Half-life period of a damped oscillation

Transcription

1 Demonstrtor # 8 Hlf-life period of dmped oscilltion TEACHER NOTES Activity title: Determintion of hlf-life period dmped oscilltions of the grvittionl pendulum, etc. Subject: Physics - Clss XI Student ge: yers Estimted durtion: 250 minutes (50 minutes, for dt collecting, 50 minutes for dt processing) Science content The Principles of Newtonin dynmics; The mechnicl lws concerning: motion, speed, the ccelertion of hrmonic oscilltions; The InLOT System; The mechnicl oscilltions; The grvittionl pendulum. Lerning objectives The lesson is vluble becuse it cretively eploits knowledge concerning Newtonin dynmics, trigonometry, mechnicl oscilltions, nd prcticl skills, through its pplicbility to non-forml lerning contets, such s sport ctivities. At the end of this lesson students will be ble: to pply the IV th principle of Newtonin dynmics nd to consolidte their understnding of the principles of Newtonin dynmics to pply the lws of oscilltory motion nd to consolidte their comprehension in wht they re concerned to understnd dmping mechnisms through the mthemticl modelling of this process to mke us of their knowledge of trigonometry to cretively use the AM system in pplied contets

2 to eplore the physics relity by testing the AM on trpeze, swing Inquiry-bsed chrcter The student will enhnce their work skills specific scientific investigtion nd discovery ctivities gered for this type of lerning: 1. Identify Questions for Scientific Investigtions Identify testble questions Refine/refocus ill-defined questions Formulte hypotheses 2 Design Scientific Investigtions Design investigtions to test hypothesis Identify independent vribles, dependent vribles, nd vribles tht need to be controlled Opertionlly define vribles bsed on observble chrcteristics Identify flws in investigtive design Utilize sfe procedures Conduct multiple trils 3 Use Tools nd Techniques to Gther Dt Gther dt by using pproprite tools nd techniques Mesure using stndrdized units of mesure Compre, group, nd/or order objects by chrcteristics Construct nd/or use clssifiction systems Use consistency nd precision in dt collection Describe n object in reltion to nother object (e.g., its position, motion, direction, symmetry, sptil rrngement, or shpe) 4 Anlyze nd Describe Dt Differentite eplntion from description Construct nd use grphicl representtions Identify ptterns nd reltionships of vribles in dt Use mthemtic skills to nlyze nd/or interpret dt 5 Eplin Results nd Drw Conclusions Differentite observtion from inference Propose n eplntion bsed on observtion Use evidence to mke inferences nd/or predict trends Form logicl eplntion bout the cuse-nd-effect reltionships in dt from n eperiment 6 Recognize Alterntive Eplntions nd Predictions Consider lternte eplntions Identify fulty resoning not supported by dt 7 Communicte Scientific Procedures nd Eplntions Communicte eperimentl nd/or reserch methods nd procedures Use evidence nd observtions to eplin nd communicte results Communicte knowledge gined from n investigtion orlly nd through written reports, incorporting drwings, digrms, or grphs where pproprite Applied technology (if ny) Pge 2 of 14

3 In order to do so the project uses n innovtive sensor dt collection tool, nmely the InLOT system ( tht consists of the following modules: SensVest - vest, equipped with vrious sensors, designed to crry components tht mesure nd trnsmit physiologicl dt to the bse sttion. Leg nd Arm Accelerometer - smll devices ttched to the leg nd/or rm tht enble the 3-D mesurement of the ccelertion for the leg nd/or rm. Bll Accelerometer - bll tht hs embedded n ccelerometer mesuring three dimensions nd communiction unit tht enbles the trnsmission of dt pckets to the bse. Bse Sttion - responsible for the collection of ll trnsmitted dt User Interfce Softwre - user friendly interfce, designed with pedgogicl frme of mind, tht enbles the process of dt nd ctions such s plotting dt on grph or creting mthemticl model to fit the dt. User detils cn be found in Anne 3.1. Mterils needed - InLOT system - PC - Physicl kit: mechnicl oscilltions - Worksheet (Annees 3.1, 3.2 nd3.3) Discussion guide Anticiption: Unit summry: Mechnicl oscilltions Essentil Question: How physics helps us to better understnd the sorrounding world? Before project pproch Before using project pproch, the high school students will review the principles of Newtonin dynmics, will discuss techniques for working with InLOT system, then write n essy bout the use of physicl knowledge in sports. Essys will be between three nd five pges nd will be noted. Essys will be evluted in terms of Newtonin dynmics hrnessing knowledge bout techniques for working with InLOT system discussed bove. After project pproch After the scenrio proposed sequence no. 3 hs been completed, indicted tht students pply the theme nd new skills to the situtions described by their essys. Students will be invited to eplore the questions: ) How physics helps us to better understnd the surrounding world? nd b) How tht gives us the performnce perspective?. Students will nlyze how science nd technology in performnce re mutully supportive nd not just thletes Building knowledge Teching strtegy The techer monitors nd dvises business groups, provides support points, support students in their pproch. Use project method Integrte knowledge nd skills chieved n dequte frmework for reflection. Reflection / Consolidtion Evlution method: gllery tour Assessment Summtive Pge 3 of 14

