Chapter 13

Week 13. Circular Motion

13.1 Lecture - Circular Motion

In this lesson, we will study rotational motion. In lecture this week, we will solve threeproblems: (Problem A) analysis of the motion of a body in uniform circular motion, and(Problem B) experimental analysis of uniform circular motion, and (Problem C) analysis ofnon-uniform circular motion using Newton’s Laws in an r✓ coordinate system.

During lab we will study the uniform circular motion case. We will verify the results forcentripetal acceleration and centripetal force.

13.1.1 Problem A. Formulate

State the Problem

We are given a mass, m, at the end of 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. As the rod Rrotates about the origin at a constant rate, the mass m moves in a circular arc as illustratedin the schematic diagram. Develop expressions for the position and acceleration of the massas a function of time.

State the Known Information

m = Known [kg] � Mass (13.1)

R = Known [m] � Length of Rod (13.2)�!V tang ⌘ v Known [m/s] � Uniform Tangential Speed of m (13.3)

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State the Desired Information

x(t) = ? [m] ↵ Displacement of m vs. time (13.4)

y(t) = ? [m] ↵ Displacement of m vs. time (13.5)�!a (t) = ? [m/s2] ↵ Acceleration of m vs. time (13.6)

13.1.2 Problem A. Assume

mrod ⇡ 0 [kg] � Neglect mass of rod (13.7)�!F f (t) ⇡ 0 [N ] � Neglect friction (13.8)

13.1.3 Problem A. Chart

A schematic diagram of the system is shown below. At some initial time, the mass m (shownin green) is traveling in a counter-clockwise circular orbit around the origin with constanttangential speed, v. The mass is connected to the mass-less rod of length R which makesan angle � with the x axis. We use two coordinate systems in the schematic diagram. Thefirst coordinate system is a traditional cartesian coordinate system with ı̂ in the positive xdirection, |̂ in the positive y direction, and k̂ in the positive z direction (out of the page).The second coordinate system is a cylindrical coordinate system, with r̂ directed radiallyoutward from the origin, �̂ measured counter-clockwise from the positive x axis, and k̂ inthe positive z direction (out of the page).

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The arc length from the x axis to the mass m is given by R�, where � is measured in[radians]. From trigonometry we can write

x(t) = R cos�(t) [m] x position (13.9)

y(t) = R sin�(t) [m] y position (13.10)

or, we can write

cos�(t) =x(t)

R[�] (13.11)

sin�(t) =y(t)

R[�] (13.12)

After some interval of time the mass will arrive at the new position (shown in red) andcontinue around the circle until it returns back to its initial position (green). A velocityvector diagram is shown below.

From trigonometry we can write

vx(t) = �v sin�(t) [m/s] x velocity (13.13)

vy(t) = v cos�(t) [m/s] y velocity (13.14)

Using Equation 13.12 in Equation 13.13 and Equation 13.11 in Equation 13.14 we can write:

vx(t) = �vy(t)

R[m/s] x velocity (13.15)

vy(t) = vx(t)

R[m/s] y velocity (13.16)

An acceleration vector diagram is shown below.

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Notice that the acceleration is directed radially inward, towards the center of the orbit. Themagnitude of the acceleration is given by

|�!a (t)| =qa2x(t) + a2y(t) [m/s2] acceleration magnitude (13.17)

13.1.4 Problem A. Execute

We take the first derivative of the velocity components given by Equations 13.15 and 13.16to determine the components of the acceleration:

ax(t) =d

dt(vx(t)) =

d

dt

⇣� v

Ry(t)

⌘

= � v

R

d

dt(y(t)) = � v

Rvy(t) [m/s2] x acceleration (13.18)

ay(t) =d

dt(vy(t)) =

d

dt

⇣ v

Rx(t)

⌘

=v

R

d

dt(x(t)) = +

v

Rvx(t) [m/s2] y acceleration (13.19)

Now, use Equation 13.16 in Equation 13.18 and Equation 13.15 in Equation 13.19 to write:

ax(t) = � v

R

v

Rx(t) = � v2

R2

x(t) [m/s2] x acceleration (13.20)

ay(t) = +v

R

⇣� v

R

⌘y(t) = � v2

R2

y(t) [m/s2] y acceleration (13.21)

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We substitute Equations 13.20 and 13.21 into the magnitude of the acceleration as given byEquation 13.17 to get

|�!a (t)| =r

v4

R4

x2(t) +v4

R4

y2(t)

|a| =r

v4

R4

R2 =

rv4

R2

=v2

R[m/s2] (13.22)

ar = �v2

R[m/s2] (13.23)

This acceleration is constant with respect to time and is directed radially inward. As shownin the schematic diagram r̂ is measured radially outward from the origin. In vector form, wewrite the acceleration in cylindrical coordinates as

�!a = �v2

Rr̂ [m/s2] (13.24)

This acceleration, which is a result of the changing direction of motion (at constant speedv), is called the “centripetal acceleration.”

