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 three
problems: (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 of
non-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 for
centripetal 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 a
bearing assembly such that it is able to pivot about a point at the origin. As the rod R
rotates about the origin at a constant rate, the mass m moves in a circular arc as illustrated
in the schematic diagram. Develop expressions for the position and acceleration of the mass
as a function of time.
State the Known Information
m = Known [kg]
R = Known [m]
!
V tang ⌘ v Known [m/s]
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Mass
Length of Rod
(13.1)
(13.2)
Uniform Tangential Speed of m
(13.3)
State the Desired Information
x(t) = ?
y(t) = ?
!
a (t) = ?
13.1.2
↵
↵
↵
Displacement of m vs. time
Displacement of m vs. time
Acceleration of m vs. time
(13.4)
(13.5)
(13.6)
[kg]
Neglect mass of rod
(13.7)
[N ]
Neglect friction
(13.8)
Problem A. Assume
mrod ⇡ 0
!
F f (t) ⇡ 0
13.1.3
[m]
[m]
[m/s2 ]
Problem A. Chart
A schematic diagram of the system is shown below. At some initial time, the mass m (shown
in green) is traveling in a counter-clockwise circular orbit around the origin with constant
tangential speed, v. The mass is connected to the mass-less rod of length R which makes
an angle with the x axis. We use two coordinate systems in the schematic diagram. The
first coordinate system is a traditional cartesian coordinate system with ı̂ in the positive x
direction, |ˆ 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 radially
outward from the origin, ˆ measured counter-clockwise from the positive x axis, and k̂ in
the 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
[radians]. From trigonometry we can write
x(t) = R cos (t)
y(t) = R sin (t)
[m] x position
[m] y position
is measured in
(13.9)
(13.10)
or, we can write
x(t)
R
y(t)
sin (t) =
R
cos (t) =
[ ]
(13.11)
[ ]
(13.12)
After some interval of time the mass will arrive at the new position (shown in red) and
continue around the circle until it returns back to its initial position (green). A velocity
vector diagram is shown below.
From trigonometry we can write
vx (t) = v sin (t)
vy (t) = v cos (t)
[m/s] x velocity
[m/s] y velocity
(13.13)
(13.14)
Using Equation 13.12 in Equation 13.13 and Equation 13.11 in Equation 13.14 we can write:
y(t)
R
x(t)
vy (t) = v
R
An acceleration vector diagram is shown below.
vx (t) =
v
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[m/s] x velocity
(13.15)
[m/s] y velocity
(13.16)
Notice that the acceleration is directed radially inward, towards the center of the orbit. The
magnitude of the acceleration is given by
q
!
| a (t)| = a2x (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 13.16
to determine the components of the acceleration:
⌘
d
d ⇣ v
(vx (t)) =
y(t)
dt
dt
R
v d
v
=
(y(t)) =
vy (t) [m/s2 ]
R dt
R
⌘
d
d ⇣v
ay (t) =
(vy (t)) =
x(t)
dt
dt R
v d
v
=
(x(t)) = + vx (t) [m/s2 ]
R dt
R
ax (t) =
x acceleration
(13.18)
y acceleration
(13.19)
Now, use Equation 13.16 in Equation 13.18 and Equation 13.15 in Equation 13.19 to write:
v v
v2
x(t) =
x(t) [m/s2 ]
RR
R2
v ⇣ v⌘
v2
ay (t) = +
y(t) =
y(t) [m/s2 ]
R
R
R2
ax (t) =
406
x acceleration
(13.20)
y acceleration
(13.21)
We substitute Equations 13.20 and 13.21 into the magnitude of the acceleration as given by
Equation 13.17 to get
r
v4 2
v4 2
|!
a (t)| =
x
(t)
+
y (t)
4
4
R
R
r
r
v4 2
v4
v2
|a| =
R
=
=
[m/s2 ]
(13.22)
4
2
R
R
R
v2
ar =
[m/s2 ]
(13.23)
R
This acceleration is constant with respect to time and is directed radially inward. As shown
in the schematic diagram r̂ is measured radially outward from the origin. In vector form, we
write the acceleration in cylindrical coordinates as
!
a =
v2
r̂
R
[m/s2 ]
(13.24)
This acceleration, which is a result of the changing direction of motion (at constant speed
v), is called the “centripetal acceleration.”
