Winding motion in a spiral-like trajectory

Transcription

1 Winin motion in spir-ike trjectory Akio Sitoh Deprtment of Cii Enineerin, Kinki Uniersity, 8--, Shinke-cho, Yo-shi, Osk 8-8, Jpn. E-mi: (Receie ; ccepte ) Abstrct In this rtice I sh escribe n esiy constructe pprtus for n experiment on winin motion in spir-ike trjectory in three imensions. The experiment resuts show how the tot time of the process epens on the initi spee, n the tot time hs its mximum ue of 6.s for spee of.67m/s. The experiment resuts were in oo reement with the theoretic preictions. The nytic soution of the probem is oriin. Keywors: Winin motion, consertion of enery, nur eocity. Resumen En este rtícuo se escribe un prto e fáci construcción pr un experimento sobre e moimiento e ire en un espir en tres imensiones. Los resutos experimentes muestrn cómo e tiempo tot e proceso epene e eoci inici y e tiempo tot que tiene su or máximo e 6,s pr un eoci e,67m/s. Los resutos experimentes se encontrn en buen concornci con s preicciones teórics. L soución nític e probem es oriin. Pbrs ce: Liquición e moimiento, conserción e enerí, eoci nur. PACS:..c,..h, ISSN I. INTRODUCTION Mny texts [,, ] contin conic penuum whose b tres in horizont circe. The b is suspene by strin. If poe sueny stns upriht into the circe, the strin wins roun the poe unti the b utimtey hits the poe. Let us consier the tot time of the process for initi spee of the b theoreticy. We cn uess the tot time wi be short when the initi spee is ery ow or ery hih. Then, et us ccute the initi spee when the tot time hs its mximum ue, n compre it the experiment resut. This probem hs not been pubishe yet. II. EXPERIMENTAL PROCEDURE AND RE- SULTS Let us ssume tht the nur eocity of the poe is the sme s tht of the b, which is owe to swin in horizont circe n so hs circur pth of rius r with constnt spee. If we ook t the pprtus from boe, we cn mesure r u sce ( ruer or simir on the bench beow). Then, u the ue of r, the ue of is ien by the formu: r ( r ) ( r ). () Here =. m, = 8. - m n is the cceertion ue to rity, 9.8 m/s. The mss of the b is 6. - k n its imeter is. - m. The enth of the poe is bout. m. In this pprtus ony the top of the poe cn rotte. A hn - ri cn be use to rotte the top u on met ro (to which the top is firmy ttche) which psses throuh tube: the ower en of the ro is he in the chuck of the hn - ri. The tube n the ri re cmpe to the ee of the bench to keep the ro upriht n to ensure it is be to rotte smoothy. The hne of the ri is turne by hn t stey rte so tht the top of the poe rottes with constnt spee. If the top stops brupty, the b moes most on qurnt with the sme constnt spee of, ce is ery sm compre with ( << ). The b tkes time t to moe on the qurnt, n t is qurter of perio of conic penuum. Then, t is ien by the foowin formu: / t ( r ). () Lt. Am. J. Phys. Euc. Vo., No., Mrch 7

2 Akio Sitoh After this qurter reoution, the b tres (urin time t ) in spir -ike trjectory s the strin wins roun the poe unti the b utimtey hits the poe. The tot time, t = t + t, is mesure by stopwtch, which cn be re ccurtey to within. s. The purpose of this experiment is to show how t epens on the initi spee. The experiment ws crrie out mny times t ifferent initi eocities ess thn.8 m/s (which correspon to the mximum spee require to keep this prticur poe from swinin ue to tension). In Fiure, the circes inicte experiment points. Fiure is stroboscopic photorph for n initi eocity of. m/s. FIGURE. The pprtus for mesurin the time t. Ony the top of the poe cn rotte. Lt. Am. J. Phys. Euc. Vo., No., Mrch 8

