Part II Oscillations and WavesChapter1 OscillationsChapter2 Mechanical wavesChapter2 Mechanical wavesChapter1 Oscillations1 Simple Harmonic Motion2 The Characteristic Quantities for SHM3 The Circle of Reference4 Energy in Simple Harmonic Motion5 Superposition of SHM1. Example of simple harmonic motion (1) The block-spring systemThe force on the particle:-restoring force (1)Newtons second law for block-spring system Denote the ratio k/m with symbol 2Take a tentative solution to Eq.(2)Dynamics equation for SHMKinematics equation for SHM1 Simple Harmonic Motion (P297,307)F=-kx(3)(2)A and arise from the integral constants(2). The simple pendulumThe force on the particle: (1)Newtons second law for the simple pendulum Let , and for small anglesWe get also a equation of motion of SHMFt=-mgsin(3)(2)(3)The Physical Pendulum (复摆)Newtons second law for rigid body:(3)(2)(1)(4) The Torsion Pendulum (扭摆)The restoring torque:(3)(2)(1)Conclusion: The simple harmonic motionThe force on the particle: (restoring force) The motion action is governed by Eq. (2)Can be described in terms of sine and cosine functionF=-kx(3)(2)(1)2 The Characteristic Quantities for SHM (P299)1. Angular Frequency , Frequency f, and Period TThe period, T, is the time for oscillator to go though one circle of motionThe frequency, f, is the number of circles in a unit of time. (SI unit: Hz)The angular frequency, , is 2 times the frequency. (SI unit: rad/s)T , f, relate to the essential nature of an oscillator, which often called natural (intrinsic) period, natural frequency, and natural angular frequency.For a block-spring oscillator:For a simple pendulum:All determined by the essential natures of two different oscillators2. The amplitude AMaximum magnitude of displacement from equilibrium3. The phase ( t + ), phase constant (or phase angle) The phase ( t + ) can reflect entirely the motion state of an oscillatorWhen t=0, reflect the initial motion state of the oscillatorA and are determined by initial conditions (How the motion starts?)When t=0, x=x0, v=v0Phase t+ State of motionMotion state PhasexmoAx = A0oxmx = 0 /2moxx = 03 /22 xmoAx= Axmo-Ax = -A a. The relationship between motion state and phaseb. Phase difference play a an important role for oscillatorTwo oscillators with phases: 1= t+ 1, 2= t+ 2Ahead in phaseLag in phaseIn phaseOut of phaseSeveral simple harmonic motion with different characteristic quantitiesDifferent Different ADifferent c.The Roles Characteristic QuantitiesThe velocity is /2 ahead in phase of the position.The acceleration is out of phase with the position.d. The relations among the position, velocity, and acceleration3 The Circle of Reference (P306)1. The corresponding relation between SHM and uniform circular motion Circle of Reference (参考圆) or Phasor (旋转矢量)Simple Harmonic Motion is the projection of uniform circular motion of phasor onto x axis.The circle in which the phasor moves so that the projection of phasors top matches the motion of the oscillating body is called the circle of reference.The phasor rotates with constant angular speed , and makes an angle t + with the x axis. When t=0, the phasor makes an angle with the x axis.For Simple Harmonic MotionFor Uniform Circular MotionAAmplitudeRadiusxDisplacementProjectionAngular FrequencyAngular Velocity = t + PhaseAngle between Phasor and x axisa. Corresponding Relation Between SHM and UCMThe simple harmonic motion is the side view of circular motion.2. Draw x-t Diagram Using Circle of Reference Homework: P317 Q6, 16, 23ox/3631x(m)ot(s)0.81x(cm)ot(s)63Example1: Find the initial phase of the two oscillationsoxab10-2x (m)t (s)2Solution:From circle of reference-22Example2: SHM: From given x-t graph, find , a, b, and the angular frequency .Solution I: locates in I or III quadrantExample3: An object of mass 4 kg is attached to a spring of k =100N.m-1. The object is given an initial velocity of v0 5m.s1 and an initial displacement of x0=1. Find the kinematics equation.Solution II: Using the phasorExample4: A particle undergoes SHM with A=4cm，f = 0.5Hz. The displacement x 2cm when t 1s, and is moving in the positive x-axis. Write the kinematics equation.Solution I: changed initial conditions: x x 0, v v 0, when t t 0.When t=1slocates in I or IV quadrant locates in I quadrant.Solution II: Using the phasor:One revolution corresponds to one period T=2s, and half a revolution corresponds to t = 1 s4 Energy in Simple Harmonic Motion (P304)1. The total mechanical energy for an isolated simple harmonic oscillatorKinetic energy:Potential energy:Total mechanical energy:Example1 Vertical SHM: Suppose we hang a spring with force constant k and suspend from it a body with mass m. Oscillation will now be vertical. Will it still be SHM?Solution I: by Newton second law The bodys motion is still SHM with the angular frequency:When the body hangs at rest, in equilibriumTake x=0 to be the equilibrium position, and take the positive x-direction to be downward.Solution II: by energy analysis When the body is at the position x, the total mechanical energy isby derivative on both sidesHomework:P319 Q365 Superposition of SHM1. An object experiences two SHMs simultaneouslyTwo SHMsResultant motion which is superposed by the two SHMs is also a SHMXYa. Superposition of SHMs using phasor diagramThe phase difference = 2 - 1.When = 2 - 1 =2k, k=0, 1, 2, The two SHMs are in phase, the resultant amplitude take its maximum.b. Superposition of SHMs under different phase differencesWhen = 2 - 1 =(2k+1), k=0, 1, 2, The two SHMs are out of phase, the resultant amplitude take its minimum.Generally, = 2 - 1 kSolution:Draw a circle of reference,Example1: x1=3cos(2 t+ )cm, x2=3cos(2 t+ /2)cm, find the superposition displacement of x1 and x2.2.Superposition of Two SHM in Same Direction With Different frequencies12a.Vibration equationb. Features of figure Amplitude modulation factorfrequencymodulationcarrier frequencyc. beatThe phenomenon that resultant oscillation increases or decreases slowly.12Beat frequencyBeat phenomenon is a very important physical phenomenon. For example, Musician use it in tuning their instruments, such as oboe, piano a. When = 2 1= k (k is an integer), b. When = ( 2k +1 ) /2 (k is an integer), xy3.Superposition of Two SHM in two perpendicular Directions with same frequency = 0 = /2 = = = 3 = 3 /2/2a. When 1/2 =m/n, m and n are integers:4. Superposition of Two SHM in two perpendicular Directions with different frequencies Lissajous figuresLissajous figuresb. When 1/2 m/n:Lissajous FiguresSummary:1.The simple harmonic motion(1) The force on the particle: (2) Dynamics equation for SHM(3) Kinematics equation for SHM(4) The total mechanical energy for SHMF=-kx(3)(2)(1)(4)2.Description for SHM(1) Motional equation for SHMa. Angular Frequency , Frequency f, and Period Tb.The amplitude Ac.The phase ( t + )(3) Circle of Reference (参考圆) or Phasor (旋转矢量)(2) x-t Diagramox/3631x(cm)ot(s)633. Superposition of SHM(1). Two SHMs simultaneously in same directionXY12(2). Two SHM in Same Direction With Different frequencies, BeatBeat frequency(3). Two SHMs in vertical Directions with same frequency (4).Two vertical SHMs with different frequencies: Lissajous figures1/2 =m/nm and n are integers