+ rotational equilibrium: a question of balance. + learning objectives problem solving: recognize...
TRANSCRIPT
+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
+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
+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
+
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
+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
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
+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