# Rotational Equilibrium: A Question of Balance

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1 Prt of the IEEE Techer In-Service Progrm - Lesson Focus Demonstrte the concept of rottionl equilirium. Lesson Synopsis The Rottionl Equilirium ctivity encourges students to explore the sic concepts of rottionl equilirium. Students work in tems to estimte nd determine the force within moile design, then groups compre results nd discuss findings. Age Levels -8. Ojectives Lern out the sic concepts of rottionl equilirium. Solve simple lgeric mnipultions. Apply grphing techniques. Lern to mke predictions nd drw conclusions. Lern out temwork nd working in groups. Anticipted Lerner Outcomes As result of this ctivity, students should develop n understnding of: rottionl equilirium sic lgeric equtions grphing mking nd testing predictions temwork Lesson Activities Students uild nd test moile to explore the principls of rottionl equilirium. Students mke predictions out the force on ech of three levels of the moile, work in tems to construct nd test their predictions, nlyze results nd compre tem results with those of the clss. Designing the moile requires the students to solve set of two liner lgeric equtions. Students solve the equtions using three different methods: y sustitution, y grphing the equtions nd finding the intersection, nd y using determinnts. Rottionl Equilirium: Pge of

2 Resources/Mterils Techer Resource Documents (ttched) Student Worksheet (ttched) Student Resource Sheet (ttched) Alignment to Curriculum Frmeworks See ttched curriculum lignment sheet. Internet Connections Alexnder Clder Foundtion ( IEEE Techer In-Service Progrm (/tispt) IEEE Virtul Museum ( ITEA Stndrds for Technologicl Litercy: Content for the Study of Technology ( McREL Compendium of Stndrds nd Benchmrks ( A compiltion of content stndrds for K- curriculum in oth serchle nd rowsle formts. Ntionl Council of Techers of Mthemtics Principls nd Stndrds for School Mthemtics ( Ntionl Institute of Stndrds nd Technology (NIST) ( Informtion out mesurements nd mesurement uncertinty. Ntionl Science Eduction Stndrds ( Recommended Reding 3,000 Solved Prolems in Physics y Alvin Hlpern (McGrw-Hill Trde, ISBN: ) Alexnder Clder nd His Mgicl Moiles y Jen Lipmn nd Mrgret Aspinwll (Hudson Hills Press, ISBN: ) Exploring the Fine Art of Moiles y Timothy Rose (Chronicle Books LLC, ISBN: ) The Essentil Alexnder Clder y Howrd Greenfeld (Hrry N Arms, ISBN: ) Optionl Writing Activity Write n essy (or prgrph depending on ge) how Alexnder Clder ( ) pplied the concepts of rottionl equilirium to his moile rt. Clder ws modern rtist known for his sculptures nd lrge scle moiles. A good exmple of Clder moile my e found t the John F. Kennedy Interntionl Airport, NY. Mny Clder resources re ville t the Ntionl Gllery of Art ( References Rlph D. Pinter Florid's West Cost USA Section of IEEE URL: Rottionl Equilirium: Pge of

3 For Techers: Alignment to Curriculum Frmeworks Note: All Lesson Plns in this series re ligned to the Ntionl Science Eduction Stndrds which were produced y the Ntionl Reserch Council nd endorsed y the Ntionl Science Techers Assocition, nd if pplicle, lso to the Interntionl Technology Eduction Assocition's Stndrds for Technologicl Litercy or the Ntionl Council of Techers of Mthemtics' Principls nd Stndrds for School Mthemtics. Ntionl Science Eduction Stndrds Grdes 5-8 (ges 0-4) CONTENT STANDARD A: Science s Inquiry As result of ctivities, ll students should develop Ailities necessry to do scientific inquiry Understndings out scientific inquiry CONTENT STANDARD B: Physicl Science As result of their ctivities, ll students should develop n understnding of Motions nd forces Trnsfer of energy Ntionl Science Eduction Stndrds Grdes 9- (ges 4-8) CONTENT STANDARD A: Science s Inquiry As result of ctivities, ll students should develop Ailities necessry to do scientific inquiry Understndings out scientific inquiry CONTENT STANDARD B: Physicl Science As result of their ctivities, ll students should develop n understnding of Motions nd forces Conservtion of energy nd increse in disorder Interctions of energy nd mtter Principls nd Stndrds for School Mthemtics (ges 6-8) Dt Anlysis nd Proility Stndrds formulte questions tht cn e ddressed with dt nd collect, orgnize, nd disply relevnt dt to nswer them. develop nd evlute inferences nd predictions tht re sed on dt. Alger Stndrds Understnd ptterns, reltions, nd functions Represent nd nlyze mthemticl situtions nd structures using lgeric symols Use mthemticl models to represent nd understnd quntittive reltionships Anlyze chnge in vrious contexts Stndrds for Technologicl Litercy - All Ages Design Stndrd 0: Students will develop n understnding of the role of trouleshooting, reserch nd development, invention nd innovtion, nd experimenttion in prolem solving. Rottionl Equilirium: Pge 3 of

