Rotational Equilibrium: A Question of Balance

1. Rotational Equilibrium: A Question of Balance Teacher In Service Program (TISP) Cape Town, South Africa Moshe Kam and Douglas Gorham IEEE Educational Activities 4 August 2006

2. 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)

3. 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

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

5. 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

6. 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

7. 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

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

9. 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.

10. 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)

11. 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)

12. 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)

13. 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)

14. Today’s activity:Build a Mobile

15. 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

16. 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

17. 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

18. Theory required • Newton’s first and second laws • Conditions for equilibrium • S F = 0 (Force Balance) Translational • St = 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

19. 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

20. 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”

23. Standing Mobile, 1937 Lobster Tail and Fish Trap, 1939, mobile Mobile, 1941 Hanging Apricot, 1951, standing mobile

24. 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.

25. Our Mobiles • Version 1 • A three-level Mobile with four weights • Tight specifications • Version 2 • An individual design under general constraints

26. Version 1 • A three-level four-weight design Level 1 Level 2 Level 3

27. 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

28. Scissors Hole Punchers Pens Wine/water glasses Binder clips 30cm Ruler Band Saw (optional) Marking pen Calculator (optional) Tools and Accessories

29. 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

30. 5 mm

31. 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 5 mm Level 1 Level 2 Level 3

32. Level 3 W x1 = W y1 x1 + y1 = 290 Level 2 2W x2 = W y2 x2 + y2 = 290 Designing the Mobile Write and solve the equations for xi And yi (i=1,2,3) 290 mm

33. Level 1 3W x3 = W y3 x3 + y3 = 290

34. Solve Equations for Level 1 By substitution 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)

35. 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

36. Solve Equations for Level 1 Using Graphics Generate points for: Y3 = 3X3 Y3 = 290 - X3

37. Numerical values for graph x3 y3 y3

38. x and y in mm The intersection is at x=72.5mm y=217.5mm

39. Graphic solution from handout

40. Activity 1: Build Version-1 Mobile • Record actual results • Compare expected values to actual values • Explain deviations from expected values

41. 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

42. 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

43. 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

44. 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?

45. Questions, comments, reflections