13.1.5
Problem A. Test
The magnitude of the centripetal acceleration is proportional to the square of the tangential
speed and inversely proportional to the radius of orbit. The centripetal acceleration is
directed 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 experimental
investigation of uniform circular motion. We will connect a pendulum to the end of an
armature, which is rotated at a uniform rotational speed by an electric motor. Given an
experimental observation of the deflection of the pendulum estimate the centripetal acceleration 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 following
information is provided for the configuration of the apparatus.
R ⇡ 5.00
r ⇡ 0.375
L ⇡ 3.9
[in]
[in]
[in]
Armature Radius
Spherical Pendulum Bob Radius
Pendulum Length
407
(13.25)
(13.26)
(13.27)
We will also assume that the deflection angle, ✓, of the pendulum can be experimentally
determined 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]
pps = Measured in Lab [pulses/sec]
count = 3200
[pulses/rev]
Pendulum Angle Displacement (13.28)
Armature Rotational Speed (13.29)
Motor Controller Resolution (13.30)
State the Desired Information
We must estimate the centripetal acceleration experienced by the bob as a function of the
rotational speed and the geometric dimensions of the apparatus.
|a| = ?
13.1.7
[m/s2 ]
↵
Centripetal acceleration
(13.31)
Problem B. Assume
We will neglect all friction e↵ects in the system for this analysis. This includes air friction
and bearing friction (in the armature bearing and the pendulum bearing). We will neglect
the mass of the pendulum rod in comparison to the mass of the pendulum bob. We will only
qualitatively 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 photograph
below. The armature is the horizontal member from which two opposing pendulums are
hung. The armature is rotated by the electric motor. The motor controller delivers a
specified pulses per second to the motor and this results in the rotational speed that the
armature is spinning at. A pendulum is hung from a free-wheeling pivot at each end of the
armature. Two pendulums are used so that the motion is well balanced and does not place
uneven stresses on the motor assembly.
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As the motor spins the armature, the pendulums are swung around. When the motor is
at rest, the pendulums hang straight down due to the action of gravity. We will observe
that the angular deflection of the pendulum increases as the rotational speed of the motor
increases. We wish to use this observation to experimentally validate the results achieved
by theory in Problem A. We will observe the moving pendulum and armature with a digital
camera. 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 armature
is at rest. The camera is not perfectly above the pivot point of the pendulum, and some
distortion is evident in the image as a result. When the armature is driven at high rates of
speed by the motor, the images may become somewhat blurred. As the rotational speed of
the armature approaches the shutter speed of the camera, the data captured from our video
images will exhibit higher uncertainty. The nomenclature for this analysis and experiment
is 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, there
is an apparent length of the armature and the pendulum rod. However, we need to be very
careful in the analysis of video and image data, since our field of view and perspective has
a 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 this
view to be at some angle relative to the x axis. From the top view, we are observing only
the projected length of the pendulum L sin ✓.
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The top view also illustrates that the horizontal projection (side view) of the armature
as perceived in the side view is R cos and the horizontal projection of the pendulum is
L sin ✓ cos ! We need to be sure to account for this in our data analysis. The side view
photograph and the nomenclature previously introduced are shown together in the next
figure.
This image helps us to understand the relationship between the pendulum length L, the
angular 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 total
radial distance from the bob to the center of rotation of the armature is thus RT otal = R+ x.
Finally, let’s draw a free body diagram for the pendulum bob. The pendulum bob has
two forces acting on it. First, the weight of the bob is W = mg directed towards the center
of the Earth. Second, the pendulum rod is in tension, with a force of T . When the system
is in uniform motion, it achieves a form of equilibrium. That is, while there continues to be
rotational motion, the angle of deflection, ✓, is constant. Thus, the vertical component of the
tension T is in static equilibrium with the weight of the bob, mg. The radial component of
the tension T exerts a force upon the bob which causes it to continuously change directions
in its circular orbit without changing the magnitude of its circumferential speed, v.
13.1.9
Problem B. Execute
Recall the governing equations:
X!
If :
F = 0 T hen : !
a =0
!
X ! d(m V )
F =
dt
!
!
F Action = F Reaction
!
