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# IntroductoryPhysicsII PHYS153

Drexel

GPA 3.54

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This 176 page Class Notes was uploaded by Vernice Schuster on Wednesday September 23, 2015. The Class Notes belongs to PHYS153 at Drexel University taught by JosephTrout in Fall. Since its upload, it has received 50 views. For similar materials see /class/212518/phys153-drexel-university in Physics 2 at Drexel University.

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Date Created: 09/23/15

Physics 153 Physics Week Two Oscillations Dr Joseph J Trout josephtroutdrexeledu 6103486495 SHM Simple Harmonic Motion Consider a mass attached to a spring with a spring constant k at rest at x00m on a frictionless table Then pull mass out to x A and release from rest XZ x00m XZA PERIOD T Time to complete one oscillation Measured in seconds FREQUENCY f Number of oscillations per seconds Measured in Hertz 1 Hz 1 1s xZ A x00m XZA 085 x x xix x xampx5xixxixim xix 2 nixix Xiix Iiitix xix m x xixitn CiIEX xEIEIEIiIEIEXEX XEX XEXEIEXbl t005 It T I Lt T ICEIEXEXEIEIEIEIEIEXEX X XEX IEIEX D i4 KEXEXEIEIEXEIEXEXEIEX 1 XXEXEXEXE 085EXEX X I XEIEIEXEI IEXEI 1 X XE2 tT I iIiX X IiliIEIEIEX X X X X I I X T I tz T I I X A x00m xA Position vs Time O 05 1 15 2 25 3 35 4 45 5 55 6 65 7 75 8 85 9 95 ts xtAcos0 Angular Frequency Position vs Time O 05 1 15 2 25 3 35 4 45 5 55 6 65 7 75 8 85 9 95 ts xtAcoswt Position vs Time Xm GLLLJIQLotwwhm Ax v Slo eofXVSt lot avg p p Position vs Time Xm GLLLJIQLotwwhm Ax v Slo eofXVSt lot avg p p Position vs Time xm IJ39IIlsdAJIIULOlwwb Ax v Slo eofXVSt lot avg p p Position vs Time Position vs Time Xm GLLLJIQLotwwhm max x v Slo eofXVSt lot avg p p Position vs Time Xm GLLLJIQLotwwhm V max max Ax v Slo eofXVSt lot avg p p Position vs Time xm Vmax A A A vavg x Slope of X VS t plot At vms 125 75 25 25 75 10 125 0 05 1 Velocity vs Time 15 2 25 3 35 4 45 5 55 6 65 7 75 8 85 9 95 ts vt vmaxsinwt am52 O 05 1 Acceleration vs Time 15 2 25 3 35 4 45 5 55 6 65 7 75 8 85 9 95 ts at amaxcoswt Consider a mass attached to a spring with a spring constant k at rest at x00m on a frictionless table Then pull mass out to x A and release from rest XZ x00m XZA EmUSK Etotz kx2 mv2 When the mass is at x A the velocity is zero EmtUSK Emtz kx2gmv2 i xA x00m Emtz kx2mv2kA2OkA2 When the mass is at x A the velocity is zero m I I X x00m XZA 1 1 Etotzakx2 mv2 EmkAcoswt2m Awsinwt2 E lkAZCOSZwtlmA2wzsin2wt ooE lot 2 2 m E 1kA2 2 1 2 k 2 tot cos wt mA sm out 2 2 m EtotzgkA2coszwtsin2wt sin29c05291 EmkA2 Energy O NLUbU39ICDV 02 04 06 08 Energy versus Time 12 14 16 18 2 22 ts 24 26 28 32 When the mass is at x 00 m there is no energy stored in the spring and the velocity is equal to the maximum velocity 1 2 1 2 1 2 k kA 2 x 2quot 2 l i X x00m XZA When the mass is at x 00 m there is no energy stored in the spring and the