+ Rotational Equilibrium: A Question of Balance

+ Learning Objectives Problem Solving: Recognize and apply geometric ideas in areas outside of the mathematics classroom Apply 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 equilibrium Systems of algebraic equations Solution techniques of algebraic equations Making and testing predictions Teamwork

+ 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 3 August 2006

+ Historical Origins Name was coined by Marcel Duchamp in 1931 to describe works by Alexander Calder Duchamp French-American artist, Associated with Surrealism and Dada Alexander Calder American artist, “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 weights Tight 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 x 1 = W y 1 x 1 + y 1 = 290 Level 2 2W x 2 = W y 2 x 2 + y 2 = 290 Write and solve the equations for x i And y i (i=1,2,3) 290 mm

+ Level 1 3W x 3 = W y 3 x 3 + y 3 = 290

+ Solve Equations for Level 1 3 W x 3 = W y 3 (1) x 3 + y 3 = 290(2) From (1): y 3 = 3x 3 (3) Substitute (3) in (2): 4x 3 = 290 or x 3 = 72.5mm (4) From (2) y 3 = 290 – x 3 or y 3 = 217.5mm(5) By substitution

Solve Equations for Level 1 3 W x 3 = W y 3 (1) x 3 + y 3 = 290(2) From (1): y 3 = 3x 3 or 3x 3 -y 3 =0(3) From (1) and (2) using Cramer’s rule Using Cramer’s Rule

+ Solve Equations for Level 1 Generate points for: Y 3 = 3X 3 Y 3 = X 3 Using Graphics

Numerical values for graph x3y3y3x3y3y3

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