Week 1 Lecture 2 Problems 2, 5. What if something oscillates with no obvious spring? What is ω? (problem set problem)
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1 Week 1 Lecure Problems, 5 Wha if somehing oscillaes wih no obvious spring? Wha is ω? (problem se problem) Sar wih Try and ge o SHM form E. Full beer can in lake, oscillaing F = m & = ge rearrange: F = m & ω = & up force due o Archimedes Principle m&&= Aρ g k w A w ρ m g + =& & down force due o graviy (where = disance from equilibrium poin and A=cross secional area) +ve This is now in SHM form ω k m so, effecive k = Aρ w g Noe ha g in his SHM equaion comes from he Archimedes conribuion 1
2 Roaional Oscillaions (review) These are sysems which oscillae roaionally abou an ais, raher han linearly Eg. Torsional Pendulum Trick here: noe ha linear variables simply change o angular ones: - all F s become τ s - all m s become I s -all s become θ s and so on. This suff is all review from firs year look a your old noes Linear: F = -k F = ma Roaional: τ = κθ τ =I & θ resoring orque orsional spring consan angular frequency, ω Combining as before: κθ = I & θ or angular displacemen κ θ = & θ I ω θ = & θ orque roaional ineria I = r dm Like he linear form ecep ha s are now θ s angular acceleraion Defining equaion for angular SHM
3 Going back o linear SHM again How do we figure ou where he block is a any given ime, i.e. wha is ()? is a nd order differenial equaion. To find as a funcion of ime (solving he differenial equaions) rewrie as: ω =& & d 0 where d + ω = ω = m k The general soluion* o his diff eq n is given by = c1 sinω+ c cosω (defining equaion for linear SHM) To evaluae consans c 1 and c, we need Boundary Condiions (BCs). (1) So, look again a our oscillaing block: Pull i ou o A and le go a = 0 So a = 0, = A (BC #1) A = ma displacemen Also a = 0, v = d/d = 0 (BC #) + equilibrium * This is somehing ha you learn in a differenial equaions course if you haven seen i ye you should soon A m = 0 3
4 Sub BC #1 ino (1) gives or A = c 1 sin( 0) + c A = c cos(0) Differeniae equaion (1), hen sub in BC # d d = ωc cosω ωc 1 sinω A = 0, d/d = 0 0 = ωc cos(0) ωc 1 c = 0 1 sin(0) So, subbing c 1 and c back ino (1) gives or = Acosω = Acos k m Noe ha his equaion only holds for he boundary condiions we assumed: a = 0, = A and v = 0. This gives us as a funcion of for his case Really imporan o keep in mind! 4
5 The Phase Consan φ Usually we wrie = Acos(ω + φ) (raher han jus =Acos ω) The value of φ depends on where we choose = 0 If we choose = 0 when = A, hen: = Acosω (hese were he boundary condiions we considered on he previous page) A However, if we choose = 0 when = 0, hen our plo of vs. looks like: The curve is shifed righ by π/ (bu of course in he general case his shif can be any value) A shifed righ by π/ = -π/ Mahemaically, his can be wrien as Or π = Acos( ω ) = Asinω (φ = - π/) Therefore, depending on where you define = 0, SHM can be represened by eiher a sin or cos funcion 5
6 1) ) 3) Therefore, here are hree equivalen ways o epress he soluion of = c ω sinω c 1 + = cosω & here we use BCs o se c 1 and c and have no phase consan. = C cos( ω + φ) = C sin( ω + φ) Use BCs o ge C and φ (noe: since C is he ampliude i will be he same for boh of hese bu φ will be 90 0 differen) Noe: We will use he soluion a lo, so we have o pick which one of he above hree we will use. We choose his one mainly because i is he form used in he ebook French. Oher books (like Pain) use he sin form more frequenly. We will mainly use: = C cos( ω + φ) Hey! Is here a relaionship Beween c 1, c and C? You be! C = + c 1 c You can see he proof of his on page 5 of he courseware or prove i wih phasors (cos and sin are 90 o ou of phase) C c c 1 Alhough we usually use he above cos form - you should be comforable wih finding each of he 3 forms of soluion for any oscillaing sysem!!! - see he eample on he ne page 6