4 formtive Anne 3.1 Using ccelerometer O y Photo ccelerometer Reference directions of ccelerometer Wht ccelerometer (AM) mesures? - The frmes of reference in which the eperiments re conducted re non-inertil, so it is necessry to simplify the model; therefore we encourge the selection of pproprite eperimentl contets secondry level pproch. - It ppers tht AM mesures, momentry, reltive ccelertion in non-inertil frmes of reference. Generlly, ccording to kinemtics in non-inertil frmes of reference: r r r r rel = bs ( cor + trnsp ) (0.1.) r r r r m rel = m bs m ( cor + trnsp ) (0.2.) m r r r rel = F + 1 F (0.3.) c -Accelerometer (AM) mesures the difference between the momentry grvittionl component (reference direction O of AM), plus centrifugl momentry ccelertion (if chnge of direction of motion) nd momentry ccelertion of movement of AM in tht direction. = g + (0.4.) cf z m 1. where F r c is supplementry forces. Prticulrly, there re situtions (eg, bll suspended t rest reltive to the erth, but reltive to mn sitting on rotting wheel, the bll ppers to be in rottion), where it my hppen tht the body viewed from S does not ny force, but still to see him moving ccelerted reltive to S' due to supplementry force, F r c : r r r bs = 0 F = 0 Fc = m (0.5.) rel An importnt clss of reference frmes is the object's own frme or frme-relted rigid object moving uniformly force from their frme (eg the mn nd the object (= S ) re resting on the rotting disc, nd the object is cught in spring). In such frmes the object is evident in the rest ( r rel = 0), lthough there is rel force F r r r r r. In this cse: F+ Fc = 0 Fc = m rel. Tht supplementry force is equl but opposite to the rel force, so it is equivlent to the Newtonin inertil force. Supplementry forces re fictitious forces tht should be dded to the rel forces to ensure the vlidity of the 2 nd principle of Newtonin mechnics in non-inertil frmes. These re not forces of interction, we cn show the body tht produces them, so it doesn t pplies the 3 rd principle of Newtonin mechnics. Pge 4 of 14

5 y z Where: - is the vlue mesured on test direction (reltive ccelertion) - g is the component of grvity ccelertion on test direction - cf is the component of centrifugl ccelertion on test direction - m is the ccelertion of movement (ccelerometer nd body together) on test direction (ccelertion of trnsport). = g + (0.6.) y z cfy cfz my = g + (0.7.) mz If the motion is mde on certin direction, reltively to the reference directions of AM, then the previous reltions re wrote on ech component of the ccelertion mesured by ccelerometer ( 0). All mesured vlues re frctions of g (grvity ccelertion), epressed reltive to the vlue of g for which ws clibrted AM. Cses: I. m = 0 (AM is t rest, set on the object whose motion is studied, or in rectiliner nd uniform motion on test is, chosen s the O is) = g + (0.8.) cf cf More if = 0 = g (0.9.) II. g = 0 ( the test is is in perpendiculr plne on verticl ) =. (0.10.) cf m In ddition if cf = 0 = (0.11.) m This is the method of determining the ccelertion of motion of AM/the object bounded on AM. Wht we cn mesure with the ccelerometer in the lbortory / prcticl pplictions? Angles: AM in resting, st longside surfce mkes n ngle α with the verticl; = g sin α α = rcsin (0.12.) g Accelertion of trnsltionl motion on: o Ais in the horizontl plne regrdless of the grvity component o Ais of the other plne, but tking into ccount the grvity component Accelertion of comple motion (rottion nd trnsltion) Pge 5 of 14

