A Case Study on Simple Harmonic Motion and Its Application

Transcription

1 Interntionl Journl of Ltest Engineering nd Mngement Reserch IJLEMR ISSN: Volume 0 - Issue 08 August 07 PP A Cse Stud on Simple Hrmonic Motion nd Its Appliction Gowri.P, Deepik.D, Krithik.S 3 Asst.Prof, Deprtment of mthemtics,sri Krishn Arts And Science College II BscMths, Deprtment of mthemtics,sri Krishn Arts And Science College 3 II BscMths,Deprtment of mthemtics,sri Krishn Arts And Science College Abstrct: In this pper, we re going to stud bout simple hrmonic motion nd its pplictions. he simple hrmonic motion of spring-mss sstem generll ehibits behior strongl influenced b the geometric prmeters of the spring. In this pper, we stud the oscilltor behior of spring-mss sstem, considering the influence of ring the erge spring dimeter Φ on the elstic constnt k, the ngulr frequenc ω, the dmping fctor γ, nd the dnmics of the oscilltions. Simple hrmonic motion nd obtins epressions for the elocit, ccelertion, mplitude, frequenc nd the position of prticle eecuting this motion. Its pplictions re clock, guitr, iolin, bungee jumping, rubber bnds,diing bords,ethqukes, or discussed with problems. Kewords: Accelertion, Amplitude, Angulr frequenc, Velocit. I. Introduction In mechnics nd phsics, simple hrmonic motion is tpe of periodic motion or oscilltion motion where the restoring force is directl proportionl to the displcement nd cts in the direction opposite to tht of displcement. Simple hrmonic motion cn sere s mthemticl model for riet of motions, such s the oscilltion of spring. In ddition, other phenomen cn be pproimted b simple hrmonic motion, including the motion of simple pendulum s well s moleculr ibrtion. Simple hrmonic motion is tpified b the motion of mss on spring when it is subject to the liner elstic restoring force gien b Hooke's Lw. he motion is usoidl in time nd demonstrtes gle resonnt frequenc. For simple hrmonic motion to be n ccurte model for pendulum, the net force on the object t the end of the pendulum must be proportionl to the displcement. his will be good pproimtion when the ngle of swing is smll. Simple hrmonic motion proides bsis for the chrcteriztion of more complicted motions through the techniques of Fourier nlsis. he motion of prticle moing long stright line with n ccelertion which is lws towrds fied point on the line nd whose mgnitude is proportionl to the distnce from the fied point is clled simple hrmonic motion. In the digrm, simple hrmonic oscilltor, consisting of weight ttched to one end of spring, is shown. he other end of the spring is connected to rigid support such s wll. If the sstem is left t rest t the equilibrium position then there is no net force cting on the mss. Howeer, if the mss is displced from the equilibrium position, the spring eerts restoring elstic force tht obes Hooke's lw. Mthemticll, the restoring force F is gien b F= -K where F is the restoring elstic force eerted b the spring in SI units: N, k is the spring constnt N m, nd is the displcement from the equilibrium position m. II. Hedings he eqution of motion of prticle eecuting simple hrmonic motion, Geogrphicl representtion of simple hrmonic motion, Composition of two simple hrmonic motions of the sme period long the sme stright line, Composition of two simple hrmonic motions of the sme period in two perpendiculr directions. 5 Pge

2 Interntionl Journl of Ltest Engineering nd Mngement Reserch IJLEMR ISSN: Volume 0 - Issue 08 August 07 PP III. Indenttions And Equtions he Eqution of Motion of Prticle Eecuting Simple Hrmonic Motion Let O be the fied point on the stright line AOB on which prticle is hing simple hrmonic motion. ke O s the origin nd OA s the X is. Let P be the position of the prticle t time t such tht OP=. he mgnitude of the ccelertion t P=μ where μ is positie constnt. As this ccelertion cts towrds O, the ccelertion t P in the positie direction of the is is -μ. he mgnitude of the ccelertion t P is proportionl to tht is the mgnitude of the ccelertion is μwhere μ is constnt. As the ccelertion is directed towrds O.i.e., in the direction of decreg. Hence the eqution of motion of P is, d Eqution is the fundmentl differentil eqution representing simple hrmonic motion. We now proceed to sole it. If V-elocit of the prticle t time t, cn be written s d d. d. d c 3 Initil lue =, =0, Put in eqution 3, c 0 c c Eqution gies the elocit corresponding to n displcement. Now s t increses, decreses. d So, is negtie. 5 Pge