Fg =g·m#
E = E2 E1 = Q1!2 W1!2
Newton’s 1st Law
(13.32)
Newton’s 2nd Law
(13.33)
Newton’s 3rd Law
(13.34)
Newton’s Law of Gravity near Earth
Work Energy Theorem
(13.35)
(13.36)
We will employ a cylindrical coordinate system for this problem. In Problem B we analyze
the radial and vertical components of motion of the device. In Problem C we will investigate
the circumferential motion of the device. We used Newton’s third law to develop the free
body diagram for the pendulum bob. In the plane defined by the intersecting lines of the
armature and the pendulum rod, we can write the sum of the forces in the radial, r, and
vertical, z, directions as
X
Fz = +T cos ✓ mg
Vertical Forces
(13.37)
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X
Fr =
T sin ✓
Radial Forces
(13.38)
In our experiment, we will observe that the angular displacement ✓ becomes constant with
time when the armature is revolved at constant speed. Thus, the vertical component of the
forces 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:
X
Fz = 0 = +T cos ✓ mg = gaz = 0
Vertical Equilibrium
(13.39)
T cos ✓ = mg
mg
T =
cos ✓
Vertical Equilibrium
(13.40)
Pendulum Rod Tension
(13.41)
Now, let’s apply Newton’s second law in the radial direction.
X
Fr = T sin ✓ = mar
Radial Motion
ar =
T sin ✓
m
Radial Acceleration
(13.42)
(13.43)
As the pendulum swings through an arc of angle the bob moves radially inward. The
centripetal acceleration is due to a change in direction, not a change in speed. Since acceleration is a vector quantity, then a force must be exerted on the body to achieve the change
in direction, even if there is no change in the magnitude of the velocity. This is a clear
confirmation of Newton’s First Law! Now, we substitute the known tension from Equation
13.41 in 13.43 to get:
ar =
ar =
13.1.10
sin ✓ mg
m cos ✓
g tan ✓
Radial Acceleration
(13.44)
Centripetal Acceleration
(13.45)
Problem B. Test
Equation 13.45 provides us with a convenient experimental means to measure the centripetal
acceleration 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 as
a function of the device geometry and tangential speed. Let’s repeat both equations here for
convenience.
ar =
ar =
v2
RT otal
g tan ✓
Theory, Equation (13.24)
Experiment, Equation (13.45)
The angle ✓ may be determined experimentally. The process to do this is explained in the
videos for the upcoming lab. How can we relate the theoretical prediction of Equation 13.24
to 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 of
our apparatus design, this radial distance is
RT otal = R + L sin ✓
Total Radial Displacement
(13.46)
The tangential speed of the pendulum is given by
v = RT otal !
Tangential Speed from Angular Speed
(13.47)
The angular speed, !, can be measured from the motor speed, pps:
1
2⇡
3200
[rad]
[pulses] [revolution] [radians]
=
[s]
[s]
[pulses] [revolution]
! = (pps)
Angular Speed from motor pulses
(13.48)
Units
Substitute Equations 13.46, 13.47 and 13.48 into Equation 13.24 to get a theoretical expression for the centripetal acceleration in term of experimentally observable quantities:
ar
[m]
[s2 ]
ar
[m]
[s2 ]
(RT otal !)2
=
RT otal
RT otal (!)2
✓
◆2
2⇡
= (R + L sin ✓)
(pps)
3200
✓
◆2
[radians] [pulses]
= [m]
[pulses]
[s]
= g tan ✓
[m]
= 2 [ ]
[s ]
ar =
Theoretical Acceleration
(13.49)
Theoretical Acceleration
(13.50)
Units
Experimental Acceleration, Eq. (13.45)
Units
We will collect data in lab for the motor speed and angular deflection. We will analyze the
data in studio to compare our theory from Equation 13.50 to our experiment using Equation
13.45.
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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 and
quantifying the motion of a motorized “fly-ball governor” system. You will again use the
video capture system to make your measurements, along with an additional image analysis
program. The purpose is to understand the concepts associated with uniform circular motion,
especially the idea of centripetal acceleration, and the notion that acceleration sometimes
causes 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 a
variety of units.
Quantity
Length
Mass
Time
Velocity
Force
Angle
Frequency
Period
Image Dimension
Basic units
[m]
[kg]
[s]
[m/s]
[kgm/s2 ]
[radians]
[radians/s]
[s]
[pixels]
Derived units
[m]
[kg]
[s]
[m/s]
[N ]
[radians]
[radians/s]
[s]
[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 and
use of all equipment and software resources.
