LAB 6: SIMPLE HARMONIC MOTION

Transcription

1 1 Name Dae Day/Time of Lab Parner(s) Lab TA Objecives LAB 6: SIMPLE HARMONIC MOTION To undersand oscillaion in relaion o equilibrium of conservaive forces To manipulae he independen variables of oscillaion: he ampliude, he frequency, and he phase To predic he equaion of moion of an oscillaing sysem To undersand conservaion of energy in an oscillaing sysem and calculae he oal energy of such a sysem. Inroducion In any sysem in which a conservaive force acs on an objec, we can define a poenial energy funcion as par of describing how ha force will affec he objec s moion. As you have learned, he poenial energy U in a one-dimensional sysem is relaed o he force by du ( ) = F( ) (1) d This ells us ha wherever here is a local minimum in a poenial energy vs. displacemen graph, he force a ha poin is zero. (Technically, his is also rue of a local maimum; maima of graviaional poenial energy are common, bu hose of oher force ypes are less likely o be encounered. You will only be dealing wih minima in oday s lab.) In an eperimen where all forces are conservaive, hen, he poin of maimum kineic energy should equal he poin of minimum poenial, and hus he poin of force equilibrium. In Fig 6-1 a righ, all forces are balanced when he objec is a poin 0. When i moves away from 0, forces will decelerae i unil i sops, reurns, and passes 0. In he absence of fricion, his process would keep on going forever, as he body will ride up he poenial curve unil all he kineic energy i had a poin 0 has been convered o poenial energy. E U() Whenever a poenial energy vs. displacemen graph 0 displays a poenial well like his, we can use a Taylor Fig 6-1 approimaion o esimae he poenial energy very close o he poin of equilibrium: U( 0) = U( 0) + U ( 0)( 0) + U ( 0)( 0) +... U( 0) + k( 0) (2) 2 2 This is a good approimaion because U'( 0) = 0, and sufficienly close o 0 he second derivaive U " is approimaely consan. We ll call ha consan k. Wihin he domain in which his holds (which mus be deermined for any paricular problem or

2 2 eperimen) he force as he body moves away from 0 is herefore du ( ) F( ) = k( 0) (3) d As you know, when a force varies direcly wih he displacemen from sable equilibrium, we call he resuling oscillaion simple harmonic moion. As his derivaion shows, any ime here is a local minimum in poenial energy, sufficienly small oscillaions will be simple harmonic moion. Oscillaion on a spring The simples seup o use for observing simple harmonic moion is a spring wih a mass suspended from one end. By Hooke s Law, he force eered by he spring varies direcly wih disance from equilibrium which is precisely he definiion of simple harmonic moion! In oher words, equaion 3 above holds for an objec on a spring a all poins, no merely wihin a small neighborhood of 0. F = k( ) (4) spring 0 As we have done before when we do spring eperimens, we se he equilibrium poin 0 as he origin ( = 0) for simpliciy. To find he displacemen-ime equaion for his siuaion, we firs find he acceleraion a any poin: d F = ma = m = k or 2 d 2 2 d k = 2 (5) d m To solve he equaion on he righ requires knowledge of advanced differenial equaions, bu he soluions ake he following general form: () = Asin( ω δ ) where k ω = (6) m A and δ are arbirary consans inroduced when he equaion is inegraed. You can verify yourself ha his is a soluion o equaion 5. The posiion of he body, when ploed agains ime, will look somehing like his: () = sinω Fig 6-2 Each of he hree variables A, ω, and δ modifies he funcion in an imporan way. As you can see from he algebra, muliplying sin ω by a consan A will srech he funcion verically. A Fig 6-3 () = Asinω

3 3 ω deermines he period of he oscillaion: Fig 6-4 2π/ω δ indicaes he iniial phase of he oscillaion. A posiive value for δ shifs he graph o he righ: 0 δ = 0 Noe: he ime 0 can be obained by noing ha sin(ω0-δ)=0, so (ω0-δ)=0, ±π, ec. This allows you o solve for 0: 0 = δ/ω, or δ = π/2 0 = (δ±π)/ω, depending on he iniial condiions. Wha is he correc choice here? δ = π Fig 6-5 As a final noe, observe he similariy beween he graphs of posiion and velociy in an oscillaing sysem: () = Asin( ω) v v () = ωacos( ω) Fig 6-6 The velociy graph is he firs derivaive of he posiion graph, as you would epec. Noe ha a =0, when posiion is also 0, he velociy is a is maimum.