3 Interntionl Journl of Ltest Engineering nd Mngement Reserch IJLEMR ISSN: Volume 0 - Issue 08 August 07 PP So, tke negtie sign in d 5 d d. Integrting,. t A Initill when t=0,=. 0 A Hence. t. t. t his result gies the position of the prticle t the end of t seconds, the time mesured from the etreme position. GEOMERICAL REPRESENAION OF SIMPLE HARMONIC MOION: Show tht if prticle describes circle with constnt ngulr elocit, hen the foot of the perpendiculr on dimeter moes with simple hrmonic motion. P A O Q A Let prticle moe long the circumference of circle of rdius with uniform ngulr elocit ω. let AA be dimeter of the circle. Let the position of the prticle t time t be P. hen AOP t. Drw PQ r to AA nd let OQ=. hen t As P moes on the circle Q moes o the dimeter AA tht is Q oscilltes between A nd A long AA. herefore the motion of Q is simple hrmonic motion. From, 53 Pge

4 Interntionl Journl of Ltest Engineering nd Mngement Reserch IJLEMR ISSN: Volume 0 - Issue 08 August 07 PP d t d t d 3 Eqution gies the elocit of the prticle Q nd 3 gies the ccelertion of Q t time. Also from 3 we note tht the motion of Q is simple hrmonic. We know tht the mplitude of the simple hrmonic motion is. he periodic time of Q= If prticle describes circle with constnt ngulr elocit then the foot of the perpendiculr from it on n dimeter eecutes simple hrmonic motion. Composition of wo Simple Hrmonic Motions of the Sme Period long the Sme Stright Line Let the two simple hrmonic motions of the sme period be gien b b t t he composition of the two simple hrmonic motions is A B A. B A.. B [ [ t t. b b t. b b ] t t. t. A I A II ] b[ t [ b t. b t. ] b t t. t. ] hen, A A A t t A t t t his eqution shows tht the composition of two simple hrmonic motions is lso simple hrmonic motion with the sme period. A is the mplitude nd is the epoch. DiidingII b I, 5 Pge

5 Interntionl Journl of Ltest Engineering nd Mngement Reserch IJLEMR ISSN: Volume 0 - Issue 08 August 07 PP A A tn b b b b Squring nd dding I nd II, A A A A A A b b b b b IV III b b b. b b[. b. ]. b III gies nd IV gies the mplitude A. Composition of wo Simple Hrmonic Motions of the Sme Period in wo Perpendiculr Directions Let prticle eecute simple hrmonic motions long two perpendiculr directions with the sme r period. ke two directions s nd es. Let the displcement of the simple hrmonic motions be gien b t t he pth of the prticle is obtined b eliminting t from nd, t t. t. t t. t. t Squring nd simplifing we get t t. t. 55 Pge

6 Interntionl Journl of Ltest Engineering nd Mngement Reserch IJLEMR ISSN: Volume 0 - Issue 08 August 07 PP Pge 3. his is conic of the form 0 b h b h k b h Eqution 3 is n ellipse. NOE :If 0 the pth is stright line. NOE :If the pth is stright line. NOE 3:If,the pth is n ellipse. NOE :If nd,the pth is circle. IV. Problems If nd be the elocities of prticle moing in SHM t distnces nd from the centre show tht the time of complete oscilltion is. In SHM,

7 Interntionl Journl of Ltest Engineering nd Mngement Reserch IJLEMR ISSN: Volume 0 - Issue 08 August 07 PP Pge he period of oscilltion= Show tht in SHM of mplitude nd period t, the elocit t distnce from the centre is gien b the reltion = - Find the new mplitude if the elocit were doubled when the prticle is t distnce from the centre, the period remining the sme. In SHM,= Eliminting, from nd 3 Let = when = 3 A =mplitude when elocit t = is doubled nd the period remins the sme. i.e.,,, Substituting in 3,

