Application of Moment of Momentum Theorem in Tie the Knot with A Hoop and A Chain

Transcription

1 5th Internatinal Cnference n Civil Engineering and Transprtatin (ICCET 2015) Applicatin f Mment f Mmentum Therem in Tie the Knt with A Hp and A Chain Zheng Shanshan1,a, Zheng Bin2,b, Qu Shuying3,c, Ha Bing 4,d 1,2,3,4 *a Yantai University, Department f Civil Engineering, Yan Tai, China zssluckyq@163.cm, bbinge199088@vip.qq.cm, cqsy_qu@163.cm, d @qq.cm Keywrds: Magic, Mment f Mmentum, Mment f Inertia, Angular Displacement, Angular Speed Abstract. The reasn why magic is mysterius is because the masters have sufficient applicatin and grasp f varius kinds f scientific principles. Here, take "tie the knt" magic fr example, carry ut several experiments f a single hp kntting with a chain, analyze the prcess f peratin, mvement and demnstratin frm the viewpint f dynamics and kinematics t discuss the principle f mechanics, giving the cmparisn and analysis between all kinds f different technique f magic shw and prcess f rganic cmbinatin between hps and chain, reveal its cgnitive rule f scientific and deepen the understanding, cgnizance and interesting in magic. Intrductin f Magic-- Tie the Knt Material Magic is wnderful, changeable, and has different subtleties frm each ther. It nt nly depends n the ingenius magic prps, but als depends n the perfrmers, manipulatin fr the prps. In the Tie the Knt magic, the irn hp and irn chain are independent and islated frm each ther. Assembled as shwn in Figure 1, the hp and chain can rub against each ther arbitrarily up and dwn, in this case, the hp and chain are unable t tie a knt. But when relaxed instantly, the hp drp dwn and knt with the chain. The ingeniusness f this magic is that the hp rtates based f the effect f external trque in the prcess f falling, entrance int the chain instantaneusly, slip dwn t the chain tail, and achieve kntting with chain. In rder t achieve the hp Figure 1. T ie the Knt magic kntting with chain by rtating and sliding, the selectin f magic prps need t meet the fllwing cnditins: (1) The material making hp and chain has a bigger specific gravity; (2) The surface f hp and chain is smth, and the cefficient f frictin is very small; (3) Every micr line f chain can be bended arbitrarily and freely, and the bending strength f chain is lw. Sizes and Specificatins f Cmpnents (1) The length f hrizntal segment f irn chain is greater than the diameter f hp and is abut 5cm, s the hp and chain can be in tuch. (2) When the rtatin angular displacement angle f hp is larger than 100, the chain belw the hp must be lng enugh fr kntting and slipping dwn The authrs - Published by Atlantis Press Figure2. T T K pefrmance Figure3. T T K pefrmance 2

2 A single hp kntting with a chain Descriptin f Magic Perfrmance 1 In perfrmance 1, as can be seen in Figure 2, Tie the Knt (TTK) magic shws the irn hp is restrained by right index finger and thumb t keep the level state naturally, the middle finger naturally bend and prtrude, the key is t apply the techniques f hand in the prcess f the hp instantly kntting with the chain. As seen in Figure 2, at the ent when the hp cllides with the middle finger in the prcess f falling, three cntinuus prcesses will be cmpleted: (1) The irn hp freely and translatinal falls displacement h1; (2) At the cllisin ent: the irn chain btains a certain ent f entum Jω ; (3) The irn hp gets an initial angular speed ω t rtate; synchrnusly it falls vertically, and achieves kntting with the irn chain [3]. Figure 4. dynamic analysis Nw, let s analyse these three stages separately, as shwn in Figure 4. Stage 1: the irn hp transnatinal falls with a displacement h1, due t gravity [4], the speed f hp befre cllides with the middle finger can be calculated as: v 2gh 1 (1) Stage 2: The irn hp has a cllisin with the prjecting middle figure. The falling displacement can be ignred due t the extremely shrt time. In this stage, the irn chain btains a certain ent f entum and a certain initial angular speed. Knwn frm the analysis f ent f entum therem [5], there is a certain relatinship between the initial angular speed ω and impulse S f the irn hp, diameter f the irn hp D and rtatinal inertia J f the irn hp. In the prcess f cllisin, the impulse ent f the irn hp equals t the ent f entum f the irn hp. D S ) J ( 2 (2) S fr the angular speed f the irn hp at the beginning f the stage 3 we have: S D 2J (3) Stage 3: During this falling prcess, the irn hp with the initial angular speed, achieves a perfect kntting after it reached the angular displacement f 100. Descriptin f Magic Perfrmance 2 In perfrmance 1, the irn hp gains enugh ent f entum and initial angular speed thrugh a cllisin, and it has a large enugh angular displacement t make the hp tie a knt with the chain. Of curse, if there is n cllisin, we culd have ther methd t achieve kntting between the hp and the chain as well [6, 7]. We make a change t make the irn hp gain enugh ent f entum t the rtatin f the mre than 100, eventually it reaches the same result as the ne befre--the hp knt with the chain. The differences between Figure 2 and 3 are: the irn hp is restrained by right index finger, middle finger and thumb t keep the level state naturally; and the directin f stretched chain is parallel t the directin f the right thumb. Figure 5. Mechanical analysis 604