7 E. A spring/mass sysem is pulled 0.5mm from equilibrium and hen given a push o give i an iniial velociy of mm/s as shown. Is resuling angular frequency is ω = rad/s. Develop an epression for (). 0.5mm v 0 = mm/s equilibrium = 0 = +0.5 mm (BC#1) and v = + mm/s (BC#) Also ω= rad/s Assume SHM applies Eamine he hree equivalen mehods: Mehod 1 uses = c sinω c 1 + cosω BC#1 gives 0.5 = C 1 sin 0 + C cos 0 C = 0.5mm BC# need o differeniae, giving v = ωc 1 cos ω ωc sin ω Subbing in BC# and w= rad/s gives = ωc 1-0 C 1 = Resul is = sin cos 7
8 Mehod uses BC 1 gives BC need o diff, gives = c cos ( ω + Φ) 0.5 = c cosφ v = ωc sin ( ω + Φ) Sub in BC and ω = rad / s gives = csin Φ ge = + csin Φ Solving he wo eq ns simulaneously gives so Φ = 1.rad c = = + 1.5cos( 1.) Mehod 3 uses BC 1 gives BC diff gives = csin ( ω + Φ) 0.5 = csin Φ v = ωccos ( ω + Φ) Ou by π/ radians (1.57) as epeced or = ccosφ c = / cosφ Solving he wo eq ns simulaneously gives Φ = 0.34 c + = 1.5 so = + 1.5sin( ) 8
9 Maple Plos for 3 Equivalen Mehods 1 := sin( ) +.5 cos( ) := 1.5 cos ( 1.) These are all idenical! 3 := 1.5 sin ( +.34) 9
10 Ploing Displacemen, Velociy and Acceleraion [back o using ()=Acos(ω+φ)] Plos of he displacemen, he velociy v, and he acceleraion a, as funcions of ime (φ = 0). A Displacemen () ( ) = Acos( ω) Noe ha v has advanced (lef) by π/ compared o () ωa v Velociy (v) d( ) & = = d v( ) = ωasin( ω) Noe ha a has advanced by π compared o () ω A a v ma a =0 a=0 a v ma a =0 a=ma a v=0 a ma Acceleraion (a) d ( ) && = = a( ) = ω Acos( ω) d 10 Hey check ou IP demo#1 for his and follow along wih he bouncing spring
11 Represening SHM as a Comple Eponenial I is useful o be able o define SHM in erms of a comple eponenial. Why? Because when we ry and solve more complicaed versions of our differenial equaion (say, ones which include damping erms or forcing erms) he mah becomes much easier if we can urn our equaion ino a comple eponenial. The comple eponenial can also be represened as a vecor (phasor). This is handy when we sar o add waves ogeher adding vecors is much easier han adding a bunch of equaions! To go from he SHM equaions o a comple eponenial, we need o go hrough a couple of seps, shown below. The ne few pages will illusrae hese seps. SHM ( = C cosω) roaing vecor comple number comple eponenial and comple vecor makes mah easier easier o add waves as vecors 11
12 Simple Harmonic Moion and Circular Moion: The Roaing Vecor Represenaion Imagine paricle P (or vecor OP) moving wih consan speed v o couner-clockwise around he circle as shown (radius A). Poin Q is he projecion ono he ais, and as he vecor OP roaes, he poin Q oscillaes in SHM from o o + o. The y posiion also ehibis SHM A any posiion: = Acosω y = Asinω y P y = Asinω A ω O Q o = Acosω Check ou he Virual Physic Laboraory websie on my webpage links. Look under Mechanics, hen Simple Harmonic moion. This will help you visualize he moion. Noe ha he only difference beween his demo and he figure above is ha his demo has he paricle oscillaing along he y ais raher han he ais. 1
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