6 Anne 3.2 ASSESSMENT TOOLS Scores for project evlution 1 = Criterion is not fulfilled 3 = Criterion is fulfilled in good mesure 2 = Criterion is met only slightly 4 = The criterion is fully met 1. All tem members undertke collbortive ctivities by completing the steps in processing id given to them nd collect dt for one of the roles within the tem 2. Ech member fulfills the role it hs in the tem. Tem members work together to chieve qulity presenttion 3. Presenttion mde meet the recommended structure. 4. Eplntion contined in the presenttion is enlightening to the public 5. Project presenttion is eloquent nd enlightening for the udience prticipting. 6. The mnner of presenttion is ttrctive nd involving public 7. Tem members re open to public questions nd formulte nswers ll questions pertinent to public 8. Introducing the tem roles demonstrtes tht members re knowledgeble in ll fields covered by the project. 9. Tem members spek out loud, communictes very cler presenttion of content, nd estblish eye contct with udience. 10. Tem members provide dditionl eplntions to the public request, using the flip chrt Completion: Note: The lesson is built vluing prior knowledge cquired in different lerning contets nd integrtes communiction skills, collbortion skills, investigtion, prcticl skills, but lso interpersonl nd socil skills, rtistic skills nd epression. Pge 6 of 14

7 Anne 3.3 AUXILIARY FOR TEACHING 3.I. Kicking life into Clssroom: Determintion of hlf-life period dmped oscilltions of the grvittionl pendulum Fig Chnges in mplitude in time for dmped oscilltions Curriculum-Frming Questions Essentil Question How would the universe pper without regulr phenomen? Unit Questions. At wht etent the lws of mechnics which re lredy known cn be pplied to periodic phenomen? Wht immedite pplictions do you see for the study of periodic phenomen in nture? Questions of content Wht periodicl mechnicl phenomen cn we identify in the nture? Wht physicl quntities re chrcteristic for the oscilltory movement? How cn we represent hrmonic oscilltor motion lws? Wht hppens to energy in motion hrmonic oscilltor? Under the ction of which type of force hrmonic oscilltory motion is present? Wht is the difference between the dmped oscilltion nd the idel one? 3.II. Into Lb with InLOT - Study dmped oscilltions. Modelling of physicl phenomen: dmped Principle method oscilltions of the pendulum grvity It ws considered tht for the hrmonic 3.II.A. Determintion of dmping oscilltion motion the oscilltor is subject only to coefficient of the grvittionl the ction of elstic type forces (F =- k). This pendulum oscilltions led to the fct tht the mplitude of the Dmped oscilltion mplitude is given by oscilltory motion remins constnt. the function of time:: It ppers, however, tht relity cnnot ignore βt A( t) = A 0 e (3.1.) the different types of brking forces tht led to deprecition of oscilltory movement. Thus the It follows tht: presence of brking forces tht led to the β A( t) mplitude is not constnt but decreses e t = (3.2.) eponentilly over time (fig.3.1.). Dmped A 0 oscilltor eqution of motion is: A0 βt π π = Ae sin ω0t = A( t) sin ω0t ( ) β t = ln (3.3.) 3.0. A t 2 ) 1 A0 β = ln (3.4.) where β is strictly positive nd is clled dmping coefficient If the dmping coefficient is lrge, the system will no longer crry n oscilltory motion, but returns to equilibrium. In such cses, we sy tht the oscilltory motion is periodic. It is not the cse nlyzed in this eperiment, the ( ) t A t dmping coefficient of the nlyzed system Pirs mesured A (t), t, nd tking into ccount the reltionship (3.8) becomes: (3.5.) Pge 7 of 14

8 dmped oscilltion is periodic, nd lmost etinguished s no. infinite oscilltions. G r n -A G r G r t -α 0 O Fig.3.2 Grvittionl pendulum In Scenrio 3, we know tht in the (3.1.) lbortory reference system, the direction of oscilltion we hve: (t) = A sin (ω 0 t-π/2), t=0, =-A v(t) = ω 0 A cos (ω 0 t-π/2) (3.2.) (t) = - ω 2 0 A sin (ω 0 t-π/2) (3.3.) note: ω 0 t-π/2= α ( t), A=R sinα 0 When the ngulr elongtion is given by lw: α (t) = α 0 sin (ω 0 t-π/2), t=0, α=-α 0 (3.4.) α& (t) =ω 0 α 0 cos (ω 0 t-π/2) (3.5.) α& & (t) =- ω 2 0 α 0 sin (ω 0 t-π/2) (3.6.) equtions dpted hrmonic oscilltions ((α 0 <5 ) of the system described in fig.3.2. A ( t) m A( t) = R g For minim 5 set of pirs A (t), t determine the dmping coefficient: β( 1) + β( 2) + β( 3) + β( 4) + β( β = ( ) 3.II.B. Determining the durtion of hlf mplitude periodic dmped oscilltions Hlf-time dmped oscilltion mplitude is determined s follows: t 1 e β = (3.7.) 2 β t = ln 2 (3.8.) 1 t 1/ 2 = ln 2 (3.9) β with β deduced bove, using eqution (3.4. nd 3.6.) Pge 8 of 14