4 4 INVESTIGATION 1: BASIC HARMONIC MOTION You will need he following maerials for his invesigaion: Harmonic moion assembly wih a verically hanging spring Two hanging masses Moion deecor Meersick Aciviy 1-1: The Equaion of Harmonic Moion 1. Find he mass of each of your mass bobs. m1 = m2 = 2. Measure how far each mass sreches he spring. Use his displacemen o calculae k, he spring consan. Remember o use correc unis (N/m). y1 = k = y2 = k = Quesion 1-1: How well do hese measuremens of k agree? You should redo he measuremens if hey differ by more han 10%. 3. Open a new aciviy in Daa Sudio. You should only have a moion deecor hooked up o he compuer. In Seup, se he deecor o read 20 daa poins per second and check posiion, velociy, and acceleraion. Open a graph of posiion vs. ime. 4. Posiion he moion deecor carefully under he hanging spring. Make sure ha he deecor is as close o verical as possible. 5. Use he ligher mass bob for his eperimen. 6. Do a pracice run on he compuer: pull he mass bob up unil he spring is fully compressed and le i fall. Make sure ha i does no hi he moion deecor. Wih he bob now oscillaing up and down, sar he Daa Sudio run and make sure ha he deecor can see he mass bob along is whole pah. Make sure he bob is oscillaing verically and has no swinging moion. 7. When you ge a clean sinusoid on your posiion graph, add plos of velociy and acceleraion on he same graph window. Prin ou his graph. Once you have he shee of paper: a. Draw verical lines connecing he maima of he posiion graph o he velociy and acceleraion. When he posiion is a is erema, wha is he velociy and acceleraion? b. On he acceleraion graph, draw he direcion of he force where he acceleraion is a is larges posiive or negaive values. Is i along he direcion of moion (velociy), or agains i? Does i have he same sign as he displacemen, or opposie?

5 5 8. Se he meersick up ne o he hanging spring. Noe he heigh of he mass bob a equilibrium as measured by he meer sick, and read off he graph he equilibrium heigh as measured by he moion sensor. These will be differen; he moion sensor value is wha will be repored : equilibrium heigh (meer sick) = equilibrium heigh (moion sensor) = Predicion 1-1: Give he equaion of he moion of his sysem if you se i o oscillaing by pulling he bob down 20 cm and releasing. Use he value of k ha you derived on he previous page and your measured m o calculae he frequency. Geing he phase righ is he hard par. Skech wha he graph will look like on he aes below. Refer o {eq.6} in he inroducion o his lab if you are confused. Label your graph wih values for he ampliude, period, and phase offse. Use posiive values for posiions above equilibrium and negaive values below. 9. To un-cluer you window, delee he velociy and acceleraion graphs. Now, pull he mass bob down 20 cm from equilibrium. Sar recording daa, hen release he mass bob. Record daa for a leas 10 seconds. Click and drag o selec your daa beginning when you released he mass bob. 10. Under Fi, selec Sine. Quesion 1-2: The sine-fiing procedure in Daa Sudio will reurn four consans: A, B, C, and D. Wha do hese consans mean? You mus find ou by double-clicking he bo conaining he fi parameers. Wrie he equaion below, using A, B, C, and D in heir proper place, and give he physical meaning (e.g. ampliude, phase, ec.) of each. You may wan o epress hese in erms of ω and δ as we have defined hem: A = B = C = D = Also, in erms of A, B, C, and/or D, wha are ω and δ? ω = δ = Quesion 1-3: The phase-offse parameer of he sine funcion (δ in {eq.6}) which your graph yields will probably no be equal o he one you prediced in Predicion 1-1. This is because you did no sar