13.1.5 Problem A. Test

The magnitude of the centripetal acceleration is proportional to the square of the tangentialspeed and inversely proportional to the radius of orbit. The centripetal acceleration isdirected radially inward.

13.1.6 Problem B. Formulate

State the Problem

Building upon the analysis presented in Part A, we now wish to conduct an experimentalinvestigation of uniform circular motion. We will connect a pendulum to the end of anarmature, which is rotated at a uniform rotational speed by an electric motor. Given anexperimental observation of the deflection of the pendulum estimate the centripetal acceler-ation experienced by the bob located at the end of the pendulum.

State the Known Information

A schematic diagram of the system is presented in the “Chart” section. The followinginformation is provided for the configuration of the apparatus.

R ⇡ 5.00 [in] � Armature Radius (13.25)

r ⇡ 0.375 [in] � Spherical Pendulum Bob Radius (13.26)

L ⇡ 3.9 [in] � Pendulum Length (13.27)

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We will also assume that the deflection angle, ✓, of the pendulum can be experimentallydetermined in the lab, and that the motor controller pulse count per second, pps [pulses/sec]may be specified. Our motor controller generates count = 3200 [pulses/revolution].

✓ = Measured in Lab [radians] � Pendulum Angle Displacement (13.28)

pps = Measured in Lab [pulses/sec] � Armature Rotational Speed (13.29)

count = 3200 [pulses/rev] � Motor Controller Resolution (13.30)

State the Desired Information

We must estimate the centripetal acceleration experienced by the bob as a function of therotational speed and the geometric dimensions of the apparatus.

|a| = ? [m/s2] ↵ Centripetal acceleration (13.31)

13.1.7 Problem B. Assume

We will neglect all friction e↵ects in the system for this analysis. This includes air frictionand bearing friction (in the armature bearing and the pendulum bearing). We will neglectthe mass of the pendulum rod in comparison to the mass of the pendulum bob. We will onlyqualitatively consider the image distortion in the digital image analysis.

13.1.8 Problem B. Chart

The apparatus that we will use in Lab and analyze here is illustrated in the photographbelow. The armature is the horizontal member from which two opposing pendulums arehung. The armature is rotated by the electric motor. The motor controller delivers aspecified pulses per second to the motor and this results in the rotational speed that thearmature is spinning at. A pendulum is hung from a free-wheeling pivot at each end of thearmature. Two pendulums are used so that the motion is well balanced and does not placeuneven stresses on the motor assembly.

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As the motor spins the armature, the pendulums are swung around. When the motor isat rest, the pendulums hang straight down due to the action of gravity. We will observethat the angular deflection of the pendulum increases as the rotational speed of the motorincreases. We wish to use this observation to experimentally validate the results achievedby theory in Problem A. We will observe the moving pendulum and armature with a digitalcamera. An image taken from the camera is shown below, when the armature is at rest.

As expected the bob hangs directly below the pivot point (due to gravity) when the armatureis at rest. The camera is not perfectly above the pivot point of the pendulum, and somedistortion is evident in the image as a result. When the armature is driven at high rates ofspeed by the motor, the images may become somewhat blurred. As the rotational speed ofthe armature approaches the shutter speed of the camera, the data captured from our videoimages will exhibit higher uncertainty. The nomenclature for this analysis and experimentis shown in the figure below. The armature is of length R and the pendulum is of length L.The pendulum bob deflects through an angle ✓ during motion.

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Now consider the same apparatus as shown in the figure below. From this side view, thereis an apparent length of the armature and the pendulum rod. However, we need to be verycareful in the analysis of video and image data, since our field of view and perspective hasa significant impact on the perceived length of objects.

A top view of the apparatus is illustrated in the next figure. The armature is shown in thisview to be at some angle � relative to the x axis. From the top view, we are observing onlythe projected length of the pendulum L sin ✓.