4 For Techers: Techer Resources Mterils Student Resource Guide nd Student Worksheet /4 x /4 y 36-inch ls wood stick, one per moile. Pennies or similr ojects of uniform weight, eight per moile. Sewing thred or light string. Mteril on which to mount the pennies: e.g. construction pper, crdord or poster ord. Mrking pen. Cellophne tpe or glue. Scissors. Ruler mrked in millimeters nd centimeters or meter stick. Topic Review Forces, Torques, Vectors, Free Body Digrms, Rottionl Equilirium, Trnsltionl Equilirium, Sttic Equilirium, Simultneous Equtions, Grphicl Solutions, Solution By Sustitution, Solution By Determinnts, Dynmic Art. Procedure. Review topics ove with clss prior to ctivity.. Provide the Student Reference Sheet to ech student. (Note: these could e distriuted s reding homework prior to the clssroom ctivity.) 3. Crete one moile set-up displyed for the clss. 4. Divide students into smll groups of 3-4 students. 5. Provide ech group with mterils list ove nd Student Worksheet. 6. Instruct student tems to predict the totl force, F, nd the positions of the lnce points. 7. Tems uild the moile nd djust the points of suspension until the moile is lnced. 8. Student tems record the ctul result, y mesuring nd recording in the tle the ctul vlues of dimensions X, Y, X, Y, X 3 nd Y 3. (see Student Worksheet) 9. Student groups compre the ctul to the predicted vlues of dimensions X, Y, X, Y, X 3 nd Y 3. 0.Results re recorded on the Student Worksheet nd shred with the group. Time Needed Two Clssroom Sessions Suggestions Provide Student Resource nd Worksheet to students to review the night efore the lesson will e completed in clss. Rottionl Equilirium: Pge 4 of

5 For Techers Techer Resources: Lesson Adpttion Options The rottionl equilirium lesson cn e esily modified to meet the needs of rnge of students. For exmple, three methods: grphicl, sustitution nd determinnts, for solving the set of equtions tht predict the lnce point for ech level re suggested. The reson for solving the prolem y more thn one method is to demonstrte tht vriety of methods cn e employed to solve given prolem, nd tht ll vlid methods result in similr nswers. However, feel free to use the lesson to demonstrte ny one of the methods tht mtch the mteril you re teching t the time or tht re pproprite for the level of your students. The second prt of the ctivity tht involves reuilding the moiles with two-penny weights to study the effect tht hevier weights hve on the differences etween the predicted nd the ctul lnce points cn, if necessry, e omitted in the interest of time. Other Considertions The nlysis presented in the lesson ignores the weights of the horizontl ls sticks from which the weights re suspended. The pproch is vlid to the extent tht the torque produced y the unlnced weight of the ls is negligile compred to the torque produced y the weight of the pennies nd the crdord cutouts. Mking the crdord cutouts hevier y using two pennies rther thn just one to weight ech crdord cutout should mke the weight of the ls less noticele. For level one, the lnce point is t the center of the ls; therefore the weight of the ls hs no effect. However, for levels two nd three, the ctul lnce points will e closer to the predicted lnce points when hevier weights re pplied to the crdord cutouts. Rottionl Equilirium: Pge 5 of

