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Rotational Equilibrium: A Question of Balance

Teacher In Service Program (TISP)

Cape Town, South Africa

Moshe Kam and Douglas GorhamIEEE Educational Activities

4 August 2006

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Who are we?

This weekend’s workshop is a joint activity of two organizational units of IEEE

The IEEE Educational Activities Board (EAB) The IEEE South Africa Section (est. 1977)

IEEE is a transnational organization dedicated to engineering, technology and science

Established in 1963 by two associations AIEE (est. 1884) and IRE (est. 1912)

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Attributes of IEEE

Largest engineering association in the world

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

Guided and lead by VOLUNTEERS

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What do you need to know about TISP? (1)

It is a program of IEEE Specifically, IEEE’s

Educational Activities Board (EAB)

It is about using IEEE volunteers to help pre-university teachers

Teachers of technology, mathematics, and science

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What do you need to know about TISP? (2)

The basic idea: present teachers with lesson plans that they can use to enhance student understanding of Engineering and Engineering Design

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

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What are we going to do today?

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

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Lesson content

We will build a Mobile to meet specifications Including basic calculations of design

parameters In teams of 2

We will develop specifications for a second Mobile and then build it

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How does this lesson align with Educational Standards in South Africa ?

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Alignment to National Curriculum Statements

Critical Outcomes As a result of the activities, all learners should develop and

demonstrate the ability to; 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.

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Learning Outcomes of Mathematics: Grade 10

As a result of the activities, all learners should develop and demonstrate the ability to;

Generate as many graphs as necessary, initially by means of point-by-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):

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)

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Learning Outcomes of Physical Science: Grade 10

As a result of the activities, all learners should develop and demonstrate the ability to;

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)

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Learning Outcomes of Mechanical Technology: Grade 10

As a result of the activities, all learners should develop and demonstrate the ability to;

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)

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Learning Outcomes of Civil Technology Grade 10

As a result of the activities, all learners should develop and demonstrate the ability to;

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)

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Today’s activity:Build a Mobile

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Focus and Objectives

Focus: demonstrate the concept of rotational equilibrium

Objectives Learn about rotational equilibrium Solve simple systems of algebraic equations

Apply graphing techniques to solve systems of algebraic equations

Learn to make predictions and draw conclusions Learn about teamwork and working in groups

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Anticipated Learner Outcomes

As a result of this activity, students should develop an understanding of

Rotational equilibrium Systems of algebraic equations Solution techniques of algebraic equations Making and testing predictions Teamwork

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Concepts the teacher needs to introduce

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

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Theory required

Newton’s first and second laws

Conditions for equilibrium F = 0 (Force Balance) Translational = 0 (Torque Balance) Rotational

Conditions for rotational equilibrium Linear and angular accelerations are zero

Torque due to the weight of an object

Techniques for solving algebraic equations Substitution, graphic techniques, Cramer’s Rule

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Mobile

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 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

http://en.wikipedia.org/wiki/Mobile_(sculpture) 3 August 2006

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Historical Origins

Name was coined by Marcel Duchamp in 1931 to describe works by Alexander Calder

Duchamp French-American artist, 1887-1968 Associated with Surrealism and Dada

Alexander Calder American artist, 1898-1976 “Inventor of the Mobile”

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Lobster Tail and Fish

Trap, 1939, mobile

Hanging Apricot,1951, standing mobile

Standing Mobile, 1937

Mobile, 1941

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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

Version 1 A three-level Mobile with four weights Tight specifications

Version 2 An individual design under general

constraints

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Version 1

A three-level four-weight design

Level 1

Level 2

Level 3

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Materials

Rods made of balsa wood sticks, 30cm long

Strings made of sewing thread or fishing string 5-cent coins 240 weight paper (“cardboard”)

Adhesive tape

Paper and pens/pencils

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Tools and Accessories

Scissors

Hole Punchers

Pens

Wine/water glasses

Binder clips

30cm Ruler

Band Saw (optional)

Marking pen

Calculator (optional)

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Instructions and basic constraints

Weights are made of two 5 cent coins taped to a circular piece of cardboard

One coin on each side If you wish to do it with only one coin it will be

slightly harder to do

Each weight is tied to a string The string is connected to a rod 5mm from the edge

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5 mm

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Level 1

Level 2

Level 3

5 mm

Rods of level 3 and 2 are tied to rods of level 2 and 1 respectively, at a distance of 5mm from the edge of the lower level rod

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Designing the Mobile

Level 3 W x1 = W y1

x1 + y1 = 290

Level 2 2W x2 = W y2

x2 + y2 = 290

Write and solve the equations for xi And yi (i=1,2,3)

290 mm

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Level 1

3W x3 = W y3

x3 + y3 = 290

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Solve Equations for Level 1

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)

By substitution

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Solve Equations for Level 1

3

0 1

290 1 29072.5

3 1 4

1 1

x

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

Using Cramer’s Rule

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3 0

1 290 870217.5

3 1 4

1 1

y

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Solve Equations for Level 1

Generate points for:

Y3 = 3X3

Y3 = 290 - X3

Using Graphics

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Numerical values for graph

0 0 29050 150 240

100 300 190150 450 140200 600 90

x3 y3 y3

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Graphic Solution

0

200

400

600

800

0 50 100 150 200

x

y

y=3x

y=290-x

The intersection is at x=72.5mm y=217.5mm

x and y in mm

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Graphic solution from handout

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Activity 1: Build Version-1 Mobile

Record actual results

Compare expected values to actual values

Explain deviations from expected values

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Hints

Sewing strings much easier to work with than fishing string

Use at least 30cm strings to hang weights

Use at least 40cm strings to connect levels

If you are very close to balance, use adhesive tape to add small amount of weight to one of the sides

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Version 2

Design a more complicated mobile More levels (say 5) Three weights on lowest rod, at least two on each one of

the other rods Different weights

First, provide a detailed design and diagram with all quantities

Show all calculations, specify all weights, lengths, etc.

Then, build, analyze and provide a short report

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Report

Description of the design, its objectives and main attributes

A free body diagram of the design All forces and lengths should be marked Key calculations should be shown and explained

A description of the final product Where and in what areas did it deviate from the design

Any additional insights, comments, and suggestions

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Questions for Participants What was the best attribute of your design?

What is one thing you would change about your design based on your experience?

What approximations did we make in calculating positions for strings? How did they affect our results?

How would the matching of design to reality change if we… 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?

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Questions, comments, reflections