velocity is equal to the maximum velocity FS kx Energy can be used to find a function for the magnitude of the velocity at any position x XZ x00m xZA 1 2 1 2l ka 2771V 2 1 2 1 2 1 2 271112 2kA ka mvzkAZ kx2 mv2kA2 x2 2 A2x2 m M2 M 5A2 x2 m Simple Harmonic Motion and Circular Motion 63ixix5xixxExixixixEXEXExixixixixim X A XZA 3 x00m A X 44 A Aa3 43 v gt v o v v v o o o v o v v ltnx 5an lt1an Vff o4 A E v XZ A x00m XZA XZ A x00m XZA 2 kxmdc dt 5 de m a t2 Solutions must be cos sin or exponential Fnetma 2 kxmdC dt 5 de m a t2 W 611 W dt dzx a t2 Acoswtclgt Aoosinoutclgt lvmaxAw szsinwt lamaxA 002 xtAcoswtqb d2x Vt Awsinwtqb kXm dtZ 2 lvmaXIZAa 2 ald f2 szsinwtqb k d x dt m x dtz lamaxlewZ Acoswtclgt AOU2COSOUZI k 2 w m k w m xtAcoswt vt vmaxsinwt Awsinwt Z39 a amaxcoswt szcoswt XZ A x00m XZA circumference 2 T39 r 2 T39 A distance 2TrA 2n A time T T Aw max x x00m XZA Recall x00m x Fnetst ma kx mbn w m a kphvum u mkw k w m x00m k a m 21T w T 21T m T 2 quot k A9 At x A 00m III1II IIIII Velocity of Mass GAILAIL mfwxfwx smm ms S m v mm VA v u v v m m m m m m I I m m g m m W M M So what we know so far xA x00m xtAcoswt vt vmaxsinwt Awsinwt at amaxcoswt Awzcoswt k w m vmaxAwA k m amaxA 22145 711 k 100 Nm Mass is released from rest at x A 025 m m 20 kg I x A 025m x200 xA025m AO25m wE JW22243 1 m 20kg 21T 2 1T 2 12813 00 2243 vmaxAwO25m22431O56ms amaxAw2O25m224512125m32 k 100 Nm Mass is released from rest at x A 025 m m 20 kg I x A 025m x200 xA025m AO25m wE JW22243 1 m 20kg 21T 2 1T 2 12813 00 2243 vmaxAwO25m22431O56ms amaxAw2O25m224512125m32 Check k IIOONm 0 5m 20kg 056ms k A amA 025mW125ms2 m 20kg vmax k100Nm x A 025m AO ix 5m 2224 5 1 w I a T 22815 SIN vmaxAw056mls amaxzAoz2125ms2 Mass is released from rest at x A 025 m m 20 kg I I xA025m x00m xtAcoswt025mcos22451t vt Awsinwt O56 mssin224s1t at Aw2cosat 125m52cos224s1t xtA coswt025mcos224s 1t vt Awsinwt 056mssin224silt aI szcoswt 125mszcos224s 1t You should be able to roughly draw out plots X A025m I In I I I 2T5625 A 025m T 2815 224ms 125ms2 125ms2 A2025m I I I I A 025 T2815 4 2T25625 m v 224ms 224mS 125mS2 125nd32 m 20 kg I I xA 025m XZODm Group Quiz Simple Pendulum Simple Pendulum Simple Pendqum Potential Energy U mgy Kinetic Energy K g m 12 l EmtUKmgymi2 C n39 I l l l I l I I I Simple Pendqum Potential Energy U mgy Kinetic Energy K m 12 EtotUImgmv2 V 1 V v I R 1h quot quot39 H U U quot max 1 00 m s Simple Pendulum Emtzmgymv2 E1E2 1 2 1 2 mgy1 mv1mgy2 mvz 900 2 V2W12gy1y2 UZUmin v vmax Pestirgl ljm Potential Energy U mgy Kinetic Energy K m 12 Em UKmgylmv2 900 2 Simple Pendulum Emtzmgymv2 E1E2 1 2 1 2 mgy1 mv1mgy2 mvz 900 2 V2 V12gy1y2 Simple Pendulum Kmax Umax Umin 1 2 mvmaxmgymaxmgymin 2 lvmaxlz V2gymaxymin 0 399 y 1 m max ymin Simple Pendulum Simple Pendulum 771 vim mgl mgl cos 9m vmaxxl gl1 cos 9m Simple Pendulum 1 quot quot Simple Pendulum mg Simple Pendulum Simple Pendulum Simple Pendulum II N CD