3 III. THEORY Winin motion in spir-ike trjectory Let us consier the time t for ien eocity of the b theoreticy, ssumin buk of the b, mss of the strin n ir resistnce re neiibe. As shown in Fi., the strin is fstene t point A. At time t, riht fter the qurter reoution, the strin is tnent to the sie of the poe, n it mkes n ne φ beow horizont ine. At n rbitrry time t + t, the position of the b is B, n the strin mkes n ne φ with respect to the horizont. If the point of contct between the strin n the poe moes from B to C in ery sm inter of time, the b moes from B to C in the sme time n its increment chne of heiht is h; t the sme time the ne φ is chne by φ. We cn then write h ( ). () FIGURE. t s function of. The soi ine is ccute, n circes ( ) re experiment points. The pu of the strin, T, oes not work, ce the ispcement is perpenicur to T t times. Hence, u the principe of consertion of enery, m h m constnt, we obtin m h m. ) The instntneous spee of the b is efine s s, () where s is the increment of ispcement which the b hs urin the short time inter. This spee cn be seprte into horizont n ertic eements represente by n, respectiey (see Fi. ); where θ is the ne A OB subtene by rc A B (s shown in the pne fiure of Fi. ), n θ is the nur ispcement in the time inter. Hence, s ( ) ( ), n eqution () cn be rewritten s. (6) From the eometry of the sitution (Fi. ), it is seen tht the istnce with - ( < ) from B to C is ien by the retion FIGURE. Stroboscopic picture of winin motion. (7) Lt. Am. J. Phys. Euc. Vo., No., Mrch 9

4 Akio Sitoh FIGURE. The strin is fstene t point A, n it ees the poe t point B n C t time t + t n t + t+, respectiey. From equtions (6) n (7) Since is ery sm s compre with, the ue of φ wi so be ery sm s compre with the ue of θ. Hence we cn neect the term / pproximtion es us to the foowin formu,. This. (8) Lt. Am. J. Phys. Euc. Vo., No., Mrch

5 Winin motion in spir-ike trjectory FIGURE. The resutnt of m n T is the centripet force m pproximtey. On the other hn, when the b is t the point B, the two forces ctin throuh the common point B re the weiht of the b m n the strin tension T, s shown in Fi.. The resutnt of m n T is the centripet force m /, pproximtey. Therefore, m tn. (9) m / In Eq. (9), we so he neecte ery sm force m ( / ) tht tens to retr the motion. Differentitin Eq. (9), we fin cot tn. () We therefore et the foowin formu from Eqs. (), (), (9) n () tn cot Intertion of both sies of eqution () ies. (). () Eq. () ies retionship between n φ. In the cse of true point prtice, we fin tht the fin ue of φ is zero by settin = in this formu. By ifferentitin eqution () with respect to time t + t, we fin tht Lt. Am. J. Phys. Euc. Vo., No., Mrch

6 Akio Sitoh ( 8 ). () On the other hn, from equtions (8), (9) n (), we fin Combinin equtions () n (), we et 6. (). () Intertion of both sies of eqution (), n u the fct tht t the fin time t = t + t n the ne φ =, we obtin 6 t. (6) U foowin eqution, cn rewrite eqution () into ( r ), we t. (7) Then, the tot time t is ien by t t 6 t. (8) Furthermore from eqution (9), the retionship between n φ is ien,. (9) tn sec U equtions (8) n (9), n the computer softwre Mthemtic [], we cn ccute numeric ues of the time t for ifferent initi eocities. The prorm is s foows:.; 9.8; 8. ; _] : f [ ( Pot [ f [ ],{, PotRne, }, {, 8}] AxesOriin Where f ] is the tot time t. [ IV. CONCLUSION * ; {, }, In Fi., the soi cure is the ccute cure which is bse on equtions (8) n (9), n it shows how the time t epens on. We cn see from this fiure tht t hs its mximum ue of 6. s for spee of.67 m/s, n the experiment resuts were in oo reement with the ccute ues. The nytic soution of the probem is oriin. In the future, we wi construct n pprtus to keep the ertic poe from swinin ue to tension t hih spee. REFERENCES [] Gmbhir, D., Introuctory Physics Vo. (Mc Grw- Hi, USA, 976), p.. [] Shortey, G. n Wiims, D., Principes of Coee Physics (Prentice-H, USA, 99), p. 68. [] Sokonikoff, I. S. n Reheffer, R. M., Mthemtics of Physics n Moern Enineerin, n e. (Mc Grw- Hi, Jpn, 966), p. 7. [] Shirishi, S., Mthemtic (Morikit, Jpn, 99), p.8. Lt. Am. J. Phys. Euc. Vo., No., Mrch