6 Techer Resources: Concepts nd Definitions Wht is Moile? A "moile" is term coined in 93 y Mrcel Duchmp to descrie erly works of Alexnder Clder. During the erly 930's, Clder experimented with sculptures tht would undulte on their own with the ir's currents. As child, Clder uilt 3-D toys out of wires. He erned degree in Mechnicl Engineering in 99 nd egn to pply engineering nd physics principls to his rt. Erly on, he strove to crete hnging sculptures of wire nd metl tht were lter known s moiles. The resulting motion nd chllenge of lnce dded interest to his work. Now moiles re used s decortive rt throughout the world, nd mde of vriety of mterils. A populr current use for moiles is to visully stimulte ies in cris. Wht is Rottionl Equilirium? When n oject is in equilirium, there is no net tendency for it to move or chnge. When no net force is cting to mke n oject move in stright line, the oject is sid to e in "trnsltionl equilirium." When no net force is cting to mke n oject turn (torque), the oject is considered to e in "rottionl equilirium." An oject in equilirium t rest is sid to e in sttic equilirium. A stte of equilirium does not men tht no forces ct on the ody though -- it mens tht the forces re lnced. Other Terms Force: A force is physicl influence tht produces chnge in physicl stte. Force equls mss times ccelertion. A force cn lso e defined s push or pull. Torque: A force tht tends to produce rottion. Torque equls force times the distnce from the force to the center of rottion. Trnsltionl Equilirium: Trnsltionl Equilirium implies tht the sum of ll externl forces pplied to n oject is zero. Equilirium: An oject in equilirium hs no resultnt force cting on it. For n oject to e in stte of equilirium it must e oth in stte of trnsltionl equilirium nd in stte of rottionl equilirium where the sum of ll of the torques equls zero. Sttic Equilirium: Sttic equilirium exists when the forces on ll components of system re lnced. Vectors: A vector is quntity tht hs two spects. It hs size, or mgnitude, nd direction. Vectors re usully drwn s rrows. Both force nd torque re vector quntities. Free Body Digrms: A free ody digrm is tool to clculte the net force on n oject. It is drwing tht shows ll the forces cting on n oject. Simultneous Equtions: Simultneous equtions re set of equtions tht contin the sme vriles. Ech solution to the set of equtions must simultneously e solution to every eqution in the set. Rottionl Equilirium: Pge 6 of

7 Techer Resources: Concepts nd Definitions (continued) Grphicl Solutions: A method of finding solutions to set of simultneous equtions y plotting on common grph the curves which represent the equtions in the set nd oserving the points which re common to ll the equtions. The coordintes of these common points or intersections re solutions to the set of equtions. Solution By Sustitution: A method of finding the solutions to set of simultneous equtions y using one eqution in the set to define given vrile in terms of ll the other vriles nd then sustituting tht defining expression into nother eqution in the set. By series of such sustitutions mthemticl expression is otined tht gives the vlue(s) tht stisfy the set of equtions for one of the vriles. Those ctul vlues re then sustituted ck into one or more of the equtions to find the vlue(s) tht stisfy the set of equtions for the remining vriles. Solution By Determinnts: A method of finding solutions to set of simultneous equtions y writing the equtions in stndrd form nd pplying the formul for solution y determinnts. For this lesson, the equtions hve only two vriles, X nd Y. The stndrd form of the equtions is: X + Y = c X + Y = c The formul for solution y determinnts is: X = c c Y = c c Dynmic Art: Ojects of rt, usully sculpture, tht involve elements tht move. The motion is sometimes powered y wind, s in the cse of wind chimes nd smll moiles, or cn e powered y sources such s electric motors, wound springs nd other mechnisms. Rottionl Equilirium: Pge 7 of

8 For Techers: Techer Resources Answer Key Level Summing torques out the point of suspension gives one reltionship (or eqution) tht must e stisfied: Eq. () W X = W Y, therefore Y = X. The dimensions of the moile provide the sis for second reltionship etween X nd Y tht must e stisfied: Eq. () X + Y = 300mm. The set of equtions cn e solved y sustituting eqution () into eqution (): X + X = 300 mm; therefore, X = 300 mm so tht X = 50 mm nd Y = 50 mm. Summing verticl forces gives: F = W + W = W. Alterntively, eqution () cn e written in stndrd form (eqution () is in stndrd form lredy) to llow for solution y determinnts. Eq. () is rerrnged: X - Y = 0 mm Eq. (): X + Y = 300mm 0 0mm 300mm 300mm X = = = 50mm 300mm 300mm Y 50mm = = = Rottionl Equilirium: Pge 8 of

9 For Techers: Techer Resources Answer Key (continued) Grphicl Solution for Level Rottionl Equilirium: Pge 9 of

10 For Techers: Techer Resources Answer Key Level Summing torques out the point of suspension gives one reltionship (or eqution) tht must e stisfied: Eq. (c) W X = W Y, therefore Y = X. The dimensions of the moile provide the sis for second reltionship etween X nd Y tht must e stisfied: Eq. (d) X + Y = 300mm. The set of equtions cn e solved y sustituting eqution (c) into eqution (d): X + X = 300 mm; therefore, 3X = 300 mm so tht X = 00 mm nd Y = 00 mm. Summing verticl forces gives: F = W + W = 3W. Alterntively, eqution (c) cn e written in stndrd form (eqution (d) is in stndrd form lredy) to llow for solution y determinnts. Eq. (c) is rerrnged: Eq. (d) X - Y = 0 mm. X + Y = 300mm. 0 0mm 300mm 300mm 300mm 600mm X = = = 00mm Y 00mm = = = 3 3 Rottionl Equilirium: Pge 0 of