9 STUDENT WORKSHEET Activity title: Determintion of hlf-life period dmped oscilltions of the grvittionl pendulum, etc. Introduction Curriculum-Frming Questions Essentil Question How would the universe pper without regulr phenomen? Unit Questions. At wht etent the lws of mechnics which re lredy known cn be pplied to periodic phenomen? Wht immedite pplictions do you see for the study of periodic phenomen in nture? Questions of content Wht periodicl mechnicl phenomen cn we identify in the nture? Wht physicl quntities re chrcteristic for the oscilltory movement? How cn we represent hrmonic oscilltor motion lws? Wht hppens to energy in motion hrmonic oscilltor? Under the ction of which type of force hrmonic oscilltory motion is present? Wht is the difference between the dmped oscilltion nd the idel one? Thinking bout the question 3.I. Kicking life into Clssroom: Determintion of hlf-life period dmped oscilltions of the grvittionl pendulum Curriculum-Frming Questions Essentil Question How would the universe pper without regulr phenomen? Unit Questions. At wht etent the lws of mechnics which re lredy known c Wht immedite pplictions do you see for the study of periodic Questions of content Wht periodicl mechnicl phenomen cn we identify in the nt for the oscilltory movement? How cn we represent hrmonic os motion hrmonic oscilltor? Under the ction of which type of for is the difference between the dmped oscilltion nd the idel one Fig Chnges in mplitude in time for dmped oscilltions Pge 9 of 14

10 3.II. Into Lb with InLOT - Study dmped oscilltions. Modelling of physicl phenomen: dmped Principle method oscilltions of the pendulum grvity It ws considered tht for the hrmonic oscilltion 3.II.A. Determintion of dmping motion the oscilltor is subject only to the ction coefficient of the grvittionl of elstic type forces (F =- k). This led to the pendulum oscilltions fct tht the mplitude of the oscilltory motion Dmped oscilltion mplitude is given remins constnt. by the function of time:: It ppers, however, tht relity cnnot ignore the βt A( t) = A 0 e (3.1.) different types of brking forces tht led to deprecition of oscilltory movement. Thus the It follows tht: presence of brking forces tht led to the β A( t) mplitude is not constnt but decreses e t = (3.2.) eponentilly over time (fig.3.1.). Dmped A 0 oscilltor eqution of motion is: A0 ( ) β t = ln (3.3.) A t βt π π = Ae sin ω0t = A( t) sin ω0t A0 2 2 β = ln (3.4.) where β is strictly positive nd is clled dmping coefficient If the dmping coefficient is lrge, the system will no longer crry n oscilltory motion, but returns to equilibrium. In such cses, we sy tht the oscilltory motion is non-periodic. It is not the cse nlyzed in this eperiment, the dmped oscilltion is periodic, nd lmost etinguished s no. infinite oscilltions. G r n -A G r G r t -α 0 O A Fig.3.2 Grvittionl pendulum ( ) t A t dmping coefficient of the nlyzed system Pirs mesured A (t), t, nd tking into ccount the reltionship (3.8) becomes: ( t) m A( t) = R (3.5.) g For minim 5 set of pirs A (t), t determine the dmping coefficient: β( 1) + β( 2) + β( 3) + β( 4) + β = ( ) 3.II.B. Determining the durtion of hlf mplitude periodic dmped oscilltions Hlf-time dmped oscilltion mplitude is determined s follows: t 1 e β = (3.7.) 2 β t = ln 2 (3.8.) 1 t 1/ 2 = ln 2 (3.9) β with β deduced bove, using eqution (3.4. nd 3.6.) Pge 10 of 14