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The top view also illustrates that the horizontal projection (side view) of the armatureas perceived in the side view is R cos� and the horizontal projection of the pendulum isL sin ✓ cos�! We need to be sure to account for this in our data analysis. The side viewphotograph and the nomenclature previously introduced are shown together in the nextfigure.

This image helps us to understand the relationship between the pendulum length L, theangular deflection ✓, and the horizontal displacement of the bob, �x. We will be able to

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measure the armature radius R and the horizontal deflection �x experimentally. The totalradial distance from the bob to the center of rotation of the armature is thus RTotal = R+�x.

Finally, let’s draw a free body diagram for the pendulum bob. The pendulum bob hastwo forces acting on it. First, the weight of the bob is W = mg directed towards the centerof the Earth. Second, the pendulum rod is in tension, with a force of T . When the systemis in uniform motion, it achieves a form of equilibrium. That is, while there continues to berotational motion, the angle of deflection, ✓, is constant. Thus, the vertical component of thetension T is in static equilibrium with the weight of the bob, mg. The radial component ofthe tension T exerts a force upon the bob which causes it to continuously change directionsin its circular orbit without changing the magnitude of its circumferential speed, v.

13.1.9 Problem B. Execute

Recall the governing equations:

If :X�!

F = 0 Then : �!a = 0 Newton’s 1st Law (13.32)

X�!F =

d(m�!V )

dtNewton’s 2nd Law (13.33)

�!F Action = ��!

F Reaction Newton’s 3rd Law (13.34)�!F g = g ·m # Newton’s Law of Gravity near Earth (13.35)

�E = E2

� E1

= Q1!2

�W1!2

Work Energy Theorem (13.36)

We will employ a cylindrical coordinate system for this problem. In Problem B we analyzethe radial and vertical components of motion of the device. In Problem C we will investigatethe circumferential motion of the device. We used Newton’s third law to develop the freebody diagram for the pendulum bob. In the plane defined by the intersecting lines of thearmature and the pendulum rod, we can write the sum of the forces in the radial, r, andvertical, z, directions as

XFz = +T cos ✓ �mg Vertical Forces (13.37)

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XFr = �T sin ✓ Radial Forces (13.38)

In our experiment, we will observe that the angular displacement ✓ becomes constant withtime when the armature is revolved at constant speed. Thus, the vertical component of theforces are in equilibrium, otherwise the angle would be changing. We thus know that az = 0.Newton’s second law in the vertical direction thus yields:

XFz = 0 = +T cos ✓ �mg = gaz = 0 Vertical Equilibrium (13.39)

T cos ✓ = mg Vertical Equilibrium (13.40)

T =mg

cos ✓Pendulum Rod Tension (13.41)

Now, let’s apply Newton’s second law in the radial direction.X

Fr = �T sin ✓ = mar Radial Motion (13.42)

ar = �T sin ✓

mRadial Acceleration (13.43)

As the pendulum swings through an arc of angle � the bob moves radially inward. Thecentripetal acceleration is due to a change in direction, not a change in speed. Since accel-eration is a vector quantity, then a force must be exerted on the body to achieve the changein direction, even if there is no change in the magnitude of the velocity. This is a clearconfirmation of Newton’s First Law! Now, we substitute the known tension from Equation13.41 in 13.43 to get:

ar = �sin ✓

m

mg

cos ✓Radial Acceleration (13.44)

ar = �g tan ✓ Centripetal Acceleration (13.45)

13.1.10 Problem B. Test

Equation 13.45 provides us with a convenient experimental means to measure the centripetalacceleration of a pendulum at the end of a swinging armature in uniform circular motion.Equation 13.24 provided us with a theoretical expression for the centripetal acceleration asa function of the device geometry and tangential speed. Let’s repeat both equations here forconvenience.

ar = � v2

RTotal

Theory, Equation (13.24)

ar = �g tan ✓ Experiment, Equation (13.45)

The angle ✓ may be determined experimentally. The process to do this is explained in thevideos for the upcoming lab. How can we relate the theoretical prediction of Equation 13.24to data that we can obtain in lab? First, we must note that the radial length referred to in

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Equation 13.24 is the distance from the origin to the center of mass of the bob. Because ofour apparatus design, this radial distance is

RTotal = R + L sin ✓ Total Radial Displacement (13.46)

The tangential speed of the pendulum is given by

v = RTotal! Tangential Speed from Angular Speed (13.47)

