2.5. The equation of the spring

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1 2.5. The equaion of he spring I moun a spring wih a weigh on i. I ie he op of he spring o a sick projecing ou he op end of a cupboard door, and fasen a ruler down he edge of he door, so ha as he spring oscillaes, he weigh moves up and down he ruler. I se he weigh oscillaing and ask he class wha he hink I am going o have hem do. The do no seem pleased wih he quesion, nor do he observe he wonderfull hpnoic moion of he spring wih childlike deligh. Their answer is more like an accusaion: You re going o ask us o find a formula for he heigh of he weigh a ime. No! No I m no, a leas no he wa ou expec. An analsis of he ssem, wih los of phsics and Newon s laws of moion and all makes for a fascinaing journe (conrar o wha ou migh hink on a Wednesda morning wih more han half he week o go) bu i reall belongs o universi. One needs o undersand calculus o do i. Wha we will do insead is work wih an analog. I ake up he biccle wheel ha we have used before, se i spinning a consan speed, and using he overhead projecor, I projec he shadow of he sick ono he wall. I ask hem o ignore he wheel, and jus wach he shadow of he sick going up and down. Then I ask hem o swich heir aenion back and forh beween he spring and he wheel. The effec is remarkable and has o be seen o be believed. The moions of he wo shadows are idenical and he sudens sare a hem mesmerized. The shadow of he sick on he wheel reall looks like i migh be on a spring. If ever high school should have an old car ire, hen i should also have a bunch of springs. M spring is some 2 cm in diameer, fairl loose, wih a period of around 2 seconds, and will easil oscillae wih an ampliude of 2-3 cm. This is experimen is well worh doing as he effec is quie sriking. Ge he kids o come up wih a spring and se he Well! We alread have a formula for he sick on he wheel. And wha he have jus seen is designed o convince hem ha he same pe of formula applies o he spring. And i does he heigh of he weigh on he end of he spring is given b an equaion of he form: = A sin 2 π. p where A is he ampliude of he oscillaion and p is he period. A p 2p -A 1

2 Suppling values for A and p. The formula for he spring is he same as he formula for he wheel! I end his pronouncemen wih a flourish, almos expecing applause a his unexpeced beneficence of he universe, ha a spring behaves he same wa as a wheel. Bu I am me insead wih blank or confused sares. How can a sine funcion describe he spring? There s no angle. Where's he angle? Tha s a remarkable commen, and one I hadn expeced. There is indeed no angle, and here is reall no reason a all o expec ha he sine funcion is wha works for general oscillaor behaviour: springs, guiar srings, ides, ec. [Though mabe ides are differen because he are generaed b he circular roaion of he moon abou he earh.] The applicabili of he sine funcion in all hese cases is reall a remarkable absracion. The ke proper of his funcion which makes i so widespread in he descripion of oscillaions isn abou angles a all, i s abou raes of change and i comes from calculus. So his observaion, ha he sine funcion describes he spring, deserves enormous respec. Jus o have some numbers, we sar he spring oscillaing and make measuremens. We pull he weigh down 3 cm from is equilibrium posiion, and le i go. We hen ime 1 ccles a 21 s for a period of p=2.1 seconds per ccle. We ge: = 3 sin 2π. 2.1 Noe ha his formula gives us = a =, ha is, i makes he assumpion ha he weigh is a is equilibrium posiion a =. Well his is a convenien place o ake he ime origin, because i keeps he equaion simple we can use a sin funcion wih no phase shif. Bu a suden a he back has a hand in he air. Doesn he ampliude decrease? Shouldn i become less han 3? In fac won he spring evenuall sop. Indeed Explaining he sine Where does he sin funcion come from? A nice quesion and here s he causal chain available o someone who undersands calculus. Le denoe he displacemen of he weigh from is equilibrium posiion. Then Hooke s Law ells us ha he resoring force exered b he spring on he weigh is proporional (and opposie in sign) o. Now since he force on he weigh is proporional o is acceleraion (Newon) ha means he acceleraion of he weigh is proporional o minus. Now if ou know calculus, ha means ha he second derivaive of is proporional o minus, and ha s a basic proper of he sin funcion. [I s rue of he cosine as well, and we could also have used ha.]

3 The effec of fricion Wha s fricion all abou? I s he endenc of hings ha are in moion and running ino one anoher o hea up. And ha hea energ is solen awa from he energ of moion (kineic energ KE). As he spring coils and uncoils, i heas up, and he ampliude of he moion decreases. This loss of KE is known as damping, and our work so far has ignored ha. The formula we ve developed so far is known as he equaion of he undamped spring. Tha s how he spring would behave in he ideal world of no fricion he ampliude would sa a 3 indefiniel. Does no he same hing happen o he biccle wheel if we don keep urning i a a consan speed? Does i no also slow down? Yes i does, and for he same reason. Le s look more closel a hese wo effecs of fricion, on he wheel and on he spring. How does fricion affec each oscillaion? I le he class wach each of he wheel and spring for a few momens and hen I ask hem o draw graphs of he moion of each objec, he sick and he weigh, which ake accoun of he loss in KE. I s an ineresing exercise because fricion affecs he moion of each in wo differen was. The wheel. For he wheel, he ampliude is unchanged bu as he wheel slows down he period p increases. We ge a graph where he disance beween successive crossings of he axis increases unil he moion sops alogeher The damped wheel The spring. This ime i s he ampliude of he oscillaion ha decreases over ime, each max is a bi lower han he previous one. Bu some quesions arise. How exacl does he max decrease over ime. The graph a he righ supposes a linear decrease, bu oher sudens drew a curvelinear decrease. Wha is he real siuaion? And wha abou he period? Does i sa he same, or does i increase, like ha of he wheel? Or does i decrease? The graph a he righ supposes ha i sas consan he spacing beween successive crossings of he axis is alwas he same. So hese are ineresing quesions exacl how does fricion affec he ampliude and he period of he spring? The damped spring 3

