Physics 2A (Fall 2012) Chapter 14: Oscillations

Transcription

1 Physics 2A (Fall 212) Chaper 14: Oscillaions The ragedy of life doesn' lie in no reaching your goal. The ragedy lies in having no goals o reach. Benjamin Mays If you have a goal in life ha akes a lo of energy, ha requires a lo of work, ha incurs a grea deal of ineres and ha is a challenge o you, you will always look forward o waking up o see wha he new day brings. Susan Polis Schulz Do i now. You become successful he momen you sar moving oward a worhwhile goal. Unknown Reading: pages ; skip secion 14.6 Ouline: equilibrium and oscillaion frequency and period simple harmonic moion Hooke s Law mass on a spring he pendulum angular frequency displacemen. velociy, and acceleraion energy in simple harmonic moion conservaion of mechanical energy frequency and period in SHM pendulum moion resonance Problem Solving Many of he problems in his chaper deal wih Hooke s law, which saes ha he resoring force of an ideal spring is given by F = -k. In his equaion, F is he force eered by he spring, k is he spring consan, and is he displacemen of he spring from is unsrained lengh. The minus sign indicaes ha he force is always in he opposie direcion of he displacemen. You should be familiar wih he definiion of he radian, which is measure of angular 36 displacemen. The relaionship beween radians and degrees is given by 1 rad = = π You should also be familiar wih he definiion of angular velociy ω. I is he angular equivalen of velociy.

2 Some problems make use of he relaionships among angular frequency, frequency, and period for simple harmonic moion: ω =2π f, and f = 1/T. Occasionally he period is given indirecly by describing a ime inerval. You mus hen know, for eample, ha he ime he oscillaor akes o go from maimum displacemen in one direcion o maimum displacemen in he oher direcion is T/2 or he ime i akes o go from maimum displacemen o zero displacemen is T/4. If hese ime inervals or ohers are given, you should be able o calculae he period, frequency, and angular frequency. You should also know how o find he maimum speed and maimum acceleraion in erms of he angular frequency and ampliude: v ma = Aω and a ma = Aω 2. Some problems require you o know he relaionship beween he angular frequency and he appropriae physical properies of he oscillaing sysem: ω = k/ m for an undamped springobjec sysem. Some problems can be solved using he principle of mechanical energy conservaion. For a spring-objec sysem, he mechanical energy E is given by: E = ½ mv 2 + ½ k 2 + mgh. Quesions and Eample Problems from Chaper 14 Quesion 1 The drawing shows idenical springs ha are aached o a bo in wo differen ways. Iniially, he springs are unsrained. The bo is hen pulled o he righ and released. In each case, he iniial displacemen of he bo is he same. A he momen of release, which bo, of eiher, eperiences he greaer ne force due o he spring? Provide a reason for your answer. Quesion 2 Suppose ha a grandfaher clock (a simple pendulum) is running slowly. Tha is, he ime i akes o complee each cycle is longer han i should be. Should one shoren or lenghen he pendulum o make he clock keep he correc ime? Why? Problem 1 The equilibrium lengh of a cerain spring wih a force consan of k = 25 N/m is.2 m. (a) Wha force is required o srech his spring o wice is equilibrium lengh? (b) Is he force required o compress he spring o half is lengh he same as in par (a)? Eplain.