The angular speed, !, can be measured from the motor speed, pps:

! = (pps)1

32002⇡ Angular Speed from motor pulses (13.48)

[rad]

[s]=

[pulses]

[s]

[revolution]

[pulses]

[radians]

[revolution]Units

Substitute Equations 13.46, 13.47 and 13.48 into Equation 13.24 to get a theoretical expres-sion for the centripetal acceleration in term of experimentally observable quantities:

ar = �(RTotal!)2

RTotal

= �RTotal(!)2 Theoretical Acceleration (13.49)

ar = �(R + L sin ✓)

✓2⇡

3200(pps)

◆2

Theoretical Acceleration (13.50)

[m]

[s2]= [m]

✓[radians]

[pulses]

[pulses]

[s]

◆2

Units

ar = �g tan ✓ Experimental Acceleration, Eq. (13.45)

[m]

[s2]=

[m]

[s2][�] Units

We will collect data in lab for the motor speed and angular deflection. We will analyze thedata in studio to compare our theory from Equation 13.50 to our experiment using Equation13.45.

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13.2 Lab - Uniform Circular Motion

13.2.1 Scope

This week you will investigate the principles of uniform circular motion by observing andquantifying the motion of a motorized “fly-ball governor” system. You will again use thevideo capture system to make your measurements, along with an additional image analysisprogram. The purpose is to understand the concepts associated with uniform circular motion,especially the idea of centripetal acceleration, and the notion that acceleration sometimescauses changes in the direction of velocity, if not the magnitude.

13.2.2 Goal

The goals of this laboratory experiment are to

1. learn how to use a video capture system for image analysis, and

2. begin to understand the physics of uniform circular motion.

13.2.3 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.

Quantity Basic units Derived units

Length [m] [m]Mass [kg] [kg]Time [s] [s]

Velocity [m/s] [m/s]Force [kgm/s2] [N ]Angle [radians] [radians]

Frequency [radians/s] [radians/s]Period [s] [s]

Image Dimension [pixels] [pixels]

Table 13.1: Units of Measurement to be used for uniform circular motion system.

13.2.4 Reference Documents

Refer to the lab videos for this week for all information regarding the proper installation anduse of all equipment and software resources.

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13.2.5 Terminology

The following terms must be fully understood in order to achieve the educational objectivesof this laboratory experiment.

Displacement Velocity SpeedAcceleration Radial TangentialCentripetal Circumferential PendulumPeriod Angle FrequencyBearing Radians DegreesPixels Rotational Speed PulseMotor Controller Count

13.2.6 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.

13.2.7 Calibration and Standardization

By now in this course, students should be in a position to conduct independent calibra-tions of hardware, and properly configure the use of all hardware, without having detailedinstructions.

The motor controller provided by the vendor sends a user-specified number of pulses persecond to the motor, resulting in circular motion. As the number of pulses per second isincreased, the rotational speed of the motor increases. We will treat the motor controller asa primary instrument, and thus will not calibrate it.

The information collected from the images will contain relative dimensions in units ofpixels. Based upon the process developed in the companion videos for the experiment, youare expected to develop your own means of calibrating the pixel dimensions to real, physicalunits of length.

13.2.8 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. Referto previous figures in this chapter for illustrations of the basic experimental apparatus,schematics, and definitions of terms used.

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13.2.9 Measurement Uncertainty

The motor controller includes a feedback control system by which one complete revolution ofthe shaft is broken into a finite number of integer “counts.” These counts may be thought ofin a manner similar to the encoder that we studied in the previous lab. Treating the motorcontroller as a primary instrument, the instrument least count of the motor controller usedin lab this week is ILCmotor = 360[�]/3200[counts] ⇡ 0.1125[degrees]. The uncertainty ofthe motor controller as a primary instrument is then taken to be

✏motor = ±1

2ILCmotor ⇡ ±1

20.1125 [degrees] ⇡ ±0.056 [degrees] (13.51)

Also, just as you are to develop your own means of calibrating the pixel dimensions, youare to consider how that calibration a↵ects the uncertainties in the corresponding physicaldimensions. Consider the blur that is evident in many of the images, and how many pixelsof variation that it may cause. You should also consider, at least somewhat quantitatively,the e↵ects of perspective on the measurements. For this, look at the set of images acquiredwhile the system was stationary.

13.2.10 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.