415
13.2.5
Terminology
The following terms must be fully understood in order to achieve the educational objectives
of this laboratory experiment.
Displacement Velocity
Speed
Acceleration Radial
Tangential
Centripetal
Circumferential
Pendulum
Period
Angle
Frequency
Bearing
Radians
Degrees
Pixels
Rotational Speed Pulse
Motor
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 YouTube
for this week’s exercise. Watch all of the available videos, and complete the online lab quiz
for the week. The videos are your best reference for the specific tasks and procedures to
follow 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 calibrations of hardware, and properly configure the use of all hardware, without having detailed
instructions.
The motor controller provided by the vendor sends a user-specified number of pulses per
second to the motor, resulting in circular motion. As the number of pulses per second is
increased, the rotational speed of the motor increases. We will treat the motor controller as
a primary instrument, and thus will not calibrate it.
The information collected from the images will contain relative dimensions in units of
pixels. Based upon the process developed in the companion videos for the experiment, you
are expected to develop your own means of calibrating the pixel dimensions to real, physical
units of length.
13.2.8
Apparatus
All required apparatus and equipment components are described and demonstrated in the
instructional videos for this exercise, or will be familiar from common or previous use. Refer
to previous figures in this chapter for illustrations of the basic experimental apparatus,
schematics, and definitions of terms used.
416
13.2.9
Measurement Uncertainty
The motor controller includes a feedback control system by which one complete revolution of
the shaft is broken into a finite number of integer “counts.” These counts may be thought of
in a manner similar to the encoder that we studied in the previous lab. Treating the motor
controller as a primary instrument, the instrument least count of the motor controller used
in lab this week is ILCmotor = 360[ ]/3200[counts] ⇡ 0.1125[degrees]. The uncertainty of
the motor controller as a primary instrument is then taken to be
1
1
✏motor = ± ILCmotor ⇡ ± 0.1125 [degrees] ⇡ ±0.056 [degrees]
2
2
(13.51)
Also, just as you are to develop your own means of calibrating the pixel dimensions, you
are to consider how that calibration a↵ects the uncertainties in the corresponding physical
dimensions. Consider the blur that is evident in many of the images, and how many pixels
of 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 acquired
while the system was stationary.
13.2.10
Preparation of Apparatus
All required equipment for conducting the laboratory exercise is made available either within
one or both of the drawers attached to the lab bench, or available from the laboratory instructor. You are expected to bring all other necessary materials, particularly your logbook and a
flash drive for storing electronic data as appropriate. You are to follow the general specifications for team roles within the lab. Although there are specific, individual expectations for
each role, you are each responsible overall to ensure that the objectives and requirements of
the 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 be
returned 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 speed
can be changed. Every group member should operate the system at two di↵erent speeds
and record a sequence of images as appropriate to complete the subsequent analyses. As
always, it is recommended that several trials be conducted at each unique setting to ensure
that valid data sets are obtained.
13.2.12
Procedure - Lab Portion
Record all observations and notes about your lab experiment in
your logbook.
417
The instructional videos for this exercise cover the specific procedures to follow as you set
up the apparatus to make measurements, and for actually collecting data with the various
devices and software interfaces. More generally, you should always observe the following
general 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 associated 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 notes
and 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 equipment
used, the actual data collected and/or notes related to the data collected (if done
electronically for instance), and any other information relevant to the reporting and
analysis of the data and understanding of the exercise itself.
5. All students should create and complete a table indicating the staffing plan for the
week (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 exercise, the location (from where was it obtained: top drawer, bottom drawer, instructor?)
and all identifying information that is readily available. If the manufacturer and serial 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 specific
identifying 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 information
as possible in case someone was to try to reproduce or verify what you did. Refer to
Table 1.3.
7. For the Lab Manager only: create a key sign-out/sign-in table for obtaining the
key to the equipment drawers, as shown in Table 1.4.
418
8. All students should create a table or series of tables as appropriate to collect his/her
own data for the exercise, as well as any specific notes related to the data collection
activities. In those cases where data collection is done electronically, there may not be
any data tables required.
9. Many of the laboratory exercises will require the use of a specific software interface
for measurements and/or control. In all cases, these will be made available on the
myCourses site unless stated otherwise.
10. The Scribe (or a designated alternative) should take a photo of each group member
performing some aspect of the laboratory exercise for inclusion in the lab report
that will be generated during the studio session. Refer to the example lab report for
more details.