11 In Scenrio 3, we know tht in the (3.1.) lbortory reference system, the direction of oscilltion we hve: (t) = A sin (ω 0 t-π/2), t=0, =-A v(t) = ω 0 A cos (ω 0 t-π/2) (3.2.) (t) = - ω 2 0 A sin (ω 0 t-π/2) (3.3.) note: ω 0 t-π/2= α ( t), A=R sinα 0 When the ngulr elongtion is given by lw: α (t) = α 0 sin (ω 0 t-π/2), t=0, α=-α 0 (3.4.) α& (t) =ω 0 α 0 cos (ω 0 t-π/2) (3.5.) α& & (t) =- ω 2 0 α 0 sin (ω 0 t-π/2) (3.6.) equtions dpted hrmonic oscilltions ((α 0 <5 ) of the system described in fig.3.2. Mterils needed - InLOT system - PC - kit physics: mechnicl oscilltions - Worksheet Sfety Follow the rules of lbour protection in the physics lbortory. Investigtion Nme of the prticipnt in the eperiment: Ctegory: student, techer ; sports, student Age:, Genre: M, F Eperimentl determintions 3.II.A. Determintion coefficient of the grvittionl pendulum dmpers. The vlue of grvittionl ccelertion site isg sttic = m/s 2 First four vlues of mimum momentry ccelertion t times 0, T, 2T, 3T nd 4 T re: m1 = m/ s 2 Action pln Determintions re mde, of course, in the reference ccelerometer with InLOT pltform: 3.II.A. Determintion dmpers coefficient of grvittionl pendulum. 1. Fit rigid pendulum ccelerometer so tht the reference is O is oriented horizontlly AM 2. The vlues recorded by the InLOT pltform nd using the reltion (2kπ) = g R A = m A=R g m (3.5.) Completed tble: Pge 11 of 14

12 m2 = m/ s 2 m3 = m/ s 2 m4 = m/ s 2 med = m/ s 2 Amplitude vlues t times 0, T, 2T, 3T, 4 T, 5T re: A 0 = m A 1 = = m A 2 = = m A 4 = = m t 0 T 2T 3T 4T 5T m A(t) A 0 = A 1 = A 2 = A 3 = A 4 = A 5 = β A (t) is clculted using eqution (3.5), nd β using reltion (3.4). The vlues obtined will be used to determine the verge of "eperimentl" ccording to the reltion (3.6) 3.II.B. Determintion of hlf-mplitude durtion of dmped periodic oscilltions. 1. Enter the dmpers coefficient of previously obtined epression (3.9) to determine the t 1/2 2. It discusses the cuses nd impct of deprecition on the vlue of t 1/2. A 5 == m Dmping coefficient vlues t the five distinct moments chosen re: β (1)= β (2)= β (3) = β (4)= β (5) = The verge vlue of dmping coefficient is β = Formulte conclusions mesurements for different eperimentl conditions: 3.II.B. Determining the durtion of hlf mplitude periodic oscilltions dmped hlf-life. Pge 12 of 14

13 Vlue of the mplitude of periodic oscilltions is written: t 1/2 = s Conclusions of discussion on the cuses nd impct of deprecition on the vlue of t 1/2 : Applied technology (if ny) Anlysis Anlyze the cuses of friction nd wht impct they hd on the outcome of the eperiment. Further investigtion 1. Relevnce. Students will reflect nd find nswers identifying possible prcticl role of the work done, the benefits of science nd technology on life in generl, the plce of science in society, the socil role of resercher. 2. Connection with the rel world. Students will reflect on the prcticl chrcter of their project, they will understnd the importnce of eperimentl dt nd the prcticl benefits of using the results. Assessment Gllery Tour: Students will prepre orl presenttions to pproprite udiences, which re ccompnied by multimedi presenttions, brochures nd websites. These products must identify current community needs nd resources nd provide cceptble solutions. Thus, the tsk turns into lerning project in support of the community, creting n uthentic purpose nd mking connection with the rel world through community. Evlution criterion: 1. All tem members undertke collbortive ctivities by completing the steps in processing id given to them nd collect dt for one of the roles within the tem 2. Ech member fulfills the role it hs in the tem. Tem members work together to chieve qulity presenttion 3. Presenttion mde meet the recommended structure. 4. Eplntion contined in the presenttion is enlightening to the public 5. Project presenttion is eloquent nd enlightening for the udience prticipting. 6. The mnner of presenttion is ttrctive nd involving public 7. Tem members re open to public questions nd formulte nswers ll questions pertinent to public 8. Introducing the tem roles demonstrtes tht members re knowledgeble in ll fields covered by the project. 9. Tem members spek out loud, communictes very cler presenttion of content, nd estblish eye contct with udience. 10. Tem members provide dditionl eplntions to the public request, using the flip chrt Pge 13 of 14

14 Pge 14 of 14