13.2.11 Sampling, Test Specimens

The basic apparatus for the fly-ball governor system is fixed, and only the rotational speedcan be changed. Every group member should operate the system at two di↵erent speedsand record a sequence of images as appropriate to complete the subsequent analyses. Asalways, it is recommended that several trials be conducted at each unique setting to ensurethat valid data sets are obtained.

13.2.12 Procedure - Lab Portion

Record all observations and notes about your lab experiment inyour logbook.

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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.

• 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.

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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.

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.

Be sure you have measured and recorded values for the armature length, the pendulumrod length, and the mass of the bob before leaving the lab.

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13.3 Studio - Matlab Simulation of Uniform CircularMotion

This week in Studio you will complete two tasks. (1) The first task will be a fun enrichmentexercise, where you will make an animation of uniform circular motion in MATLAB. Usingcode that has been written for you, you will create a baseline animation, from which youcan alter the code to see the e↵ect on the movie. (2) The second task involves comparingthe centripetal acceleration you measure in lab to the theoretical value. The theoreticalacceleration will require an experimental estimate of the angle theta that the bob rises fromthe horizontal. Note that theta will be di↵erent for each of your angular speeds. You willcreate a plot of centripetal acceleration versus angular velocity. This plot will include boththe theoretical value and the experimental value. You may use either Excel or MATLAB todo your analysis for Task 2. Since the plot will contain only 3 points, one for each rotationalspeed you tested, it is our recommendation that you use Excel for Task 2. You will evaluatehow well the experiment compares to theory and write up this analysis in your your weeklyreport.

13.3.1 Calculation and Interpretation of Results

The equations needed for Studio this week were derived in the lecture portion of the text. Be-low is a brief summary of the key relationships that are needed for this analysis of centripetalacceleration.

ar = �g tan ✓ Experimental Acceleration (13.52)

[m]

[s2]=

[m]

[s2][�] Units

ar = �(RTotal!)2

RTotal

= �RTotal(!)2 Theoretical Acceleration (13.53)

[m]

[s2]= [m]

[rad]

[s2][�] Units

! = (pps)1

32002⇡ Speed From Motor Pulses (13.54)

[rad]

[s]=

[pulses]

[s]

[revolution]

[pulses]

[radians]

[revolution]Units

RTotal = R + L sin ✓ Total Radial Displacement (13.55)

[m] = [m] + [m][�] Units

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13.3.2 Procedure - Studio Portion

Videos

As discussed previously, MATLAB o↵ers numerous training videos that can assist users withunderstanding how to accomplish certain tasks in MATLAB. There are two videos that youmay find particularly useful for Studio this week. You may access these videos from the Helpmenu within MATLAB, or at the MATLAB Youtube channel.

Please view the video “Using basic plotting functions.”Please view the video “Creating basic plot interactively.”

Studio Pre-work

Prior to arriving at Studio, each student should have acquired the necessary data in lab,recorded data in your notebook and stored data on a thumb drive. You should also have acorresponding schematic that clearly identifies where each measurement was made in sym-bolic notation. You will need to know the radius length, R, the rod length L, and the angle✓ for each motor angular speed.

Please complete at least steps 1-6 of Task 1 prior to coming to Studio. Youmay type your script into a text file using a simple text editor (such as Notepad) if youdon’t have MATLAB on your own computer. Note that the software can be downloaded forfree following the instructions on the home page of our myCourses site. You will need toeventually run the code in MATLAB to debug it. You will receive a quiz grade based on thecompleteness of your submission.

Task 1: Animation of Uniform Circular Motion - Studio Pre-Work

1. CREATE A SCRIPT FILE: Please complete this step as part of your pre-work beforearriving at Studio, so that we can spend time together doing debugging. If you havenot chosen to purchase a student license for MATLAB, Steps 1 through 6 may becompleted in a text editor such as Notepad. If you have purchased a student versionof MATLAB, or if you wish to use the open access Studio hours, you may completeSteps 1 through 6 directly in MATLAB.

If you are using MATLAB, then from within the MATLAB environment, use the pull-down command to execute “File - New - Script.” After the script editor window opens,use the pull down command to execute “File - Save As” and save this file to your thumbdrive, in a folder named studio12 and a file named Lastname uniform circular motion.m.

If you are using Notepad, then from within the Notepad editor, use the pull-downcommand to execute “File - Save As” and save this file to your thumb drive, in a foldernamed studio12 and a file named Lastname uniform circular motion.m.

