+Rotational

Equilibrium: A Question of

Balance

+Learning Objectives

Problem Solving: Recognize and apply geometric ideas in areas

outside of the mathematics classroomApply and adapt a variety of appropriate

strategies

Communication: Communicate mathematical thinking

coherently and clearly to peers, teachers, and others

2

+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

+Today’s activity: Build a Mobile

+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

+Anticipated Learner Outcomes

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

Rotational equilibriumSystems of algebraic equationsSolution techniques of algebraic

equations Making and testing predictionsTeamwork

+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

+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

+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

+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”

+

Lobster Tail and Fish Trap, 1939, mobile

Hanging Apricot,1951, standing mobile

Standing Mobile, 1937

Mobile, 1941

+ 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.

+Our Mobiles

Version 1

A three-level Mobile with four weightsTight specifications

Version 2

An individual design under general constraints

+ Version 1 A three-level four-weight design

Level 1

Level 2

Level 3

+Materials

Rods made of balsa wood sticks, 30cm long

Strings made of sewing thread or fishing string

Coins

240 weight paper (“cardboard”)

Adhesive tape

Paper and pens/pencils

Tools and Accessories

Scissors

Hole Punchers

Pens

Wine/water glasses

Binder clips

30cm Ruler

Band Saw (optional)

Marking pen

Calculator (optional)

+Instructions and basic constraints

Weights are made of two standard

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

5 mm

+

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

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

+Level 1

3W x3 = W y3

x3 + y3 = 290

+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

Solve Equations for Level 1

3

0 1

290 1 29072.5

3 1 4

1 1

x

3

3 0

1 290 870217.5

3 1 4

1 1

y

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

+Solve Equations for Level 1

Generate points for:

Y3 = 3X3

Y3 = 290 - X3

Using Graphics

Numerical values for graph

0 0 29050 150 240

100 300 190150 450 140200 600 90

x3 y3 y3

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

+Graphic solution from handout

+Activity 1: Build Version-1 Mobile

Record actual results

Compare expected values to actual values

Explain deviations from expected values

+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

+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

+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

+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

Questions, comments, reflections