Development of Mathematical Formulas Correlating the Normal and Tangential Components of Acceleration with Its Rectangular (x-y) Components

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1 Developme of Mhemil Formuls Correlig he Norml Tgeil Compoes of Aelerio wih Is Regulr (-) Compoes Rogelio Free C. Del Cruz 1* 1 Ml Colleges Lgu, PuloDiezmo Ro, Cbuo Ci, Lgu, Philippies. * Correspoig uhor. Tel.: ; f: ; emil: rfelruz@ml.eu.ph Musrip submie Jue, 014; epe Deember 15, 014. oi: /ijpm Absr: I his su, opi i Egieerig Mehis ws emie. The mhemil formuls orrelig he orml geil ompoes,, of elerio wih is regulr ompoes,, were erive. This su will provie egieerig priioers, ehers, sues wih iiol ools i solvig problems i urvilier moios. Ke wors: Curvilier moio, orml geil ompoes of elerio, regulr ompoes. 1. Irouio I urvilier moios, i is ver useful o resolve he elerio io ompoes, eiher io is orml geil ompoes, whih re orml ge o he ph of he moio, or io is regulr ompoes, ommol kow s ompoes. These ompoes sepre eoe respeivel he re of hge of mgiue of ireio of veloi. The re priulrl useful whe i is eee o rele veloi elerio irel wih he ph iself [1]. I he oe of his su, elerio is he re whih he veloi of bo hges per ui of ime. Veloi is he re of lier moio of bo i priulr ireio. The mgiue of veloi, kow s spee, is usull epresse i erms of ise overe per ui of ime. The mhemil efiiio of veloi is, ime [1] [5]. s v, for elerio, v, where v is veloi, is elerio, s is posiio, is The mi objeive of his su is o erive he mhemil formuls orrelig he orml geil ompoes,, of elerio wih is regulr ompoes,. The speifi objeives re: 1) To evelop mhemil formuls orrelig he orml geil ompoes of elerio wih is regulr ompoes whe he gle θ is mesure bewee. ) To evelop mhemil formuls orrelig he orml geil ompoes of elerio wih is regulr ompoes whe he gle θ is mesure bewee. For he sope limiio, he su overe he erivio of mhemil formuls orrelig he orml geil ompoes of elerio wih is regulr ompoes. All isussios referre o urvilier moio i ple, bu m lso be pplible o spil moio. The sigifie of his su is o provie lerive soluios o problems i urvilier moios. These 39 Volume 5, Number 1, Jur 015

2 formuls will serve s iiol ools for egieerig priioers, ehers, sues.. Curvilier Moio I urvilier moio, he posiio of he prile is speifie b he oorie s, whih is he ise mesure log he ph from fie referee poi. As he prile moves from A o B, see Fig. 1, urig ifiiesiml ime iervl, i res r of riusρ ifiiesiml legh s [] [8]. where, s hee, s = ρθ Fig. 1. Moio ph of prile. Fig.. Norml geil ompoes of elerio. where θ is i ris, ρ is lle he rius of urvure of he ph A. If he equio of he ph is kow, is rius of urvure be ompue from: 3 / 1 or 3 / 1. The orml geil ompoes of elerio, see i Fig., re epresse s follows, Norml ompoe of elerio, Tgeil ompoe of elerio, Also, v v s v. v. 40 Volume 5, Number 1, Jur 015

3 I is ver useful o resolve he elerio io ompoes, eiher io is orml geil ompoes, whih re orml ge o he ph of he moio, or io is regulr ompoes, ommol kow s ompoes. I his regr, i is impor o esblish he orrelio bewee hese ompoes..1. Whe he Agle θ Is Mesure bewee Derivio of mhemil formuls orrelig he orml geil ompoes of elerio wih is regulr ompoes whe he gle θ is mesure bewee, see i Fig. 3. where: is he geil ompoe of elerio is he orml ompoe of elerio is he -ompoe of elerio is he -ompoe of elerio is he resul elerio See Fig. 4, we ge 1. (1) 1. (). (3) os () (b) Fig. 3. Moio ph of prile. Fig. 4. Aelerio ompoes polgo.. (4) 1 41 Volume 5, Number 1, Jur 015

4 1 =. (5) = θ. (6) 1 = siθ. (7) ( θ ) siθ =. (8) 1 = siθ siθ θ +. (9) osθ 1 = siθ siθ θ. (10) osθ si θ 1 = siθ. (11) osθ osθ os θ = siθ. (1) osθ = siθ + osθ. (13) = +. (14) = +. (15) =. (16) ( si θ θ ) + os = +. (17) = si θ si θ osθ + os θ. (18) ( 1 si θ ) si θ osθ + ( 1 os θ ) =. (0) = os θ siθ osθ + si θ. (1) ( osθ θ ) = si. () = osθ siθ. (3) =. (4) osθ = θ. (5) 1 = θ. (6) 1 = siθ. (7) ( θ ) siθ =. (8) 1 = siθ siθ θ. (9) 1 4 Volume 5, Number 1, Jur 015