2. CREATE A TITLE BLOCK: Please complete this step as part of your pre-work be-fore arriving at Studio. In the editor window, create a title block for your computer

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simulation program. In the ME department, your title block should always includethe name of the author, the academic term, the name of the course, and a descriptivetitle. You may add comments to explain the use and limitations of your scripts. Overyour career, as you develop a library of scripts, this will be an important way for youto build upon previous knowledge. In MATLAB, the special character % is used toindicate that any text on the current line to the right of the character is considered acomment, and is not considered a command to MATLAB. It is good practice to starteach script with commands to clear all variables from workspace memory, and to clearthe contents of the command window.

% Author: Edward Hensel Fall 213-1% 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 workspace, removing old junk from memoryclc % clear command window

3. CREATE A LIST OF KNOWN INFORMATION: Please complete this step as partof your pre-work before arriving at Studio. In the editor window, create a numberof scalar variables that contain known information for numerical simulation. Theseexpressions should look similar to those shown below, but should be replaced withnumerical values appropriate for your experiment. By using the same table of constantsas your experiment, you will be able to compare your simulated oscillation plots withyour 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.

% Set the known information and simulation parameters% for the simulation

%%%%%%%%%%%%%%%%%% GEOMETRY %%%%%%%%%%%%%%%%%%%RTotal = 0.20 ; % length of rodRadius = 0.015 ; % radius of bob

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4. CREATE A LIST OF SIMULATION PARAMETERS: Please complete this step aspart of your pre-work before arriving at Studio. Add the following lines of code to yourscript in the editor window. The simulation parameters have to do with making themovie. We will not go into all the detail about each line of code. As you enter thiscode, please make sure you understand what you are typing. List any questions thatyou have about the script in your logbook, so that we can review them during Studio.Later, during Studio when you have the simulation running, you can experiment andvary the parameters to see their a↵ect on your movie.

Num_Angle = 100 ; % [frames] to be used in the movie

%%%%%%%%%%%%%%%%%% PLOT AREA %%%%%%%%%%%%%%%%%%%

Diameter = 2.0 * Radius;xmax = RTotal + Diameter ;ymax = xmax;xmin = -xmax;ymin = -ymax;xlen = xmax - xmin ;ylen = ymax - ymin ;

%%%%%%%%%%%%%%%%%% VIDEO SETUP %%%%%%%%%%%%%%%%

% setup a plotting area to make a video in MATLABwriterObj = VideoWriter(’LastnameCircularMotion.avi’);writerObj.FrameRate = 10;open(writerObj);figure(1);set(gca,’nextplot’,’replacechildren’);set(gcf,’Renderer’,’zbuffer’);% define the limits of the plotting areaaxis( [xmin xmax ymin ymax] );axis ( ’off’ );% capture the current image from the screenframe=getframe;% use the current image to the begin the videowriteVideo(writerObj,frame);

Be sure to use your own “Lastname” in the VideoWriter command.

5. LOOP OVER ALL TIME STEPS: Please complete this step as part of your pre-workbefore arriving at Studio. In the editor window, we will create a “For” loop. The “For”

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loop in MATLAB is a programming construct that easily allows us to compute veryrepetitive information. In the example below, the “For” loop uses an index (or timestep counting) scalar variable named Nt, which will vary from 2 to Num Angle witha default increment of 1. Every command between the For statement and the endstatement will be executed in sequence each time the time step is incremented. Whatwe are doing here is essentially creating a cartoon.

% Loop over each angle step, Using the simulation equations% Move the device through the angles of one period of revolutionfor Nt = 0:Num_Angle

% paint the background of the screen in a uniform color% to begin with a blank canvas for the current imagerectangle(’Position’,[xmin,ymin,xlen,ylen],...

’Curvature’,[1,1],’FaceColor’,’w’,...’LineStyle’,’none’);

% compute the angle of the pendulum at this point in rotationTheta = 2.0 * pi * Nt / Num_Angle ;% compute the position of the end of the pendulum rodX = RTotal * cos ( Theta );Y = RTotal * sin ( Theta );% coordinates of the rod, from origin to rod-endXLine = [ 0.0, X ] ; % From the origin to XYLine = [ 0.0, Y ] ; % From the origin to Y% draw pendulum rod on teh canvasline(’XData’, XLine, ’YData’, YLine, ’LineWidth’,3 );% draw the pendulum bob using a curved rectangleX1 = X - Radius; % Lower x corner of the bobY1 = Y - Radius; % Lower y corner of the bobrectangle(’Position’,[X1,Y1,Diameter,Diameter],...