11. Record all relevant data and observations in your logbook, even those that may not
have been explicitly requested or indicated by the textbook or videos. If in doubt
about 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 been
returned to the proper place. Log out of the computer, and straighten up everything
on the lab bench as you found it. Put the lab stools back under the bench and out of
the way.
13. Prepare for the upcoming studio session for the week by carefully read and understand
sub-Section 3 of the textbook, and complete the Studio pre-work prior to your arrival
at Studio.
Be sure you have measured and recorded values for the armature length, the pendulum
rod length, and the mass of the bob before leaving the lab.
419
13.3
Studio - Matlab Simulation of Uniform Circular
Motion
This week in Studio you will complete two tasks. (1) The first task will be a fun enrichment
exercise, where you will make an animation of uniform circular motion in MATLAB. Using
code that has been written for you, you will create a baseline animation, from which you
can alter the code to see the e↵ect on the movie. (2) The second task involves comparing
the centripetal acceleration you measure in lab to the theoretical value. The theoretical
acceleration will require an experimental estimate of the angle theta that the bob rises from
the horizontal. Note that theta will be di↵erent for each of your angular speeds. You will
create a plot of centripetal acceleration versus angular velocity. This plot will include both
the theoretical value and the experimental value. You may use either Excel or MATLAB to
do your analysis for Task 2. Since the plot will contain only 3 points, one for each rotational
speed you tested, it is our recommendation that you use Excel for Task 2. You will evaluate
how well the experiment compares to theory and write up this analysis in your your weekly
report.
13.3.1
Calculation and Interpretation of Results
The equations needed for Studio this week were derived in the lecture portion of the text. Below is a brief summary of the key relationships that are needed for this analysis of centripetal
acceleration.
ar = g tan ✓
[m]
[m]
= 2 [ ]
2
[s ]
[s ]
(RT otal !)2
ar =
= RT otal (!)2
RT otal
[m]
[rad]
= [m] 2 [ ]
[s2 ]
[s ]
1
! = (pps)
2⇡
3200
[rad]
[pulses] [revolution] [radians]
=
[s]
[s]
[pulses] [revolution]
RT otal = R + L sin ✓
[m] = [m] + [m][ ]
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Experimental Acceleration
(13.52)
Units
Theoretical Acceleration
(13.53)
Units
Speed From Motor Pulses
(13.54)
Units
Total Radial Displacement
Units
(13.55)
13.3.2
Procedure - Studio Portion
Videos
As discussed previously, MATLAB o↵ers numerous training videos that can assist users with
understanding how to accomplish certain tasks in MATLAB. There are two videos that you
may find particularly useful for Studio this week. You may access these videos from the Help
menu 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 a
corresponding schematic that clearly identifies where each measurement was made in symbolic 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. You
may type your script into a text file using a simple text editor (such as Notepad) if you
don’t have MATLAB on your own computer. Note that the software can be downloaded for
free following the instructions on the home page of our myCourses site. You will need to
eventually run the code in MATLAB to debug it. You will receive a quiz grade based on the
completeness 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 before
arriving at Studio, so that we can spend time together doing debugging. If you have
not chosen to purchase a student license for MATLAB, Steps 1 through 6 may be
completed in a text editor such as Notepad. If you have purchased a student version
of MATLAB, or if you wish to use the open access Studio hours, you may complete
Steps 1 through 6 directly in MATLAB.
If you are using MATLAB, then from within the MATLAB environment, use the pulldown 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 thumb
drive, 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-down
command to execute “File - Save As” and save this file to your thumb drive, in a folder
named 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 before 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 include
the name of the author, the academic term, the name of the course, and a descriptive
title. You may add comments to explain the use and limitations of your scripts. Over
your career, as you develop a library of scripts, this will be an important way 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 character is considered a
comment, and is not considered a command to MATLAB. It is good practice to start
each script with commands to clear all variables from workspace memory, and to clear
the 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
clc
% clear up the workspace, removing old junk from memory
% clear command window
3. CREATE A LIST OF KNOWN INFORMATION: Please complete this step as part
of your pre-work before arriving at Studio. In the editor window, create a number
of scalar variables that contain known information for numerical simulation. These
expressions should look similar to those shown below, but should be replaced with
numerical values appropriate for your experiment. By using the same table of constants
as your experiment, you will be able to compare your simulated oscillation plots with
your experimental measurements.