Simple Pendulum max mg sine sin ONO for small 0 max mg9 i max mg l Simple Pendulum Fnetma 2 d s mgsm9m a39z2 519 dsld9 dt dt dzsld29 a39z2 a39z2 E mph Pendulum Fnetma 2 cis wgn9m c112 99maxcoswl d9 E 9mastmwt 03929 F 9maxw2coswt Simple Pendulum Fnelma mgsin m c112 sin99 dzs 9 g 12 9ld29 g 12 03929 12 g9maxcoswtl9mmw2COS wt g9l glw2 99maxcoswt 2 ZTe emwsinwz T21Tg Teemw2coswz Simple Pendulum Physical Pendulum Physical Pendulum CM Physical Pendulum quot In u Physical Pendulum PhYSica pendulum Ph i IPendulum gt 5 TrXF Trfsin0 ZTI0 2 a m dsin9l d29L d9 I d12 I T2Tr L RECALL Hula Hoop on a Peg O ICMf rzdm M 10sz dm 0 dm ICMMR2 IICMMD2 IMR2MR2 M I2MR2 Hula Hoop on a Peg ICMfjdm ICMR2fdm 0 2 ICMMR IICMMD2 dm IMR2MR2 I2MR2 M T227T L ng 2 T227T 221139 2 R MgR V g dmAdx dmAdx 1 2 3 2 I M Ml Ml Ml 12 l 12 T2Tr L ng 2 T2n 13Ml2T 2L Mgl2 3 g 1 2 3 2 I M Ml Ml Ml 12 l 12 FnetFS wkAy mg0 kAymg k yoy1mg Equilibrium Position FmPywkAy mg0 kAymg f k yo y1mg 31 Fnetkyy0 mg FmPywkAy mg0 kAymg f k yo y1mg 5 Fmkyy0 mg makyy0 mg makyy0 k yoyl ma ky kyo ky0 ky1 malglg0lg0lg1 ma kbjq FnetFS wkAy mg0 kAymg t k yo y1 mg lt a FneIZ ky yo mg ma k y yll Set y1 as new equilibrium position y1 0 ma 2 ky Equilibrium Position y All the equations we used for the horizontal spring on a frictionless table can be used here as long as we take the equilibrium position to be y 0 FnetFS wkAy mg0 kAymg t k yoy1mg lt a FneZZ ky y0 mg ma k y yll Set y1 as new equilibrium position y1 0 ma 2 ky Equilibrium Position y All the equations we used for the horizontal spring on a frictionless table can be used here as long as we take the equilibrium position to be y 0 y Equilibrium Position VVVQVOVV39 39V i A 311 A O 1 Damped Harmonic Motion a t m w in Am nbm an F w e 1w X w w NT 53 W 2 E Figure 1111 may m Hlvs lquot 39I39Itquotr mun Inn mu Is r IIluilln B m IEu v 3H 1 in u 3H1 395 I Illll M il 9i II IIIun JIHE 1quot I linl39 quotll7i39iugc a Lug b 4mk C a Underdamped b lt be b Critically Damped b bc C Overdamped bgt be Under Damped Harmonic Motion ll Xt m 10 k 100 b 6 we 316 w 315 Forced Damped Harmonic Motion dzx dx kx bEF0s1nwt m dtz D 14gk 1 a 1 391 15 A y r 39v39v v39v v o o 1 xtAsinwtclgt A F0 m2w2 wi quot 192002 V39V39 D KIAE K 1 1 u a 71 39 A x If h m f A 6 quot 6 kx bvFosinwtma xtAsinwtltlgt A F0 m2w2 wib2w2 When 00 000 kx bvFosinwtma W 6 6 e xtAsinwtltlgt A F0 m2w2 wib2w2 l Vquot 9quot 9 quot When 00 000 When the driving frequency equals the natural frequency the amplitude reaches a maximum b2kgs b3kgs b10kgs Under Damped Harmonic Motion 1 125 15 1 75 2 225 25 2 75 3 325 35 3 75 4 425 45 4 75 5 525 W Waves Types of Waves or Pulses Transverse Wave Velocity of wave is perpendicular to motion of the particles of the medium Displacement r 7 1E Velocity of Propagation Transverse waves marr occur on a string on the surface of a liquid and throughout a solid Longitudinal Wave Velocity of wave is parallel