6 6 recording daa a ime = 0, bu held he mass bob unil you knew he recording had already begun. The DaaSudio sine fi has an opion ha will shif he fi over and ake he offse ou for you. Double click on he fi parameer bo again, and selec Fi wih firs seleced daa poin a X=0. You have o do his EVERY TIME YOU FIT. (I s simpler ha way ) Calculae values of Ampliude, ω and δ from he fi parameers, and give hem here: Quesion 1-4: How well did you predicions mach your eperimenal resuls? 11. Prin ou a copy of your graph, including your fi curve and daa, o urn in a he end of lab. INVESTIGATION 2: ADJUSTING DEPENDENT AND INDEPENDENT VARIABLES There are only hree variables in he classical harmonic-moion equaion (6), and four in he sine-fi rouine which Daa Sudio applies o an eperimen. The fourh is merely a verical ranslaor: in (6), i is assumed ha he sysem is a equilibrium a posiion = 0, while in he eperimens you are running oday, he zero posiion is occupied by he moion deecor. As you saw in he previous invesigaion, wo of hese variables are compleely conrolled by he eperimener: ampliude and phase. The hird, period, a funcion of ω, is compleely dependen on he eperimenal seup. This means ha you should be able o adjus each of hese variables and no affec any of he ohers. You will es his hypohesis in his invesigaion. You will use your daa from he previous invesigaion, and compare wih he daa which resuls when you change one of hese variables a a ime. Aciviy 2-1: Varying Ampliude 1. In his eperimen you will record he moion of he mass bob wih differen ampliudes. You will once again use he ligher of he wo masses. Predicion 2-1a: Predic he equaion for he posiion-ime graph which would resul if you pulled he mass bob down o only 10 cm. Draw i on he following aes, wih your predicions for he graph s ampliude, period, and phase offse.

7 7 2. Perform he eperimen using he eac same procedure you did in he previous invesigaion, ecep his ime, you will pull i down only 10 cm. 3. Using he sine fi, find he equaion for he bob s moion. Make sure ha you have seleced daa wih he mouse, and ha you ell he fi o sar wih he firs poin a =0. Quesion 2-1: Compare his equaion wih he one you deermined in Invesigaion 1. Are your values for ω he same? For δ? Why or why no? Quesion 2-2: Compare your values of ω and δ in Invesigaion 1 and his aciviy: how did changing he funcion s ampliude affec he period and phase? Aciviy 2-2: Varying ω In he las aciviy, we played wih he ampliude of our moion and asked if changing i affeced he res of he eperimen. In his aciviy, we will play wih he quaniy ω, which he heory says should k equal. As we are using only one spring, we will vary ω by using differen masses. m 1. Take he heavier mass and hang i from he spring. Predicion 2-2: Predic he equaion of he posiion-ime graph which will resul from releasing he heavier mass bob from a spring eension of 20 cm. Draw i on he following aes, wih your predicions for he graph s ampliude, period, and phase-offse. 2. Hold he mass bob cm below equilibrium, sar recording daa, and release he bob. 3. Fi your sine curve o he daa. Record i below. Make sure you shifed he =0 of he fi. 4. Prin ou a copy of your graph o urn in a he end of lab.

8 8 Quesion 2-4: How well did he daa mach your predicions? Eplain any discrepancies. Quesion 2-5: Compare he bob s moion in his eperimen o is moion in Invesigaion 1: was i affeced in any way? Was he ampliude or he phase affeced by he change in mass? Aciviy 2-3: Varying Phase In his aciviy we will compare wha happens when you sar he bob oscillaing from differen poins in is rajecory. 1. Use he ligher mass bob for his aciviy. Predicion 2-3: Predic he equaion of he posiion-ime graph which will resul from releasing he mass bob from 20 cm above equilibrium. Draw i on he following aes, wih your predicions for he graph s ampliude, period, and phase-offse. 2. Hold he mass bob 20 cm above equilibrium. Sar recording daa, hen release he bob. Record daa for a leas 10 seconds. 3. Fi a sine curve o your daa. Make sure you shif he firs poin o =0. Record is equaion: 4. Prin ou a copy of your graph o urn in a he end of lab. Quesion 2-6: Was your predicion correc? Eplain any discrepancies. Quesion 2-7: Compare his equaion o he equaion from Invesigaion 1. Wha is differen? 5. For he las eperimen in his invesigaion, you will sar he mass bob oscillaing from he equilibrium poin by giving i a push up or down. The ampliude of he oscillaion should be close

