Jerk and Hyperjerk in a Rotating Frame of Reference
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1 Jek an Hypejek in a Rotating Fame of Refeence Amelia Caolina Spaavigna Depatment of Applie Science an Technology, Politecnico i Toino, Italy. Abstact: Jek is the eivative of acceleation with espect to time an then it is the thi oe eivative of the position vecto. Hypejeks ae the n-th oe eivatives with n>3. This pape escibes the elations, fo jeks an hypejeks, between the quantities measue in an inetial fame of efeence an those obseve in a otating fame. These elations can be inteesting fo teaching puposes. Keywos: Rotating fames, Physics eucation eseach. 1. Intouction Jek is the ate of change of acceleation, that is, it is the eivative of acceleation with espect to time [1,]. In teaching physics, jek is often completely neglecte; howeve, as emake in the Ref.1, it is a physical quantity having a geat impotance in pactical engineeing, because it is involve in mechanisms having otating o sliing pieces, such as cams an genevas. Fo this eason, jeks ae commonly foun in moels fo the esign of obotic ams. Moeove, these eivatives of acceleation ae also consiee in the planning of tacks an oas, in paticula of the tack tansition cuves [3]. Let us note that, besies in mechanics an tanspot engineeing, jeks can be use in the stuy of seveal electomagnetic systems, as popose in the Ref.4. Jeks of the geomagnetic fiel ae investigate too, to unestan the ynamics of the Eath s flui, ion-ich oute coe [5]. Jeks an hypejeks - jeks having a n-th oe eivative with n>3 - ae also attactive fo eseaches that ae stuying the behaviou of comple systems. In fact, in 1997, Linz [6] an Spott [7] genealize jeks into the jek functions, functions which ae use to escibe systems able of isplaying a chaotic behaviou. In [8], it ha been shown that the hypejek systems ae pototypical eamples of comple ynamical systems in a highimensional phase space. The jek is impotant fo engineeing because it is able evaluating the estuctive effects of motion on a mechanism o the uneasiness feeling cause to the passenges in vehicles [9]. In fact, when esigning a tain, enginees consie to keep the jek less than metes pe secon cube fo the passenges comfot. It is theefoe impotant, in teaching physics fo engineeing stuents, to evote some time to the iscussion of jek. Hee we ae poposing, in paticula, the iscussion of the elations fo jeks an hypejeks, between the quantities measue in an inetial fame of efeence an those obseve in a otating fame. This is a subject that, to the autho s best knowlege, ha eceive no attention fom physics teaches. Figue 1: Catesian inetial fame an otating fame, having the same oigin at est.
2 . Jeks an Hypejeks Let us consie two fames of efeence having the same oigin at est. One is an inetial fame, (, y, z), the othe a otating fame (, y,z ), which is otating about z=z with constant angula velocity =kˆ (see Figue 1). î,ĵ, kˆ an î,ĵ,kˆ ae the unit vectos of aes. Fo any point in the space, we have: = Let us eive (1): î+ yĵ+ zkˆ = î + yĵ + z kˆ (î+ y ĵ+ zkˆ) = ( î + yĵ + z kˆ) () Accoingly, the velocity is: (1) v= î+ y ĵ+ z kˆ = î + y z ĵ + kˆ + î + ĵ y + kˆ z = v + Let us stess that the unit vectos eivatives of unit vectos: (3) î,ĵ,kˆ ae otating. Theefoe, in (3), we use the î ĵ kˆ = î ; = ĵ ; = kˆ (4) So we have the well-known equation between the velocities obseve in the two fames: v= v + (5) Let us eive this equation again: (v We can easily fin: î+ v y ĵ+ vzkˆ) = ( v + ) (6) a= a + v + (7) In (7), we have the Coiolis an centipetal tems. These ae the acceleations we can ceate by means of the vectos at ou isposal, which ae, v an. The jek is the eivative of the acceleation an then:
3 3 J = a = (a î+ a a î + y y ĵ+ a z a ĵ + kˆ) = z kˆ + a ( a + v + ) î + a y ĵ + a z kˆ v + + To avoi any confusion with the unit vecto of y-ais, hee the symbol of the jek is witten using the uppe-case lette. The fist of tems efeing to the otating fame is the jek seen in it. î ĵ kˆ v J = J + a + a y + a z + + (9) Using (4) again, we have the equation linking jeks in the two fames: J= J + a + a + v + v + = J + 3 a + 3 v + In (10), we have one tem moe, because, besies, v an, we have also a. In the Figue we can see the iection of vecto, fo instance. (8) (10) Figue : One of the tems in Eq.10. Accoing to [9], thee is no univesally accepte name fo the fouth eivative, which is the ate of the change of jek. The tem jounce has been use, but this tem has the awback of using the same initial lette as jek. Anothe suggestion is snap (symbol s). The cackle (symbol c) an pop (symbol p) ae use fo 5-th an 6-th eivatives espectively. Let us etemine the elation fo the snaps. We have to eive again: Then: J = ( J + 3 a + 3 v + ) (11)
4 4 S= S + J + 3 J + 3 a + 3 a + 3 v + v + (1) The equation between the snaps obseve in the two fames is: S= S + 4 J + 6 a + 4 v + (13) In the Figue 3 we can see the iection of vecto. Figue 3: One of the tems in Eq.13. We can easily continue fo the following eivatives. Fo cackles, the elation is: C= C + 5 S + 10 J + 10 a + 5 v + 3. Two simple eamples Let us apply in two simple cases, the elations that we have peviously iscusse. The simplest possible eamples ae those of paticles at est. We can stat fom the inetial fame: let us consie a paticle at est on -ais at istance R fom the oigin. We have that: (14) = R î = R cos( t) î R sin( t) ĵ = R C î R S ĵ (15) In (15), we efine C = cos(t),s sin( t). We have velocity an acceleation: v= 0 v = = kˆ = Fom the elation a= a + v + : ( RC î R S ĵ ) = RS î RC ĵ a = 0 a = v = = RC î + R S (16) ĵ (17)
5 5 An, of couse, we have fom (10), a jek: J = 0 J = 3 a 3 v = = 3 RS î + 3 R C (18) ĵ Jek eists because the paticle, in the otating fame, has a centipetal acceleation, which is a vecto having constant moulus but a iection changing with time. Also in the case of a paticle in unifom motion o in unifomly acceleate motion, in the otating fame we obseve this paticle having a jek. Snap, cackle, pop an so on ae iffeent fom zeo. In the same manne, we can consie a paticle at est in a otating fame: = R cos( t) î+ R sin( t) ĵ= R C î+ R S ĵ= Rî We have velocity an acceleation: v = 0 v= = kˆ a = 0 a = = ( R î ) = R ĵ ( R ĵ ) = R î (19) (0) (1) In (0) an (1), we easily ecognize the well-known velocity an acceleation of the cicula unifom motion. Howeve, we have also a jek: J = 0 J = = 3 ( Rî ) = R ĵ () An cackle an pop: C = 0; P = 0; C= = P= = 3 4 ( R ĵ ) = Rî 4 5 ( Rî ) = R ĵ (3) As we have seen in this eample, jek eists because the paticle has a centipetal acceleation, which is a vecto having constant moulus but a iection changing with time. We can popose, fo instance, some poblems on caousels o chai-o-planes, asking the stuents to evaluate the maimum value of angula spee to keep the jek less than metes pe secon cube. Of couse, these two popose eamples ae quite simple: they can be use by stuents fo pacticing with otating fames an coss poucts.
6 Refeences 1. Schot, S. H. (1978). Jek: The time ate of change of acceleation, Am. J. Phys. 46, Sanin, T. R. (1990). The jek, The Physics Teache, 8, Vv. Aa. (015). Jek (physics), Wikipeia. 4. Xu, X. X., Ma, S. J., & Huang, P. T. (009). New concepts in electomagnetic jeky ynamics an thei applications in tansient pocesses of electic cicuit. Pogess In Electomagnetics Reseach M, 8, Bloham, J., Zatman,, & Dumbey, M. (00). The oigin of geomagnetic jeks, Natue 40, Linz, S. J. (1997). Nonlinea ynamical moels an jeky motion, Am. J. Phys., 65, Spott, J. C. (1997). Some simple chaotic jek function, Am. J. Phys., 65, Chlouveakis, K.E., & Spott, J.C. (006). Chaotic hypejek systems, Chaos, Solitons & Factals, 8(3), Gibbs, P., & Gaget, S. (1998). 6
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