to motion of the particles of the medium Velocity of propagation p i i iiiillllllilliii l i ll iiiilllliliii Displacement ltgt Wavelength l Crest IAmplitude Undisturbed Trough state Amplitude Maximum displacement of medium from the equilibrium position ie the height of a crest or depth of a trough Wavelength The length of a wave Period The time for one wave to pass by Frequency The number of waves per second Superposition The resultant wave of two or more waves moving through a medium is just the algebraic sum of each wave Interference If two or more waves encounter each other they seem to pass through one another totally unaffected Pond 2A Constructive Interference Superposition and Interference Destructive Interference Superposition and Interference What is the speed of a pulse or wave on a spring D W Linear Density A 711 mm H A 1 total A m mtotal u A S i A m total i I i I gt i gt T I 39 I T g 9 quot R a I I f a I I I I Q 7 HAmmt0tal I 39 Al I A m m A S i F quot F T x If T I x 9 i R 2 11 H O C U GD puu l Horizontal Components Cancel Vertical components both point toward the center sin99 for ltlt9 9 FT AsR29 AmuAsuR29 FCAmaC v2 2F 9A Tsm mR D a sin 9 N 9 for ltlt 9 1 9 PT A 32R 2 9 R A m J A s J R 2 9 F C A m ac 2 v 2FTsm9AmE 2 2FT9429R AS Am A quotM r IJ tum sin99 for ltlt9 Fr PT A 32R 2 9 9 11 AmuAsuR29 FCAmaC 2 v 2FTsm9AmE 2 2FT9J29R FTZIJVZ FT 112 J Velocity of a wave on a string U I I 1 FTng 8 FT mg In General v elastic property F vmedium p inertial property V T t 1 s rmg J REFLECTIONS ZVIOITDEFEFI Incident Wave or Pulse Hard Surface 1 V Hard Surface Reflected Wave or Pulse Waves reflected off a Hard surface are 180 out of phase with respect to the incident wave REFLECTIONS ZVIOITDEFEFI Incident Wave or Pulse Hard Surface 1 Hard Surface lt Reflected Wave or Pulse Waves reflected off a Hard surface are 180 out of phase with respect to the incident wave REFLECTIONS ZVIOITDEFEFI v39 Incident Wave or Pulse A Soft Surface i Soft Surface Reflected Wave or Pulse Waves reflected off a Soft surface are in phase 0 out of phase with respect to the incident wave REFLECTIONS F H1ltH2 gtv1 V 7 V1gtV2 Incident Wave Hard Surface or Pulse Transmitted Wave or Pulse A v1 Hard Surface Reflected Wave or Pulse Waves reflected off a Hard surface are 180 out of phase with respect to the incident wave REFLECTIONS WP u1gtu2 u V1 V1 lt V2 Incident Wave soft Surface or Pulse Transmitted Wave or Pulse v gt 1 v 2 Reflected Soft Surface Wave or Pulse Waves reflected off a Soft surface are in phase with respect to the incident wave Waves Phenomena 125 Interference and Superposition Two Waves that differ by a phase shift Phase Shift I radians 900 Two Waves Differ by a Phase Shift ym y1 1 SM sum 127 Phase Shift I radians 450 Two Waves Differ by a Phase Shift ym y1 1 SM sum 128 Phase Shift 1 0 radians 00 Two Waves Differ by a Phase Shift ym y1 1 SM sum 129 Phase Shift 1 