9 9 o 20 cm, hough i need no be eac. You may need o ry a few imes before you ge a good oscillaion. Predicion 2-4: Predic he equaion of he posiion-ime graph which will resul from saring he mass bob s oscillaion a he equilibrium poin. Draw i on he following aes, wih your predicions for he graph s ampliude, period, and phase-offse. 6. Sar recording daa, hen se he sysem oscillaing as described above. Try o se he bob oscillaing as sharply as possible (i.e. ake as lile ime as possible in acceleraing he bob), because i is only when you release he bob ha i will ehibi rue sinusoidal moion. 7. When you ge an ampliude close o 20 cm, click and drag o selec daa from when you se he bob in moion. Record he he equaion of he curve (make sure you make he firs poin =0): 8. Prin ou a copy of your graph o urn in a he end of lab. Quesion 2-8: Was your predicion correc? Eplain any discrepancies. Quesion 2-9: Compare your wo graphs from his Aciviy and he graph from Invesigaion 1. As you varied he iniial phase of he oscillaion, how did he equaion change?

10 10 INVESTIGATION 3: ENERGY IN AN OSCILLATING SYSTEM Le s go back o he idea of he poenial well discussed in he inroducion. The shape of such a well indicaes wha kind of force we are dealing wih: graviy, which is (almos eacly) consan over shor disances, yields a sraigh-line poenial graph wih slope g. A spring force, which varies linearly wih disance from equilibrium, yields a parabolic poenial graph wih he vere a he equilibrium poin. This is imporan, because only a parabolic poenial funcion will yield rue sinusoidal harmonic moion a all ampliudes. Oher poenials will approimae sinusoidal moion a small ampliudes, bu as he body swings furher from equilibrium, is moion will begin o diverge from a rue sine curve more and more. The eperimenal seup you are using oday gives resuls which map o sine funcions wih very lile deviaion. This indicaes ha he poenial energy under which he bob is acing is quadraic wih respec o disance from equilibrium. As i urns ou, we can disregard graviy in calculaing he poenial energy of a spring supporing a hanging mass because of he linear naure of a spring force. Wheher your spring is siing a is rue equilibrium or suppors a mass bob which sreches i a disance of 0 o a new equilibrium, sreching or compressing ha spring an addiional disance y will resul in he same force ky being eered on he mass. The effecive poenial energy a equilibrium is 0 (his is precisely wha equilibrium means); he graviaional poenial downward is balanced by he spring s poenial energy, essenially he chemical poenial energy holding he meal ogeher and pulling he end of he spring upward when i is sreched. As he sring sreches farher, EGP does no increase, bu Uspring does, meaning he ne endency is back oward equilibrium; if he spring is compressed pas equilibrium, he spring s poenial decreases (relaive o is rue equilibrium poin), and he graviaional poenial pulls he mass bob back down. Aciviy 3-1: Kineic Energy from Spring Poenial 1. Under Seup, selec he checkboes ne o posiion and velociy under he moion sensor. 2. Using he calculaor, program wo equaion measuremens ino he eperimen: kineic energy and spring poenial energy. Remember ha, in giving he compuer he equaion for poenial energy, you mus find he equilibrium poin and define an equaion o find he bob s disance from ha poin. 3. Define a hird funcion composed simply of he sum of he oher wo funcions: his is he oal mechanical energy of he sysem. Ener your equaions here: KE =

11 11 U = ME = 4. You may use whichever mass bob you like for his Invesigaion. Make sure you ener he correc mass in your equaions. Predicion 3-1: Predic he maimum and minimum kineic energies of he sysem if i is se ino oscillaion wih an ampliude of 20 cm. Predicion 3-2: Predic he behavior of he sysem s mechanical energy as you have defined i in your calculaed funcion. Will i be consan, or will i vary? If i varies, predic wha paern i will follow. 5. Tes your predicions. Se he sysem in moion by holding he mass bob 20 cm below equilibrium, saring he daa recorder, and releasing. 6. Prin ou graphs of posiion and velociy on one shee and he hree calculaed energy funcions on anoher shee. Scroll he aes so you can clearly see he poenial and kineic energies oscillaing back and forh. Quesion 3-1: Were your predicions correc? Did your poenial and kineic energy calculaions display conservaion of energy? Why or why no? Eplain any discrepancies wih your predicions. End-of-lab Checklis: Make sure you urn in: Your lab manual shees wih all predicions and quesions answered and daa filled in. Graphs for he following aciviies: 1-1, 2-2, 2-3 (2 graphs), 3-1 (2 shees)