SIMPLE HARMONIC MOTION

Transkript

1 SIMPLE HARMONIC MOTION 1. Periodic oion, kineaics of sipe haronic oion oion inear, circuar, o-and-fro = vibraion/osciaion osciaion can be irreguar, bu we wi discuss jus he reguar one caed SIMPLE HARMONIC MOTION which is conneced wih a sead circuar oion ( r =, T ) can be described b he eqn: a = k ax acceeraion and dispaceen fro equiibriu a he sae ie insan. Quaniies ha describe s.h.., phasor diagras hp://fzika.jreich.co/index.php?sekce=browse&page=156 a) dispaceen r T T 4 π π 3T T 4 3π π equiibriu ie ange = rsin ω = sinω π ω = π f = T 1. A ass suspended on a spring provides a s.h.. of period seconds. Assue he axiu dispaceen fro equiibriu 1 c and he beginning of he easureen when he objec oves up passing he equiibriu posiion; cacuae he dispaceen of he objec afer one second and 3 seconds SIMPLE HARMONIC MOTION

2 . Wrie he eqn. = f() 5 c. s b) veoci and acceeraion a c ω veoci and acceeraion of he objec perforing a s.h.. are he side projecions of he siiar quaniies conneced wih he suiabe s.c.. v v cos ω = ωrcosω = ωcosω = size of a = a sinω = ω rsinω = ω sinω, bu opposie direcion o he dispaceen so fina c a = ω sin ω = ω The equaions can be derived as he firs and second derivaive of he dispaceen according o he ie v = / = d d = a = v / = dv d = = d d advanage sipe, we can work ou +/- fro he eqns isef disadvanage jus ahs, for everone no aware of he connecion wih reai - - SIMPLE HARMONIC MOTION

3 3. Copare wih he firs definiion eqn. and discuss. 4. Skech siiar figures for quadrans II. IV. Reae he direcion of he veoci wih he direcion of oion. Sae fro he uua direcion of he veoci and acceeraion if he oion is acceeraed or deceeraed. Find he poins wih he axiu vaues of veoci (acceeraion) and zero vaues of he sae quaniies. c) -, v-, a- graphs I. II. III. IV. dispaceen ie/ange veoci ie/ange acceeraion ie/ange SIMPLE HARMONIC MOTION

4 3. Iniia phase iporan when we do no sar o easure he ie when he objec passes he equiibriu posiion up appropriae ange/ie shoud be added or subraced o ge he copee sine or cosine curve φ ϕ = ω = sin( ω + ϕ ) when he ange is fro π o π, iniia phase can be subraced: v = a = φ v = a = = sin( ω ϕ ) 5. Cacuae he iniia phase when a) =.5 s, T = 4 s b) =.1 s, T = s L4/1-14, X15, SIMPLE HARMONIC MOTION

5 4. Mass on a spring springs obe Hooke s aw k... spring consan = he force needed o produce uni exension (=1) F k = [ k ] = N 1 I. II. k g k(+) equiibriu posiion g I. F = R F S = F G k = g II. F F R R = F S F G F R = k( + ) g = k + k k = k The resuan causes acceeraion according o Newon s nd Law sizes: a = k, bu acceeraion and dispaceen have opposie direcions, so: a = k k a = copare: a = ω hence k ω = ake ω = π T T = π or k T, f... naura period, frequenc f = 1 π k 6. A igh spira spring is oaded wih a ass of 5 g and i exends b 1 c. Assue he axiu dispaceen fro equiibriu of 5 c, zero iniia phase and cacuae: a) he period of sa verica osciaions b) he veoci a equiibriu c) acceeraion c above equiibriu d) he ie aken o ge c above equiibriu SIMPLE HARMONIC MOTION