Tl39 radians 1800 Two Waves Differ by a Phase Shift ym y1 1 SM sum 130 The study of waves vary by a phase shift has applications in the acoustics of buildings I39 2 Arr1 r2 Constructive A rn A n012 Destructive A rg A n135 Two VVaves that traveled the same distance wee One wave travels and extra 12 of a wavelength Standing Waves Two Identical Waves moving in two different directions 134 t02 s START Standing Waves 15 A ym 20 o 2 4 6 8 10 1AM y2mgt Xm y1y2 135 m Y 75 Ill 33 AA t00 5 Standing Waves ym xm y2m y1y2 137 t01 s f 75 Ill 33 AA t02 5 WW Standing Waves A A y1mgt y2mgt y1y2 139 t03 5 Standing Waves x 3 A I g I I I ll v V V t04 5 Standing Waves ym xm y2m y1y2 141 t05 5 Standing Waves ym xm y2m y1y2 142 t06 5 Standing Waves ym xm y2m y1y2 143 t07 5 Standing Waves ym y1mgt y2mgt y1 y2 O N 4 C 00 O xm 144 t08 5 WW Standing Waves xm y1m y2m y1y2 145 t09 5 Standing Waves ym xm y2m y1y2 146 t10 5 Standing Waves ym xm y2m y1y2 147 t11 5 Standing Waves ym xm y2m y1y2 148 t12 5 Standing Waves ym xm y2m y1y2 149 t13 5 Standing Waves ym xm y2m y1y2 150 t14 5 Standing Waves t15 5 Standing Waves ym y1mgt y2mgt y1y2 152 t16 5 Standing Waves ym xm y2m y1y2 153 Standing Waves Hun yyy 1 xm MM 0 5 0 5 0 5 0 5 0 2 1 1 1 2 t18 5 Standing Waves ym xm y2m y1y2 155 t19 s STO P Standing Waves ym xm y2m y1y2 156 Resonance Small amplitude driving force produces large amplitude waves 157 Oscillations in Buildings Seismic records from a 1984 earthquake show how much stress different parts of a roof may endure Collapse of Masonry Church On December 7 1988 a magnitude 6 9 earthquake shook northwestern Armenia and was followed four minutes later by a magnitude 5 8 aftershock The earthquakes affected an area 80 km in diameter This earthquake devastated the cities of Spitak and Leninakan where 25000 deaths occurred This photo illustrates the collapse of an old stone masonry Armenian church in Leninakant Churches are vulnerable to earthquake damage because of their high unsupported roofst Many such historical buildings either collapsed totally or sustained severe damage 159 160 Taipei Taiwan Beats Two Waves with different frequencies 161 Wm Two Waves with different frequencies Beats 162 Wm Two Waves with different frequencies Beats ts 163 Wm Two Waves with different frequencies Beats ts 164 Wm Two Waves with different frequencies Beats ts 165 Wm Two Waves with different frequencies Beats ts 166 More on Standing Waves Two Identical Waves moving in two different directions 167 What is the speed of a pulse or wave on a spring Linear Density A 711 mm A 1 total 2 III What is the speed of a pulse or wave on a spring Mnl i Linear Density A 711 mm A 1 total 2 III What is the speed of a pulse or wave on a spring Mwli 39 Linear Density A 711 mm A 1 total 2 III E I g n Endpoints MUST be nodes Relationship 71 v distance A Af time T X f2

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