4 The effec of damping on he period. The effec of fricion is o decrease he ampliude, so wha we reall wan o deermine here is he effec of ampliude on period. If he oscillaions are small, is he period an differen (shorer or longer?) han if he oscillaions are large? We collec some daa, iming 1 oscillaions for a range of saring ampliudes. The daa show quie convincingl ha he period is alwas he same. There s some small variaion around 2.1 seconds/ccle, bu here s no rend in his variaion ha is, here s no up- or down-paern and so here s no reason o suppose he period changes wih he ampliude. In fac, a heoreical argumen will show ha he period of a spring is independen of he ampliude. As he moion of he spring decas he period remains consan. So he above graph, ha shows a consan period as he ampliude decreases, seems o be correc. saring ampliude ime for 1 ccles Thus he wheel and he spring exhibi an ineresing pair of opposie behaviours in response o fricion, summarized in he able a he righ. Here p() is he period of he wheel and A() is he ampliude of he spring and he boh depend on ime he period increases and he ampliude decreases. Bu he precise form of hese funcions is unknown. Wheel Spring ampliude consan period increases ampliude decreases period consan 2π = 3sin p() 2π = A() sin 1. 2 Our nex ask will be o r o ge hold of he precise naure of he funcion A(). This urns ou o be an ineresing quesion. Problems 1. A speck of pain is on he ouside of m bike ire. Le be he heigh of he speck above he road. Suppose I sar a res on m bike and accelerae o some maximum speed which I hold consan. Draw a graph of agains which illusraes his acceleraion, assuming ha a he beginning (=) he speck is in conac wih he ground. 2. You push our bab siser on a swing making her go higher and higher wih each oscillaion unil she is high enough and our pushing simpl susains her a a consan ampliude. (a) Draw a graph of he angle he swing rope makes wih he verical agains ime.. B he wa, i urns ou ha he period of a swing is pre close o being independen of he ampliude, a leas for bab-siser pe oscillaions. (b) On he same se of axes draw he graph of he verical disance beween he swing sea and he ground. 3. I am pumping up m bike ire. M moion is periodic, each ccle being a sequence of wo srokes, up and down, however he up-sroke is differen from he down-sroke. Draw a graph of he heigh of m hand agains ime, showing hree ccles. 4

5 4. A spring wih a weigh on he end is suspended from he ceiling of he room and i oscillaes up and down wih an ampliude of 6 cm making 15 complee ccles ever minue. Suppose he moion is undamped (no fricion) and le denoe he heigh of he weigh above he equilibrium posiion. (a) Draw a careful graph of agains assuming ha a = he weigh is a he equilibrium poin = and moving upwards. Show 2 full ccles. (b) Now find an equaion for our curve, ha is find a formula for a an ime. Use radian mode. (c) Locae on our graph all imes a which he weigh has heigh =3 cm, and use our formula of (b) o calculae hese exacl. 5. A spring wih a weigh on he end is suspended from he ceiling of he room and i oscillaes up and down wih a period of 3 s and an ampliude of 75 cm. Suppose he moion is undamped (no fricion) and a is low poin, he weigh is 5 cm above he floor of he room. Le denoe he heigh of he weigh above he floor. (a) Draw a careful graph of agains assuming ha a = he weigh is a is high poin. (b) Now find he equaion of our curve, ha is find a formula for a an ime. Use radian mode. (c) Find he firs momen a which he heigh of he weigh above he floor is =1 cm. 6. A spring wih a weigh on he end is suspended from he ceiling of a large freigh elevaor which runs up and down he side of a all building. Suppose he weigh oscillaes up and down wih a period of 4 s and an ampliude of 1 m. Suppose he moion is undamped (no fricion) and a is low poin, he weigh jus ouches he floor of he elevaor. Suppose ha he elevaor is ascending a a consan speed of 1 m/s. I urns ou ha his won affec he dnamics of he oscillaion inside he elevaor. In fac i follows from he laws of phsics ha here is no experimen an observer inside he elevaor can do o decide wheher he elevaor is saionar or is moving a consan speed. Your job, deailed below, is o draw he graph of he heigh of he weigh wih respec o he building agains ime. You should use squared paper o draw our graphs wih a scale so ha one square represens 1 m b 1 s. Indeed, le's inroduce hree variables. Le x denoe he heigh of he floor of he elevaor above he ground, le denoe he heigh of he weigh above he floor of he elevaor, and le z denoe he heigh of he weigh above he ground. We suppose ha a =, he floor of he elevaor is a ground level (x=) and he weigh is on he floor of he elevaor (=). (a) Begin b ploing separael he graphs of x and agains. Then ake our x-graph and mark z-poins on his graph a 1-second inervals, using he fac ha z=x+. Finall draw he z-curve hrough hese poins. (b) A he momens when is a is exreme poins = and =2, he weigh is saionar wih respec o he elevaor, and is speed relaive o he building mus be he same as he speed of he elevaor. Discuss he geomeric significance of hese poins on our z-graph. (c) Now find he equaion of our z-curve, ha is find a formula for z a an ime. Use radian mode. 5