11 For Techers: Techer Resources Answer Key (continued) Grphicl Solution for Level Rottionl Equilirium: Pge of

12 For Techers: Techer Resources Answer Key Level 3 Summing torques out the point of suspension gives one reltionship (or eqution) tht must e stisfied: Eq. (e) 3 W X 3 = W Y 3, therefore Y 3 = 3X 3. The dimensions of the moile provide the sis for second reltionship etween X 3 nd Y 3 tht must e stisfied: Eq. (f) X 3 + Y 3 = 300mm. The set of equtions cn e solved y sustituting eqution (e) into eqution (f): X 3 + 3X 3 = 300 mm; therefore, 4X 3 = 300 mm so tht X 3 = 75 mm nd Y 3 = 5 mm. Summing verticl forces gives: F = 3W + W = 4W. Alterntively, eqution (e) cn e written in stndrd form (eqution (f) is in stndrd form lredy) to llow for solution y determinnts. Eq. (e) is rerrnged: Eq. (f): 3X 3 - Y 3 = 0 mm. X 3 + Y 3 = 300mm mm 300mm 300mm 300mm 900mm X 3 = = = 75mm Y 5mm 3 3 = = = Rottionl Equilirium: Pge of

13 For Techers: Techer Resources Answer Key (continued) Grphicl Solution for Level 3 Rottionl Equilirium: Pge 3 of

14 For Techers: Techer Resources Answer Key (continued) Dt nd Results Tle - Results Level Level Level 3 Predicted X 50 mm 00 mm 75 mm Predicted Y 50 mm 00 mm 5 mm Predicted force, F W 3 W 4 W Single penny results Mesured X Mesured Y Difference X, % Difference Y, % Two-penny results Mesured X Mesured Y Difference X, % Difference Y, % Record ll dimensions to the nerest whole millimeter. Rottionl Equilirium: Pge 4 of

15 Student Resource: Concepts nd Definitions Wht is Moile? A "moile" is term coined in 93 y Mrcel Duchmp to descrie erly works of Alexnder Clder. During the erly 930's, Clder experimented with sculptures tht would undulte on their own with the ir's currents. As child, Clder uilt 3-D toys out of wires. He erned degree in Mechnicl Engineering in 99 nd egn to pply engineering nd physics principls to his rt. Erly on, he strove to crete hnging sculptures of wire nd metl tht were lter known s moiles. The resulting motion nd chllenge of lnce dded interest to his work. Now moiles re used s decortive rt throughout the world, nd mde of vriety of mterils. A populr current use for moiles is to visully stimulte ies in cris. Wht is Rottionl Equilirium? When n oject is in equilirium, there is no net tendency for it to move or chnge. When no net force is cting to mke n oject move in stright line, the oject is sid to e in "trnsltionl equilirium." When no net force is cting to mke n oject turn (torque), the oject is considered to e in "rottionl equilirium." An oject in equilirium t rest is sid to e in sttic equilirium. A stte of equilirium does not men tht no forces ct on the ody though -- it mens tht the forces re lnced. Other Terms Force: A force is physicl influence tht produces chnge in physicl stte. Force equls mss times ccelertion. A force cn lso e defined s push or pull. Torque: A force tht tends to produce rottion. Torque equls force times the distnce from the force to the center of rottion. Trnsltionl Equilirium: Trnsltionl Equilirium implies tht the sum of ll externl forces pplied to n oject is zero. Equilirium: An oject in equilirium hs no resultnt force cting on it. For n oject to e in stte of equilirium it must e oth in stte of trnsltionl equilirium nd in stte of rottionl equilirium where the sum of ll of the torques equls zero. Sttic Equilirium: Sttic equilirium exists when the forces on ll components of system re lnced. Vectors: A vector is quntity tht hs two spects. It hs size, or mgnitude, nd direction. Vectors re usully drwn s rrows. Both force nd torque re vector quntities. Free Body Digrms: A free ody digrm is tool to clculte the net force on n oject. It is drwing tht shows ll the forces cting on n oject. Simultneous Equtions: Simultneous equtions re set of equtions tht contin the sme vriles. Ech solution to the set of equtions must simultneously e solution to every eqution in the set. Rottionl Equilirium: Pge 5 of