8 Interntionl Journl of Ltest Engineering nd Mngement Reserch IJLEMR ISSN: Volume 0 - Issue 08 August 07 PP Substituting in, New mplitude= 3 APPLICAIONS PROBLEMS: he speed of wes in prticulr guitr string is 5m/s. Determine the fundmentl frequenc of the string if its length is 76.5cm. V=5m/s L=76.5cm=0.765m F =? Welength=*length =*.765m =.53m Speed=frequenc we length F =S/W 5m / s F =.53m =78 Hz. Determine the length of guitr string required to produce fundmentl frequencfirst hrmonic of 56 Hz. he speed of wes in prticulr guitr string is known to be 05m/s. V=05 m/s 58 Pge

9 Interntionl Journl of Ltest Engineering nd Mngement Reserch IJLEMR ISSN: Volume 0 - Issue 08 August 07 PP F =56 Hz S=f*w W=s/f W= 05m / s 56Hz W=.58m Length=/*welength =0.79 m. 3A guitr string with length of 80.0cm is plucked. he speed of we in the string is 00m/s. clculte the frequenc of the first, second nd third hrmonic. L=80.0cm=0.80 m V=00m/s =welength =*L =*0.80 =.6 m V=f* F = 00m / s.6m 50Hz F n =n*f F =500Hz F 3 =750hz A point on the string of iolin moes up nd down in simple hrmonic motion with n mplitude of.mm nd frequenc of 875hz. wht is the mimum speed of tht point in SI units? bwht is the mimum ccelertion of the point in SI units? Amplitude =.mm Frequenc=875hz Mimum speed=πfa = =683.8mm =6.8 m/s Mimum ccelertion= πf A = =3783 =3.750 m/s. 5Most grndfther clock he pendulums with djustble lengths one such clock loses 0 min per d when the length of its pendulum is 30in with wht length pendulum will this clock keep perfect time? A reltionship between the period nd length of the pendulum must be deeloped for the two situtions. g GM According, circulr frequenc of pendulum is gien b. therefore, the period P= L R L L R GM Diiding this eqution for period b nother period with length gies the necessr reltionship for this problem: 59 Pge

10 Interntionl Journl of Ltest Engineering nd Mngement Reserch IJLEMR ISSN: Volume 0 - Issue 08 August 07 PP P P L L his gien informtion is s follows: L =30 Assuming the pendulum eecutes n ccles. he pendulum eecutes n ccles per d P =0/m Since the pendulum tkes 0min longer to crr out the sme number of ccles, the period when the pendulum loses 0min is, P =50/mins L = L P P = 9.6 in Conclusion In this cse stud simple hrmonic motion nd its pplictions. Different pplictions problems re soled nlticll with ect eqution of simple hrmonic motion. We cn clculte the periodic time lue of oscillting object from origin b this methods. References Journl Ppers: []. Bli.N.P Dnmics lmi Publictions P Ltd. []. Duripndin, Lmi Duri pndin mechnics Publictions Chnd.S nd compn Ltd. [3]. Kushl kumr gh Dnmics Publictions Asoke Ghosh.K.PHI Lerning Prite limited, New delhi. []. R.M nd Shrm.G.C Dnmics Publiction Chnd.S nd Compn prite limited, New Delhi. [5]. Vittl.P.R nd Annth Nrnnn.V dnmics published b Mrghm Publictions. [6]. Venktrmn.M.K. published b Agstur Publictions. Websites nd Serch Engines [7]. [8]. [9]. [0]. []. []. [3]. Books []. A.P.French,Vibrtions nd wesnoton,new York,96 [5]. P.Mohzzbi nd J.P.McCrickrd,Am.J.Phs989 [6]. R.Resnick,D.Hllid nd K.S.Krne,Phsics for students in science nd EngineeringNew York 60 Pge