption. Refer to the text for examples.
Note: Figure captions are required for every plot and diagram in the report,except for the title page. Figure captions are placed below the figures, and arenumbered sequentially beginning with Figure 1 for the first figure in the report.
• PAGE 3: The heading on this page should read Results. Include a table withinitial angle of displacement, damping coe�cient and natural frequency valuesfor each team member as illustrated in Figure 11.7. You should use the valuesfor ⇠ and ! that you obtained by fine-tuning your MATLAB script. Be sure toinclude uncertainty estimates for all results.
Remember that any measured data point or value calculated from measure datahas an uncertainty. At the top of the table, include a table caption, for exampleTable 1. A brief table caption. Refer to the text for examples.
We will include only tables and plots with no accompanying text. Thus, it isimportant that your tables, graphs and captions clearly communicate to thereader what the data represents.
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Figure 11.7: Summary results table format for week 11 lab report.
Note: Table captions are required for every table in the report, except for thetitle page. Unlike figure captions, table captions are placed above the tables, andare numbered sequentially (independent of figure caption numbering) beginningwith Table 1 for the first table in the report.
• PAGE 4: No heading is needed on this page, since it is a continuation of theResults section. Present plots of of angular displacement as a function of time,one plot for each team member. Format the plot according to the guidelinesshown in previous chapters. Arrange the plots so that they are easily compared onto another. Be sure that each plot contains both the theoretical and experimentalresults.
• PAGE 5: The heading on this page should read Conclusions. Here you willstate the major conclusions that can be drawn from this analysis. In otherwords, you will qualitatively and quantitatively answer the questions posed bythe experiment. Consider the following guiding questions when preparing yourconclusion. Do any of your results violate Newton’s Laws or the Work EnergyTheorem, within uncertainty limits? In evaluating your estimates for angularposition, consider if there were any systematic bias present in your results. Whatare the most significant contributors to uncertainty, and how would you mitigatethem? Finally, comment on whether your experimental results support the WorkEnergy Theorem within reasonable uncertainty.
Your conclusion should be NO LONGER than 1/2 a page when typed in 12 ptfont.
• The final report should be collated into one document with page numbers anda consistent formatting style for sections, subsections and captions. Before up-loading the file, you must convert it to a pdf. Non-pdf version files may notappear the same in di↵erent viewers. Be sure to check the pdf file to make sureit appears as you intend.
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11.4 Recitation
Recitation this week will focus on problem solving. Please come prepared, with yourattempts at the homework problem already in your logbooks.
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11.5 Homework Problems
1 Using your experimental data from your pendulum, estimate the local accelera-tion of Earth’s gravity, g. Estimate the uncertainty in this value. Now, comparethis value of g and its uncertainty with the value determined earlier in the coursewith the dropping ball experiment. Which method provides a better estimate ofg? Justify your logic.
2 Using your experimental data from your pendulum, estimate the damping ratio⇠ for your pendulum system. Estimate the uncertainty in this value.
3 Using your experimental data from your pendulum, estimate the natural fre-quency ! for your pendulum system. Estimate the uncertainty in this value.
4 Using your experimental data from your pendulum, estimate the phase angle �0
for your pendulum system. Estimate the uncertainty in this value.
5 Using your computed values of ⇠, ! and �0
and the observed initial disturbance ✓0
predict the angular position of the pendulum bob as a function of time. Estimatethe uncertainty in your prediction.
✓(t) = ✓0
e�(⇠)t cos(!t� �0
) (11.54)
6 A famous pendulum was displayed in the Washington D.C. Smithsonian Institu-tion until 1998. The pendulum had a length of R = 52[ft] and a hollow brassbob weighing about mg ⇡ 240[lbf ]. Estimate the period, T , of oscillation of thispendulum. How many oscillations would this pendulum undergo in a 24[hr] day?Using resources available on-line, http://www.si.edu/Encyclopedia_SI/nmah/pendulum.htm, explain how this pendulum was used to prove experimentally thatthe Earth rotates about its axis.
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