16 Student Resource: Concepts nd Definitions (continued) Grphicl Solutions: A method of finding solutions to set of simultneous equtions y plotting on common grph the curves which represent the equtions in the set nd oserving the points which re common to ll the equtions. The coordintes of these common points or intersections re solutions to the set of equtions. Solution By Sustitution: A method of finding the solutions to set of simultneous equtions y using one eqution in the set to define given vrile in terms of ll the other vriles nd then sustituting tht defining expression into nother eqution in the set. By series of such sustitutions mthemticl expression is otined tht gives the vlue(s) tht stisfy the set of equtions for one of the vriles. Those ctul vlues re then sustituted ck into one or more of the equtions to find the vlue(s) tht stisfy the set of equtions for the remining vriles. Solution By Determinnts: A method of finding solutions to set of simultneous equtions y writing the equtions in stndrd form nd pplying the formul for solution y determinnts. For this lesson, the equtions hve only two vriles, X nd Y. The stndrd form of the equtions is: X + Y = c X + Y = c The formul for solution y determinnts is: X = c c Y = c c Dynmic Art: Ojects of rt, usully sculpture, tht involve elements tht move. The motion is sometimes powered y wind, s in the cse of wind chimes nd smll moiles, or cn e powered y sources such s electric motors, wound springs nd other mechnisms. Rottionl Equilirium: Pge 6 of

17 Student Worksheet: Mterils Student Resource Guide nd Student Worksheet /4 x /4 y 36-inch ls wood stick, one per moile. Pennies or similr ojects of uniform weight, eight per moile. Sewing thred or light string. Mteril on which to mount the pennies: e.g. construction pper, crdord or poster ord. Mrking pen. Cellophne tpe or glue. Scissors. Ruler mrked in millimeters nd centimeters or meter stick. Step One: Prepre the mterils The horizontl memers of the moile re mde from ¼ x ¼ inch lswood. Cut three pieces thirty-one centimeters (or 30 mm) long. For convenience, mrk the center point of ech horizontl memer nd, eginning in the center nd going in ech direction, mrk ech centimeter nd hlf centimeter long the length of the horizontl memer. The spcing etween weights will in ll cses e 30 centimeters or 300 millimeters. Mke ech weight from poster ord or crdord cutout. Tpe or glue single penny to ech cutout. Use only pennies were tht were minted fter 983 in order to ensure uniform weights of the pennies. Use lightweight string or thred to suspend the cutouts from the horizontl memers. Step Two: Tem Predictions Predict the totl force, F, nd the position of the lnce points. Before uilding the moile, estimte the dimensions X, Y, X, Y, X 3 nd Y 3 nd estimte the forces, F, F or F3 in the min supporting strings in terms of W. Ignore the weight of the horizontl memers nd of the string or thred in mking these preliminry estimtes. Drw free-ody digrms for ech level nd show ll work. Enter your estimtes in the tle. The solutions for "X" nd "Y" involve set of two simultneous equtions. Check your nswers for ech level of the moile y plotting on the grph pper provided the liner function defined y ech eqution. The solution is given y the coordintes of the intersection of the two lines tht represent the two equtions. Rottionl Equilirium: Pge 7 of

18 Student Worksheet: (continued) Step Three: Build Your Moile Build the moile nd djust the points of suspension until the moile is lnced. Step Four: Record Your Actul Results Mesure nd record in the tle the ctul vlues of dimensions X, Y, X, Y, X 3 nd Y 3. Step Five: Anlyze Your Results. Compre the ctul to the predicted vlues of dimensions X, Y, X, Y, X 3 nd Y 3. Clculte the differences, expressed in percentges of the predicted vlues, etween the predicted nd the ctul mesured dimensions X, Y, X, Y, X 3 nd Y 3. Show ll work. Explin the differences. Would you expect your predicted nd ctul lengths to e closer or fther prt if the weights were hevier? Test your nswer y dding second penny to ech of the cutouts nd repeting your mesurements. Record your new results in the tle. Rottionl Equilirium: Pge 8 of

19 Student Worksheet: (continued) Dt nd Results Tle - Results Predicted X Level Level Level 3 Predicted Y Predicted force, F Single penny results Mesured X Mesured Y Difference X, % Difference Y, % Two-penny results Mesured X Mesured Y Difference X, % Difference Y, % Record ll dimensions to the nerest whole millimeter. Rottionl Equilirium: Pge 9 of

20 Student Worksheet: (continued) Grph of Equtions for Level Rottionl Equilirium: Pge 0 of

21 Student Worksheet: (continued) Grph of Equtions for Level Rottionl Equilirium: Pge of

22 Student Worksheet: (continued) Grph of Equtions for Level 3 Rottionl Equilirium: Pge of

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