6 5. Sipe penduu a sa bob of ass suspended b a igh inexensibe sring of engh fro a fixed poin for sa ange on (ess han 5 degrees), when he dispaceen can be aos a sraigh ine weigh of he bob... g g cos Θ... is ension coponen baanced b he sring force g sinθ... is angenia coponen = resoring force, unbaanced nd NL!!! Find hese forces in he foowing figure: copare g sin Θ = a g = a g a = for he sizes, bu acceeraion and dispaceen have opposie direcions, so: g a = a = ω hence g ω = ake Θ T = π or g f = 1 π g T, f... naura period, frequenc 7. Expain wh he period of a sipe penduu does NOT depend on he ass of he oad. Prove i! 8. A ba of ass 4 g hangs on a sring 5 c ong. Then i is pushed and i sars o ove o-and-fro having axiu disance fro equiibriu posiion c. When we sar o easure he ie a axiu dispaceen, cacuae: a) he frequenc of he osciaions b) ie aken o ge o equiibriu c) he dispaceen when he speed is jus haf of he axiu one d) skech he dispaceen-ie graph L4/9-34, SIMPLE HARMONIC MOTION

7 6. Dnaics of s.h.. s.h.. is acceeraed a = ω he acceeraion is caused b he resuan force acing on he osciaing objec ( nd N.L.) F = a = ω rises wih he dispaceen bu i is awas poined owards equiibriu! 9. Discuss F in boh previous exapes of s.h.. a) Which forces ac on he ass on a spring (sipe penduu) and wha is heir resuan? b) How is he size of he resuan reaed o he dispaceen? 1. Look a a of he previous cacuaions and if ou have enough inforaion, cacuae he axiu resuan force and he force a paricuar dispaceens fro he exapes. If ou canno do ha, sae which daa are issing. 7. Energ of s.h.. assue FREE (undaped) osciaion, where he echanica energ is no convered ino oher pes he oa echanica energ reains consan, on he vaues of poenia and kineic energ can change o obe he forua E = E + E ech k p a equiibriu - kineic energ is... (he objec has he axiu speed ), poenia energ is zero a axiu dispaceen kineic energ is... (he objec sops here), poenia energ is axiu 1 1 E kax = v = ω r = Epax = E ech energ dispaceen graph energ ie graph energ... energ energ... energ... energ... energ... energ... energ ie -r r On he ie axis abe he fracions and uipes of period (assue zero iniia phase) SIMPLE HARMONIC MOTION

8 8. Free, daped and forced osciaions, resonance free osciaions echanica energ is conserved idea siuaion on daped osciaion echanica energ is convered ino oher pes rea siuaion - he apiude of he osciaions gradua decreases, THE PERIOD STAYS THE SAME (see he eqn) hp:// hp://paws.keering.edu/~drusse/deos/sho/dap.h - igh daped penduu in he air - heavi daped no osciaion, he objec jus reurns o equiibriu - criica daped = heavi daped during he shores possibe ie T/4 shock absorbers hp://en.wikipedia.org/wiki/shock_absorber hp://auo.howsuffworks.co/car-suspension.h SIMPLE HARMONIC MOTION

9 forced osciaions rea siuaion where an exerna force is used o keep he osciaions e.g. a swing or Baron s penduus he force appied is no he on iporan quani, he force shoud be appied in suiabe ie inervas (driving frequenc) o do iniu work (in. energ needed o keep osciaions) = resonance apiude of vibraion f driving frequenc driver penduu L4/ Look a a of he previous cacuaions and if ou have enough inforaion, cacuae he oa echanica energ sored in he osciaing sse, axiu kineic and axiu poenia energ. Assue free osciaions. If ou canno do ha, sae which daa are issing. Answers: 1. a) b)..5sin(157) 5. a).5 rad = 45º b).1 rad = 18º 6. a).63 s; b).5 s -1 ; c) s - ; d).4 s 8. a).7 Hz; b).36 s; c) ±1.73 c N;.1 N N; 13.4 N J J SIMPLE HARMONIC MOTION