3 Figure 3 shws the initial preparatin f the kntting prcess. Once the hp is released, it will instantly set int the chain. The key pint is the use f hand techniques, at the ent when a hp falls, first relax yur thumb naturally and make the hp d the fixed axis rtatin arund the middle finger which is nt relaxed (since it's a very shrt time, the falling displacement f the hp in the n-relaxatin side during this prcess can be neglected), rtatinal angular displacement is larger than 100. Then the hp disaffiliated frm middle finger with a vertical dwnward accelerating translatin, finishing the knt with chain eventually (Figure 5). The size f the irn hp and irn chain as shwn in Figure 6. In the falling prcess f the irn hp fixed-axis rtatin, the speed f irn hp: v 2gr sin (4) In stage f rtatin, the irn hp gains ent f entum and a certain angular speed at the same time, accrding t the ent f entum therem: Figure 6: Sizes f hp and chain The angular speed f the irn hp: mvd 2J (6) In the whle prcess f the hp s rtatin, the displacement in vertical directin and the frictin between the irn hp and train is ignred due t the extremely shrt rle time. Analyzing this prcess accrding t Figure 5, by the applicatin f cnservatin f energy therem [8]: (7) When θ is mre than 100, the irn hp and chain will cmplete a set f knt, then the hp drps dwn with freely translatin. The key t success fr kntting is that the angular displacement θ f the irn hp is larger than 100. S hw shuld we select materials and size f the magic prps, in rder t guarantee the cmpletin f the magic smthly? Here we need t calculate and cmparatively analysis the selectin f hp and chain parameters. First, calculatin accrding t the knwn size f the hp and chain in this article, accrding t cnservatin f energy therem: ( 2l ) m gr (1 sin 100 ) mgr cs100 (8) Get the maximum length f the chain that guarantees the chain kntting with the hp. In the prcess f the irn hp translatinal falling, gravity and frictinal resistance f chain affect mvement f hp; the equatins f mtin [9, 10] fr the hp can be used t calculate the minimum length f the chain: (9) We need t analyze and cmpare the parameters f the irn hp. As can be seen frm the frmula (2), the material with larger mass, bigger ent f inertia, smther surface and lwer frictin factr can be kntted. Other materials like plastic, aluminium r wd, is difficult t knt with chain. Table 1. Parameters fr Different Types f Material D d ~ ~ ~ ~ 100 m g (5) 605

4 Based n the table abve, frictin factr f the materials selected are relatively lw. Because heavier material is easy t get a larger angular displacement and achieve set f knt with chain, and the lighter material gets the smaller angular displacement, and the prbability f cmpleting perfrmance is lw. Cnclusin "Tie the Knt" magic described frm tw single hps and chains junctin, tw hps verlapping sets f ndes, tw hps dislcatin superimpsed set f ndes and three r mre hps tacking, it s prcess and results als were analyzed in this paper. Reach the fllwing cnclusin: 3.1. The length f chain, the size f the hp and the prprtin f hp are imprtant factrs that influence the magic is success r nt, nly within the scpe f pssible values can ensure the success f magic By explring the scientific principles f magic, a magic phenmenn can be achieved by a variety f magic tricks t explre the experiment in tw rings and chain Magic innvates cnstantly in explring, it enriches peple's lives and prvides a steady stream f pwer t scial prgress and human develpment. Acknwledgement This wrk is supprted by Higher Educatin Refrm in Shandng Prvince Prject (SD ) and Graduate Innvative Educatin Funds f Shandng Prvince (Grant N. SDYY11194). Special thanks t Zhuang Biazhng fr his direct cntributin n the specific wrks. References [1] Lu Chen, effect f Magic cnsciusness n the develpment f magic, China Acrbatics and Magic, N.3, pp.56~57, 2009 [2] Lu Yunting, The Games Between Magic and Math: The Aesthetic Dimensin f Art Sccer, Jurnal f Sprts and Science, Vl.34, N.4, pp.39~47, 2013 [3] Zhuang Biazhng, Sun Chengqi, Wu Lixiang, The third dynamic analysis f magic --Tw r Mre Irn Hps Kntting with Irn Chain, Mechanics in Engineering, Vl.31, N.3, pp.108~109, 2009 [4] Qia Hngzhu, Theretical Mechanics, Beijing: China Building Industry Press, pp.204~261, 2009 [5] Zhu Yanb, Theretical mechanics curse, Beijing: Higher Educatin Press, pp.150~210, 2009 [6] Qu Shuying, Zheng Shanshan, Cnstructin f Experiment Teaching Demnstratin Centre with Interests Driven, Independent Experiments, and Explratry Innvatin, BIJ, Vl.10, N.20, pp.12032~12036, 2014 [7] Lin Gngshi, selecting prject f riginal innvatin scientific research, Science f Science and Management, pp.36~39, 2004 [8] Theretical Mechanics Teaching and Research Sectin f Harbin Industrial University, Theretical Mechanics (Ⅱ), Beijing: Higher Educatin Press, 2009 [9] Ray Clugh, structural dynamics, Beijing: Higher Educatin Press, N.3, pp.13~15, 1979 [10]Liu Hezhi, Zhang Lng, simple analysis f the magic hp falling stress prcess, science and technlgy infrmatin, N.7, pp.253,

5 607