The semicolon at the end of each line inhibits MATLAB from printing intermediate
results to the screen. It is good engineering practice to include a comment on each
line, indicating the engineering units associated with each assignment statement. Note
that it is your responsibility, as the engineer, to verify that the units of every equation
and constant are correct, since the simulation tools (both Excel and MATLAB) and
the program have no concept of units associated with the mathematical expressions
and assignment statements.
% Set the known information and simulation parameters
% for the simulation
%%%%%%%%%%%%%%%%%% GEOMETRY %%%%%%%%%%%%%%%%%%%
RTotal
= 0.20 ; % length of rod
Radius
= 0.015 ; % radius of bob
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4. CREATE A LIST OF SIMULATION PARAMETERS: Please complete this step as
part of your pre-work before arriving at Studio. Add the following lines of code to your
script in the editor window. The simulation parameters have to do with making the
movie. We will not go into all the detail about each line of code. As you enter this
code, please make sure you understand what you are typing. List any questions that
you 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 and
vary 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 MATLAB
writerObj = VideoWriter(’LastnameCircularMotion.avi’);
writerObj.FrameRate = 10;
open(writerObj);
figure(1);
set(gca,’nextplot’,’replacechildren’);
set(gcf,’Renderer’,’zbuffer’);
% define the limits of the plotting area
axis( [xmin xmax ymin ymax] );
axis ( ’off’ );
% capture the current image from the screen
frame=getframe;
% use the current image to the begin the video
writeVideo(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-work
before 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 very
repetitive information. In the example below, the “For” loop uses an index (or time
step counting) scalar variable named Nt, which will vary from 2 to Num Angle with
a default increment of 1. Every command between the For statement and the end
statement will be executed in sequence each time the time step is incremented. What
we 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 revolution
for Nt = 0:Num_Angle
% paint the background of the screen in a uniform color
% to begin with a blank canvas for the current image
rectangle(’Position’,[xmin,ymin,xlen,ylen],...
’Curvature’,[1,1],’FaceColor’,’w’,...
’LineStyle’,’none’);
% compute the angle of the pendulum at this point in rotation
Theta = 2.0 * pi * Nt / Num_Angle ;
% compute the position of the end of the pendulum rod
X = RTotal * cos ( Theta );
Y = RTotal * sin ( Theta );
% coordinates of the rod, from origin to rod-end
XLine = [ 0.0, X ] ; % From the origin to X
YLine = [ 0.0, Y ] ; % From the origin to Y
% draw pendulum rod on teh canvas
line(’XData’, XLine, ’YData’, YLine, ’LineWidth’,3 );
% draw the pendulum bob using a curved rectangle
X1 = X - Radius; % Lower x corner of the bob
Y1 = Y - Radius; % Lower y corner of the bob
rectangle(’Position’,[X1,Y1,Diameter,Diameter],...
’Curvature’,[1,1],’FaceColor’,’b’,...
’LineStyle’,’none’);
% capture the current image from the screen
frame=getframe;
% append the current image to the end of the video
writeVideo(writerObj,frame);
end
% close the video file
close(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 green
arrow in the ribbon of the editor window or typing the m file’s name in the Command
Window.
When you execute the script successfully, you should see a window pop-up, containing
your movie.
Do not expect your script to execute properly the first time. It is nearly inevitable
that you will need to do some debugging to correct errors in your script. This trouble
shooting process is a normal part of programming and is a powerful engineering skill
to develop. Periodically, as you work, save your script file to your USB drive, so that
you have a convenient recovery point in the event of a significant error.
Remember to use the information provided in the error messages in the Command
Window to guide you debugging process. Also, it is helpful to click on variables in the
editor window to see if they are highlighted to help identify typos. If you get stuck and
can’t determine why the code won’t run, simply upload your script file to the weekly
Dropbox. We will spend the first few minutes of Studio reviewing this script and can
help with any final debugging then.
7. OBSERVATIONS AND ANALYSIS: After you have your movie working, experiment
with code to determine how to make the rod longer, thinner or the bob larger, or
whatever you would like to do. Adjust your simulation script to produce a visually
appealing rendering of the uniform circular motion experiment.
