Rotational Equilibrium:
A Question of Balance
Teacher In Service Program (TISP)
Cape Town, South Africa
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Moshe Kam and Douglas Gorham
IEEE Educational Activities
4 August 2006
Who are we?
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This weekend’s workshop is a joint
activity of two organizational units of IEEE
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The IEEE Educational Activities Board (EAB)
The IEEE South Africa Section (est. 1977)
IEEE is a transnational organization
dedicated to engineering, technology and
science
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Established in 1963 by two associations
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AIEE (est. 1884) and IRE (est. 1912)
Attributes of IEEE
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Largest engineering association in the
world
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360,000 members in 150 countries
Major publisher and organizer of conferences
Major developers of standards
Provider of communication and networking
opportunities for engineers, scientists, and
technology practitioners
A public charity, dedicated to serving the
public
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Guided and lead by VOLUNTEERS
What do you need to know about TISP? (1)
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It is a program of IEEE
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It is about using IEEE
volunteers to help preuniversity teachers
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Specifically, IEEE’s
Educational Activities Board
(EAB)
Teachers of technology,
mathematics, and science
What do you need to know about TISP? (2)
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The basic idea: present teachers with lesson
plans that they can use to enhance student
understanding of Engineering and Engineering
Design
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The ultimate outcome is classroom activities with
students about Engineering
We are concentrating, however, on interacting with the
teachers
Success = teachers take our lesson plans to their
classrooms
All TISP lesson plans need to be aligned with national
curriculum standards
What are we going to do today?
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Simulate a TISP activity
Provide an opportunity for volunteers to
experience first hand what we are trying to
do with teachers
Motivate IEEE volunteers to conduct TISP
sessions with educators throughout the
pre-university educational system in
South Africa
Lesson content
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We will build a Mobile to meet
specifications
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Including basic calculations of design
parameters
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In teams of 2
We will develop specifications for a
second Mobile and then build it
How does this lesson align with
Educational Standards in South Africa ?
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Alignment to National Curriculum
Statements
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Critical Outcomes
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As a result of the activities, all learners should develop and
demonstrate the ability to;
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identify and solve problems and make decisions using critical and
creative thinking;
work effectively with others as members of a team, group,
organisation and community;
organise and manage themselves and their activities responsibly
and effectively;
collect, analyse, organise and critically evaluate information;
communicate effectively using visual, symbolic and/or language
skills in various modes;
use science and technology effectively and critically showing
responsibility towards the environment and the health of others;
and
demonstrate an understanding of the world as a set of related
systems by recognising that problem solving contexts do not exist
in isolation.
Learning Outcomes of
Mathematics: Grade 10
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As a result of the activities, all learners should develop and
demonstrate the ability to;
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Generate as many graphs as necessary, initially by means of pointby-point plotting, supported by available technology, to make test
conjectures and hence to generalise the effects of the parameters a
and g on the graphs of the functions.(10.2.2)
Investigate, generalise and apply the effect of the following
transformations of the point (x; y):
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A translation of p units horizontally and q units vertically;
A reflection in the x-axis, the y-axis or the line y = x.
(10.3.4)
Demonstrate an appreciation of the contribution to the history of the
development and use of geometry and trigonometry by various
cultures through a project. (10.3.7)
Learning Outcomes of
Physical Science: Grade 10
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As a result of the activities, all learners should develop and
demonstrate the ability to;
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plan and conduct a scientific investigation to collect data
systematically with regard to accuracy, reliability and the need to
control one variable. (10.1.1)
seek patterns and trends in information collection and link it to
existing scientific knowledge to help draw conclusions. (10.1.2)
Communicate information and conclusions with clarity and precision
(10.1.4)
Apply scientific knowledge in familiar, simple contexts. (10.2.2)
Learning Outcomes of
Mechanical Technology: Grade 10
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As a result of the activities, all learners should develop
and demonstrate the ability to;
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present assignments by means of a variety of communication
media. (10.2.5)
describe the functions of appropriate basic tools and equipment
(10.3.2)
explain the use of semi-permanent joining applications (10.3.5)
distinguish between different types of forces found in engineering
components by graphically determining the nature of these forces
(10.3.6)
Learning Outcomes of Civil
Technology Grade 10
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As a result of the activities, all learners should develop and
demonstrate the ability to;
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present assignments by means of a variety of communication media.
(10.2.5)
describe the properties and the use of materials in the built
environment. (10.3.2)
describe functions, use and care of basic tools and equipment.