’Curvature’,[1,1],’FaceColor’,’b’,...’LineStyle’,’none’);

% capture the current image from the screenframe=getframe;% append the current image to the end of the videowriteVideo(writerObj,frame);

end

% close the video fileclose(writerObj);

6. EXECUTE AND DEBUG THE SCRIPT:

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Once you are done typing the code in an m file, run it by either pressing the greenarrow in the ribbon of the editor window or typing the m file’s name in the CommandWindow.

When you execute the script successfully, you should see a window pop-up, containingyour movie.

Do not expect your script to execute properly the first time. It is nearly inevitablethat you will need to do some debugging to correct errors in your script. This troubleshooting process is a normal part of programming and is a powerful engineering skillto develop. Periodically, as you work, save your script file to your USB drive, so thatyou have a convenient recovery point in the event of a significant error.

Remember to use the information provided in the error messages in the CommandWindow to guide you debugging process. Also, it is helpful to click on variables in theeditor window to see if they are highlighted to help identify typos. If you get stuck andcan’t determine why the code won’t run, simply upload your script file to the weeklyDropbox. We will spend the first few minutes of Studio reviewing this script and canhelp with any final debugging then.

7. OBSERVATIONS AND ANALYSIS: After you have your movie working, experimentwith code to determine how to make the rod longer, thinner or the bob larger, orwhatever you would like to do. Adjust your simulation script to produce a visuallyappealing rendering of the uniform circular motion experiment.

8. SUBMIT YOUR WORK: After you have completed the movie and made it visuallyappealing, please upload (a) your debugged MATLAB script m-file and (b) your movieanimation file to the myCourses dropbox.

Task 2: Analysis of Angular Acceleration - Studio Work

9. GATHER IMAGE ANALYSIS INFORMATION: If you did not already do this in lab,you need to perform image analysis on your video images that were collected in labto determine the relative positions of the governor fly-balls at your di↵erent rotationalspeeds. Watch the lab video titled “Uniform Circular Motion Image Analysis” for in-structions on how to do this. You will also need to download the appropriate LabVIEWVI from the myCourses site. Record all the necessary information in your logbook.

10. CENTRIPETAL ACCELERATION FROM EXPERIMENT: Using your experimentaldata, program Excel (or MATLAB) to determine the centripetal acceleration based onyour experimental data. Be sure to apply the high level formatting used throughoutthe course to make your spreadsheet or code readable.

11. CENTRIPETAL ACCELERATION FROM THEORY: Using Excel (or MATLAB)calculate the predicted centripetal acceleration for your measured data.

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12. PLOT: Compare theory and experiment on and xy plot using proper plotting formatsand documentation you have demonstrated in previous analysis in this class. You mayprepare your plot either in Excel or MATLAB, using the tool that you determine tobe most convenient.

13. 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 12.1, 13.3.1 and 13.2.9 in your work.

(a) In your logbook, compare your predictive simulation results with your experi-mental results. Fully explain the similarities and di↵erences between your exper-imental observations and your theoretical predictions.

(b) Discuss the uncertainty in your calculated and theoretical values.

(c) How does your experiment demonstrate the validity of Newton’s 2nd Law?

14. SUBMIT YOUR WORK: Remember to remove your USB drive from the computer,and take it with you when you leave the Studio. Save your analysis files to the USBdrive. 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 work to yourindividual Week 12 dropbox on myCourses before leaving the room, or within 24 hrs.

15. CONGRATULATIONS! You have just completed the Studio portion for week 12.

16. WRITE THE REPORT: Please refer to section 13.3.3 Report on details for the reportsubmission. Before leaving Studio, decide on a date and time to meet with your teammates to prepare the report.

13.3.3 Report

Please use the same task distribution for writing the report that was outlined in Week 1.This week we have added a theory section, which should be completed by the Team Manager.The scribe is responsible for compiling the report, however all team members are responsiblefor ensuring that the report is uploaded correctly and on time.

Prepare a report to include only the following components:

• TITLE PAGE: Include the title of your experiment, “Uniform Circular Motion”,Team Number, date, authors, with the scribe first, the team member’s role for theweek, and a photograph of each person beginning to initiate their trial, with a labelbelow each photo providing team member’s name.