5 = si θ si θ θ +. os θ (30) 1 = siθ siθ θ. osθ (31) siθ siθ 1 = siθ. osθ osθ (3) si θ 1 = siθ. osθ (33) os θ = siθ. osθ (34) = siθ + osθ. (35) =. (36) ( si θ θ ) + os = +. (37) = si θ si θ osθ + os θ. (38) ( 1 si θ ) si θ osθ + ( 1 os θ ) =. (39) = os θ si θ osθ + si θ. (40) ( osθ θ ) = si. (41) = osθ siθ. (4).. Whe he Agle θ Is Mesure bewee Derivio of mhemil formuls orrelig he orml geil ompoes of elerio wih is regulr ompoes whe he gle θ is mesure bewee, see i Fig. 5. Fig. 5. Moio ph of prile. See Fig. 6, we ge = 1 +. (43) 43 Volume 5, Number 1, Jur 015

6 1. (44) 1. (45) os 1 1 si. (46) 1 os si. (47). (48) 1. (49) si. (50) si. (51) os si si. (5) os si si. (53) os 1 si si. (54) os os si. (55) os si os. (56). (57) si os. (58) si si os os. (59) 1 si si os 1 os. (60) os si os si. (61) os si. (6) os si. (63). (64) os. (65) 1 1 si. (66) 44 Volume 5, Number 1, Jur 015

7 Subsiue (64) (70) i(44), hee. (67) 1 si. (68) 1 si si. (69) 1 si si. (70) os 1 si si. (71) os os 1 si si. (7) os si os. (73) si os. (74) si si os os. (75) 1 si si os 1 os. (76) os si os si. (77) os si. (78) os si. (79) Fig. 6. Aelerio ompoes polgo. 3. Colusio Direios for Fuure Use The mhemil formuls erive orrelig he orml geil ompoes of elerio wih is regulrompoes whe he gle θ is mesure bewee, see Fig. 3, re he followig: si os. (80) 45 Volume 5, Number 1, Jur 015

8 os si. (81) os si. (8) si os. (83) A he mhemil formuls orrelig he orml geil ompoes of elerio wih is regulrompoes whe he gle θ is mesure bewee, see Fig. 5, re s follows: si os. (84) os si. (85) si os. (86) os si. (87) The reserher reommes he use of he bove meioe formuls s lerive meho i solvig problems ivolvig elerios i urvilier moios. Akowlegme This reserh ws ispire b he formuls evelope b F. L. Siger i his book [1]. I hk him for h. Also, I woul like o hk he followig: m wife, Floreliz R. Del Cruz m ugher, MelieFlor Iree R. Del Cruz, for he pig job rwigs; o m eerig e, Dr. Jo G. Hofileñ, for her oiuous wrm suppor h ispire me i oig his pper. I woul like o epress m mos siere griue o Ml Colleges Lgu, Philippies, for fuig he publiio preseio of his pper, bove ll, I hk Go, he Fher Almigh, wihou Him, I m ohig. This work is eie o m moher, Iree C. Del Cruz, for her oiuous eer lovig re o m moher-i-lw, Filipis R. Ruiz for lws believig i me. Referees [1] Siger, F. L. (1980). Egieerig Mehis: Sis Dmis (3r e.). (pp ). New York, USA: Hrper & Row, Publishers I. [] Pel, A., & Kiusls, J. (011). Uersig Egieerig Mehis: Dmis (SI e.). (pp. 69-8). Sigpore: Cegge Lerig Asi Pe. L. [3] Beer, F. P., & Johso Jr., E. R. (007). Veor Mehis for Egieers: Dmis (8h e.). (hper 11). New York: The MGrw-Hill Co. [4] Hibbeler, R. C. (006). Priiples of Sis Dmis (10h e.). (pp ). New York: Preie-Hll. [5] Hibbeler, R. C. (006). Egieerig Mehis: Priiples of Sis Dmis (pp. 3-7). New Jerse: Perso Preie Hll. [6] Robers, A. P. (003). Sis Dmis wih Bkgrou Mhemis. New York: Cmbrige Uiversi Press. [7] MGill, D. J., & Kig, W. W. (1995). Egieerig Mehis (3r e.). Boso: PWS. [8] Sous-Lile, R. W., Im, D. J., & Bli, D. S. (008). Egieerig Mehis: Dmis (Compuiol e.). (pp ). USA. 46 Volume 5, Number 1, Jur 015

9 Rogelio Free C. Del Cruz is grue of he bhelor of siee i ivil egieerig from Si Louis Uiversi, Bguio Ci, Philippies (1985) he mser of siee i mhemis from Pmsg Cbuo (Uiversi of Cbuo), Cbuo Ci, Philippies (013). He worke i he Deprme of Publi Works Highws L Uio Egieerig Disri ( ), he joie S Miguel Pkgig Speiliss I. Righpk Pl ( ). He lso worke i Msour Corig Comp i A Dmmm, KSA (008) ler i Vzquez Mrigl & Co., I. He sre i he fiel of eme s pr-ime ful member i Lgu College of Busiess Ars ( ), full-ime ful i Colegio e S Ju e Ler Clmb (009-01), pr-ime ful i Leum of he Philippies Lgu (010), pr-ime ful i Uiversi of Perpeul Help Ssem Lgu (010-01), pr-ime ful i Pms g Cbuo (01), urrel, he is ss. professor polluio orol offier i Ml Colleges Lgu, Philippies (01-prese). Egr. Del Cruz is member of Philippie Isiue of Civil Egieers (PICE), Mhemil Soie of he Philippies CALABARZON Chper (MSPC), Polluio Corol Assoiio of he Philippies, I, (PCAPI). 47 Volume 5, Number 1, Jur 015