8. SUBMIT YOUR WORK: After you have completed the movie and made it visually
appealing, please upload (a) your debugged MATLAB script m-file and (b) your movie
animation 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 lab
to determine the relative positions of the governor fly-balls at your di↵erent rotational
speeds. Watch the lab video titled “Uniform Circular Motion Image Analysis” for instructions on how to do this. You will also need to download the appropriate LabVIEW
VI from the myCourses site. Record all the necessary information in your logbook.
10. CENTRIPETAL ACCELERATION FROM EXPERIMENT: Using your experimental
data, program Excel (or MATLAB) to determine the centripetal acceleration based on
your experimental data. Be sure to apply the high level formatting used throughout
the 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 formats
and documentation you have demonstrated in previous analysis in this class. You may
prepare your plot either in Excel or MATLAB, using the tool that you determine to
be most convenient.
13. OBSERVATIONS AND ANALYSIS: Write responses to the following questions in your
logbook. 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 crossreference 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 experimental results. Fully explain the similarities and di↵erences between your experimental 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 USB
drive. You may want this file in the future! Please be sure to sign and date your
engineering logbook before you leave the studio, and to submit your work to your
individual 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 report
submission. Before leaving Studio, decide on a date and time to meet with your team
mates 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 responsible
for 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 the
week, and a photograph of each person beginning to initiate their trial, with a label
below 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 the
theory related to the experiment and simulation. Be sure to define every variable in
the equations, with units and include equation numbers.
• PAGE 2: The heading on this page should read Experimental Set-up. Create a
diagram 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, including
each key component and where measurements were taken. The important information
to communicate are the variable names, distances, axis and datums that relate to your
measurements and results. It is a good practice to add a legend that defines any
variables or components of the schematic that are not obvious. At the bottom of the
figure include a figure caption, for example Figure 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 are numbered
sequentially beginning with Figure 1 for the first figure in the report.
• PAGE 3: The heading on this page should read Results. Include the table with
experimental data for each team member, for all cases measured; motor speed, angular velocity, angle that the bob rises from the vertical, theoretical acceleration and
experimental acceleration.
Remember that any measured data point or value calculated from measure data has
an 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 important
that your tables, graphs and captions clearly communicate to the reader what the data
represents.
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 numbered
sequentially (independent of figure caption numbering) beginning with Table 1 for the
first table in the report.
• PAGE 4: No heading is needed on this page, since it is a continuation of the Results
section. On a single page, include plots of centripetal acceleration as a function of
angular speed. Put all team member’s data on a single page if possible. Format the
plots according to the guidelines shown in previous chapters. Arrange and format the
plots so that they are easily compared one to another.
• PAGE 5: The heading on this page should read Conclusions. Here you will state
the major conclusions that can be drawn from this analysis. In other words, you
will 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 uncertainty
limits? In evaluating your estimates for angular velocity, consider if there were any
systematic bias present in your results. What are the most significant contributors to
uncertainty, and how would you mitigate them? Finally, comment on whether your
experimental 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 consistent formatting style for sections, subsections and captions. Before uploading the
file, you must convert it to a pdf. Non-pdf version files may not appear the same in
di↵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 attempts
at 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 about
the origin as shown in the figure below. At what value(s) of is the vertical component
of the position vector, ry , greatest in magnitude? At what value(s) of is the vertical
component of velocity, vy , the greatest in magnitude? At what value(s) of is the
vertical 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] around
a flat (not inclined) circular track of radius r = 1, 000 [f t]. The static coefficient of
friction between the tires and the track is µ = 0.8[ ]. What is the maximum speed
that the vehicle can travel at without losing control? What will happen to the vehicle
when the limit is exceeded?
3. Use on-line resources to determine estimates for the nominal mass of the Earth and
it’s moon, Luna. Similarly, locate an estimate for the mean distance between Earth
and Luna. Use your understanding of uniform circular motion to explain the orbit of
the moon around the planet. Estimate the circumferential speed of the moon in it’s
orbit. Estimate the centripetal acceleration of the moon.
4. A satellite is in geosynchronous orbit (the satellite remains above a single point on
the Earth) about the Earth, centered above a point on the equator. The satellite has
a mass of 1000[kg]. Use your understanding of uniform circular motion to explain
the orbit of the satellite around the planet. Estimate the circumferential speed of
the satellite in it’s orbit. Estimate the centripetal acceleration of the satellite. What
distance must the satellite orbit at, as measured from the surface of the Earth?
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