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• PAGE 1: The heading should read Theory. In no more than one page, describe thetheory related to the experiment and simulation. Be sure to define every variable inthe equations, with units and include equation numbers.

• PAGE 2: The heading on this page should read Experimental Set-up. Create adiagram of the experimental set-up. We will include only the diagram and its caption.Thus, is it important that your diagram clearly communicate the set-up, includingeach key component and where measurements were taken. The important informationto communicate are the variable names, distances, axis and datums that relate to yourmeasurements and results. It is a good practice to add a legend that defines anyvariables or components of the schematic that are not obvious. At the bottom of thefigure include a figure caption, for example Figure 1. A brief figure caption. Referto the text for examples.

Note: Figure captions are required for every plot and diagram in the report, exceptfor the title page. Figure captions are placed below the figures, and are numberedsequentially beginning with Figure 1 for the first figure in the report.

• PAGE 3: The heading on this page should read Results. Include the table withexperimental data for each team member, for all cases measured; motor speed, an-gular velocity, angle that the bob rises from the vertical, theoretical acceleration andexperimental acceleration.

Remember that any measured data point or value calculated from measure data hasan uncertainty. At the top of the table, include a table caption, for example Table 1.A brief table caption. Refer to the text for examples.

We will include only tables and plots with no accompanying text. Thus, it is importantthat your tables, graphs and captions clearly communicate to the reader what the datarepresents.

Note: Table captions are required for every table in the report, except for the title page.Unlike figure captions, table captions are placed above the tables, and are numberedsequentially (independent of figure caption numbering) beginning with Table 1 for thefirst table in the report.

• PAGE 4: No heading is needed on this page, since it is a continuation of the Resultssection. On a single page, include plots of centripetal acceleration as a function ofangular speed. Put all team member’s data on a single page if possible. Format theplots according to the guidelines shown in previous chapters. Arrange and format theplots so that they are easily compared one to another.

• PAGE 5: The heading on this page should read Conclusions. Here you will statethe major conclusions that can be drawn from this analysis. In other words, youwill qualitatively and quantitatively answer the questions posed by the experiment.Consider the following guiding questions when preparing your conclusion. Do any of

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your results violate Newton’s Laws or the Work Energy Theorem, within uncertaintylimits? In evaluating your estimates for angular velocity, consider if there were anysystematic bias present in your results. What are the most significant contributors touncertainty, and how would you mitigate them? Finally, comment on whether yourexperimental results support the Newton’s second law.

Your conclusion should be NO LONGER than 1/2 a page when typed in 12 pt font.

• The final report should be collated into one document with page numbers and a con-sistent formatting style for sections, subsections and captions. Before uploading thefile, you must convert it to a pdf. Non-pdf version files may not appear the same indi↵erent viewers. Be sure to check the pdf file to make sure it appears as you intend.

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13.4 Recitation

Recitation this week will focus on problem solving. Please come prepared, with your attemptsat the homework problem already in your logbooks.

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13.5 Homework Problems

1. Consider the mass m moving in uniform circular motion with speed v in orbit aboutthe origin as shown in the figure below. At what value(s) of � is the vertical componentof the position vector, ry, greatest in magnitude? At what value(s) of � is the verticalcomponent of velocity, vy, the greatest in magnitude? At what value(s) of � is thevertical component of acceleration, ay, the greatest in magnitude?

2. A vehicle with a weight of W = 3, 500 [lbf ] travels at constant speed v [mph] arounda flat (not inclined) circular track of radius r = 1, 000 [ft]. The static coe�cient offriction between the tires and the track is µ = 0.8[�]. What is the maximum speedthat the vehicle can travel at without losing control? What will happen to the vehiclewhen the limit is exceeded?

3. Use on-line resources to determine estimates for the nominal mass of the Earth andit’s moon, Luna. Similarly, locate an estimate for the mean distance between Earthand Luna. Use your understanding of uniform circular motion to explain the orbit ofthe moon around the planet. Estimate the circumferential speed of the moon in it’sorbit. Estimate the centripetal acceleration of the moon.

4. A satellite is in geosynchronous orbit (the satellite remains above a single point onthe Earth) about the Earth, centered above a point on the equator. The satellite hasa mass of 1000[kg]. Use your understanding of uniform circular motion to explainthe orbit of the satellite around the planet. Estimate the circumferential speed ofthe satellite in it’s orbit. Estimate the centripetal acceleration of the satellite. Whatdistance must the satellite orbit at, as measured from the surface of the Earth?

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