(10.3.3)
demonstrate an understanding of applicable terminology. (10.3.5)
distinguish between different types of forces found in load bearing
structures. (10.3.6)
list different manufacturing process or construction methods. (10.3.7)
identify quantities of materials for small projects. (10.3.9)
explain the use of different joining applications. (methods) (10.3.10)
Today’s activity:
Build a Mobile
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Focus and Objectives
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Focus: demonstrate the concept of rotational
equilibrium
Objectives
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Learn about rotational equilibrium
Solve simple systems of algebraic equations
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Apply graphing techniques to solve systems of algebraic
equations
Learn to make predictions and draw conclusions
Learn about teamwork and working in groups
Anticipated Learner Outcomes
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As a result of this activity, students
should develop an understanding of
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Rotational equilibrium
Systems of algebraic equations
Solution techniques of algebraic equations
Making and testing predictions
Teamwork
Concepts the teacher needs to
introduce
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Mass and Force
Linear and angular acceleration
Center of Mass
Center of Gravity
Torque
Equilibrium
Momentum and angular momentum
Vectors
Free body diagrams
Algebraic equations
Theory required
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Newton’s first and second laws
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Conditions for equilibrium
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Translational
Rotational
Conditions for rotational equilibrium
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Linear and angular accelerations are zero
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Torque due to the weight of an object
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Techniques for solving algebraic equations
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S F = 0 (Force Balance)
S t = 0 (Torque Balance)
Substitution, graphic techniques, Cramer’s Rule
Mobile
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A Mobile is a type of kinetic sculpture
Constructed to take advantage of the principle of
equilibrium
Consists of a number of rods, from which
weighted objects or further rods hang
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The objects hanging from the rods balance each other,
so that the rods remain more or less horizontal
Each rod hangs from only one string, which gives it
freedom to rotate about the string
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http://en.wikipedia.org/wiki/Mobile_(sculpture)
3 August 2006
Historical Origins
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Name was coined by Marcel Duchamp in 1931 to
describe works by Alexander Calder
Duchamp
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Alexander Calder
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French-American artist, 1887-1968
Associated with Surrealism and Dada
American artist, 1898-1976
“Inventor of the Mobile”
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Standing Mobile, 1937
Lobster Tail and Fish
Trap, 1939, mobile
Mobile, 1941
Hanging Apricot,
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1951, standing mobile
Alexander Calder on building a mobile
"I used to begin with fairly complete drawings,
but now I start by cutting out a lot of shapes....
Some I keep because they're pleasing or
dynamic. Some are bits I just happen to find.
Then I arrange them, like papier collé, on a table,
and "paint" them -- that is, arrange them, with
wires between the pieces if it's to be a mobile, for
the overall pattern.
Finally I cut some more of them with my shears,
calculating for balance this time."
Calder's Universe, 1976.
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Our Mobiles
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Version 1
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Version 2
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A three-level Mobile with four weights
Tight specifications
An individual design under general
constraints
Version 1
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A three-level four-weight design
Level 1
Level 2
Level 3
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Materials
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Rods made of balsa wood sticks, 30cm long
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Strings made of sewing thread or fishing string
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5-cent coins
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240 weight paper (“cardboard”)
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Adhesive tape
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Paper and pens/pencils
Tools and Accessories
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Scissors
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30cm Ruler
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Hole Punchers
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Band Saw (optional)
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Pens
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Marking pen
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Wine/water glasses
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Calculator (optional)
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Binder clips
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Instructions and basic constraints
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Weights are made of two 5 cent coins
taped to a circular piece of cardboard
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One coin on each side
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Each weight is tied to a string
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If you wish to do it with only one coin it will be
slightly harder to do
The string is connected to a rod 5mm from the edge
5 mm
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Rods of level 3 and 2 are tied to rods of level 2 and 1 respective
at a distance of 5mm from the edge of the lower level rod
5 mm
Level 1
Level 2
Level 3
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Designing the Mobile
Write and solve the equations for xi And yi
(i=1,2,3)
Level 3
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W x1 = W y1
x1 + y1 = 290
290 mm
Level 2
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2W x2 = W y2
x2 + y2 = 290
Level 1
3W x3 = W y3
x3 + y3 = 290
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Solve Equations for Level 1
By substitution
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3 W x3 = W y3
(1)
x3 + y3 = 290
(2)
From (1): y3 = 3x3
(3)
Substitute (3) in (2): 4x3 = 290 or x3 = 72.5mm
(4)
From (2) y3 = 290 – x3 or y3 = 217.5mm
(5)
Solve Equations for Level 1
Using Cramer’s Rule
3 W x3 = W y3
(1)
x3 + y3 = 290
(2)
From (1): y3 = 3x3 or 3x3-y3=0
(3)
From (1) and (2) using Cramer’s rule
0
x3 
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1
290 1
290

 72.5
3 1
4
1 1
3
0
1 290 870
y3 

 217.5
3 1
4
1 1
Solve Equations for Level 1
Using Graphics
Generate points for:
Y3 = 3X3
Y3 = 290 - X3
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Numerical values for graph
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x3
y3
0
50
100
150
200
0
150
300
450
600
y3
290
240
190
140
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Graphic Solution
800
y
600
y=3x
400
y=290-x
200
0
0
50
100
x
x and y in mm
The intersection is at
x=72.5mm y=217.5mm
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150
200
Graphic solution from handout
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Activity 1: Build Version-1 Mobile
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Record actual results
Compare expected values to actual
values
Explain deviations from expected
values
Hints
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Use at least 30cm strings to hang weights
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Use at least 40cm strings to connect levels
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Sewing strings much easier to work with than
fishing string
If you are very close to balance, use adhesive
tape to add small amount of weight to one of the
sides
Version 2
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Design a more complicated mobile
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First, provide a detailed design and diagram with
all quantities
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More levels (say 5)
Three weights on lowest rod, at least two on each one of
the other rods
Different weights
Show all calculations, specify all weights, lengths, etc.
Then, build, analyze and provide a short report
Report
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Description of the design, its objectives and main
attributes
A free body diagram of the design
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A description of the final product
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All forces and lengths should be marked
Key calculations should be shown and explained
Where and in what areas did it deviate from the design
Any additional insights, comments, and
suggestions
Questions for Participants
What was the best attribute of your design?
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What is one thing you would change about your design based on
your experience?
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What approximations did we make in calculating positions for strings?
How did they affect our results?
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How would the matching of design to reality change if we…
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Used heavier weights
Used heavier strings
Used strings of different lengths connected to the weights
Used heavier rods
To educators: Can you implement this
lesson plan in your classroom?
Questions, comments, reflections
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