3 SUMMARY The goal of Chaper 14 has been o undersand sysems ha oscillae wih simple harmonic moion. GENERAL PRINCIPLES Resoring Forces SHM occurs when a linear resoring force acs o reurn a sysem o an equilibrium posiion. Mass on spring Pendulum k (F ne ) =-k m The frequency of a mass on a spring depends on he mass and he spring consan: (F ne ) =-a mg L bs The frequency of a pendulum depends on he lengh and he free-fall acceleraion: s L Energy If here is no fricion or dissipaion, kineic and poenial energies are alernaely ransformed ino each oher in SHM, wih he sum of he wo conserved. E = 1 2 mv k 2 = 1 2 mv ma 2 = 1 2 ka2 A All kineic All poenial A f = 1 k 2pA m f = 1 g 2pA L IMPORTANT CONCEPTS Oscillaion An oscillaion is a repeiive moion abou an equilibrium posiion. The ampliude A is he maimum displacemen from equilibrium. The period T is he ime for one cycle. We may also characerize an oscillaion by is frequency f. Simple Harmonic Moion (SHM) SHM is an oscillaion ha is described by a sinusoidal funcion. All sysems ha undergo SHM can be described by he same funcional forms. Posiion-versus-ime is a cosine funcion. A Velociy-versus-ime is an invered sine funcion. v ma v Acceleraion-versus-ime is an invered cosine funcion. a ma a A 2A T f = 1 T T 2T T 2T 2A 2v ma () = A cos(2pf) v () =-v ma sin(2pf) ma = A v ma = 2pfA 2a ma T 2T a () =-a ma cos(2pf) a ma = (2pf ) 2 A APPLICATIONS Damping Simple harmonic moion wih damping (due o drag) decreases in ampliude over ime. The ime consan deermines how quickly he ampliude decays. A A/e 2A Resonance A sysem ha oscillaes has a naural frequency of oscillaion f. Resonance occurs if he sysem is driven wih a frequency f e ha maches his naural frequency. This may produce a large ampliude of oscillaion. Oscillaion ampliude f f e Physical pendulum A physical pendulum is a pendulum wih mass disribued along is lengh. The frequency depends on he posiion of he cener of graviy and he momen of ineria. mg The moion of legs during walking can be described using a physical pendulum model. f = 1 mgd 2pB I d Momen of ineria = I

4 Problem 2 An air-rack glider aached o a spring oscillaes beween he 1 cm mark and he 6 cm mark on he rack. The glider complees 1 oscillaions in 33 s. Wha are he (a) period, (b) frequency, (c) ampliude, and (d) maimum speed of he glider? Problem 3 A compuer o be used in a saellie mus be able o wihsand acceleraions of up o 25 imes he acceleraion due o graviy, In a es o see wheher i mees his specificaion, he compuer is boled o a frame ha is vibraed back and forh in simple harmonic moion a a frequency of 9.5 Hz. Wha is he minimum ampliude of vibraion ha mus be used in his es?

5 Problem 4 A block of mass m =.75 kg is fasened o an unsrained horizonal spring whose spring consan is k = 82. N/m. The block is given a displacemen of +.12 m, where he + sign indicaes ha he displacemen is along he + ais, and hen released from res. (a) Wha is he force (magniude and direcion) ha he spring eers on he block jus before he block is released? (b) Find he frequency of he resuling oscillaory moion. (c) Wha is he maimum speed of he block? (d) Deermine he magniude of he maimum acceleraion of he block? Problem 5 The shock absorbers in he suspension sysem of a car are in such bad shape ha hey have no effec on he behavior of he springs aached o he ales. Each of he idenical springs aached o he fron ale suppors 32 kg. A person pushes down on he middle of he fron end of he car and noices ha i vibraes hrough 5 cycles in 3. s. Find he spring consan of eiher spring.

6 Problem 6 A 2 g mass aached o a horizonal spring oscillaes a a frequency of 2. Hz. A one insan, he mass is a = 5. cm and has speed v = -3 cm/s. Deermine: (a) The period. (b) The ampliude. (c) The maimum speed. (d) The oal energy. Problem 7 An archer pulls he bowsring back for a disance of.47 m before releasing he arrow. The bow and sring ac like a spring whose spring consan is 425 N/m. (a) Wha is he elasic poenial energy of he drawn bow? (b) The arrow has a mass of.3 kg. How fas is i raveling when i leaves he bow?

7 Problem 8 A kg block is resing on a horizonal fricionless surface and is aached o a horizonal spring whose spring consan is 124 N/m. The block is shoved parallel o he spring ais and is given an iniial speed of 8. m/s, while he spring is iniially unsrained. Wha is he ampliude of he resuling simple harmonic moion? Problem 9 A.4 kg mass is aached o a spring wih a force consan of 26 N/m and released from res a disance of 3.2 cm from he equilibrium posiion of he spring. Wha is he speed of he mass when i is halfway o he equilibrium posiion?

8 Problem 1 Asronaus on a disan plane se up a simple pendulum of lengh 1.2 m. The pendulum eecues simple harmonic moion and makes 1 complee vibraions in 28 s. Wha is he acceleraion due o graviy? Problem 11 If he period of a simple pendulum is o be 2. s, wha should be is lengh?