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Chapter 12 Oscillations

Chapter 12 Oscillations. Mechanical oscillations:. pendulum, string of a guitar, vocal cords, …. More general oscillations:. Electrical, optical, atomic, …. Oscillations: motions that repeat themselves An object moves back and forth repeatedly around the equilibrium position .

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Chapter 12 Oscillations

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  1. Chapter 12 Oscillations

  2. Mechanical oscillations: pendulum, string of a guitar, vocal cords, … More general oscillations: Electrical, optical, atomic, … Oscillations: motions that repeat themselves An object moves back and forth repeatedly around the equilibrium position. Or a physical quantity changes repeatedly around a fixed value.

  3. Analysis of oscillations Simple harmonic motion (SHM) is the most simple and basic oscillation. Every oscillation can be composed of SHM.

  4. Oscillations of a spring Horizontal Spring-mass system ( Hooke’s law ) 1-D motion, origin o is the equilibrium position x is the displacement of the object relative to o F is called restoring force x>0: stretched, F<0 x<0: compressed, F>0

  5. Dynamic equation of SHM Any vibrating system with a restoring force exhibits a simple harmonic motion (SHM) where m is the mass which is oscillating angular frequency This is called the dynamic equation of SHM 5

  6. x x o h Vibrating cube Example1: A woody cube can float with height h under water. If it is pushed down and released, there will be a SHM. Determine the angular frequency. Solution: Assume a mass m and length l With displacement x 6

  7. Thinking question Thinking: A mass M is connected to two springs (k1, k2). What is the angular frequency? x1 x2 F k1 k2 x M 7

  8. A and : integral constants → initial conditions Motional equation of SHM motional equation 1)Amplitude A:the maximum magnitude of displacement from equilibrium 2) Angular Frequency ω: rate of the vibration determined by the system 8

  9. 3) : the phase of SHM at time t, note as θ : phase at t=0, initial phase or phase angle Quantities in SHM Period T: time required for one complete cycle Frequency f: number of complete cycles per sec natural frequency Phase: key point in oscillations or waves 9

  10. Phase difference 1) For one SHM at different time 2) For two SHM with same ω at same time >0: x2 is in front of x1 =0: x1 and x2 are in phase <0: x2 is behind of x1 =π: x1 and x2 are inverse phase 10

  11. Velocity and acceleration Can also be treated as SHM. Phase difference? 11

  12. A and : integral constants → initial conditions Determine motional equation determined by the system When t=0: 12

  13. Spring oscillator Example2: A 4kg mass is attached to a spring of k=100N/m. It has an initial velocity v0=-5m/s and initial displacement x0=1.0m. Motional equation? Solution: Angular frequency Amplitude Phase angle 13

  14. (+/2)  M A t+φ x o p x (-/2) Rotational vector method SHM Uniform circular motion OM =A OM rotates with constant  counterclockwise () (0) p is the projection of M on x axis, the position of P: This is a SHM! Rotational vector Angle t+φis the phase 14

  15. x Geometric description UCM two SHMs in x and y direction Show rotational vectors 1) x=-A, t+φ=  2) x=0, v>0: t+φ= -/2 t+φ= /3 3) x=A/2, v<0: 15

  16. x Rotational vector in SHM Example3: T=2s, A=0.12m. When t=0, x0=0.06m, the object moves in positive direction. Determine: a) motional equation Solution: a) b) the phase when x=-0.06m, v<0 c) minimum time required from position b to E.P. rotation angle 16

  17. x (cm) 6 3 o 2 t (s) ( )cm  /3 o x Graph of function Example4: Determine the motional function. Solution: A = 6cm , t=2s t=0 17

  18. Energy in SHM Kinetic energy Potential energy where Total mechanical energy Total mechanical energy is conserved in SHM 18

  19. m k F x o s Pushing on a spring Example5: No friction, k=16N/m, m=4kg. A force F=10N pushes mmoving s=5cm, and we let t=0 at the time when m reaches the far left. Determine: a) motional function Solution: a) Vibration energy is from work done by F b) at what position Ek = Ep ? 19

  20. dl l O x Massive spring system Example6: A massive spring ms (k, L), attached by a mass M, determine the angular frequency. (no fr) Solution:We have already obtained ( Ch7, P15 ) Conservation of total mechanical energy M 20

  21. o  h  C mg Physical pendulum Extended body oscillates in vertical plane For a small angle 21

  22. m, l  l M, R  l Simple pendulum and more Example7: Determine the period of pendulums: a) b) c) Solution:a)Simple pendulum b) c) ? Measure of I Natural walking pace 22

  23. Longest tunnel Thinking: If there is a tunnel that extends from one side of the earth, through its center, to the other side. Determine the time required for a travel through it. Solution: 23

  24. ω A2 A1 2 A 1 O x  Superposition of SHM Superposition: the sum of several motions 1. Same Direction, Same Frequency SHM with same frequency  can also be known, but it is not important 24

  25. Constructive & destructive superposition 1) in phase constructive superposition 2) inverse phase destructive superposition 25

  26. x General superposition Example8: Determine the superposition of x1 and x2 : x1=3cos(t+/2) ；x2=3cos(t+) Solution: 26

  27. Homework The superposition of x1 and x2 makes a new SHM x withamplitude A=17.3cm, where A1=10cm, and , determine amplitude A2. 27

  28. Different frequency 2. Same Direction, Different Frequency For a special case: Amplitude varying oscillation —— Beats where Amplitude modulation factor 28

  29. Amplitude modulation x1 x2 x Amplitude modulation: AM Frequency modulation: FM 29

  30. 1 : 2 1 : 1 1 : 1 4 : 3 4 : 3 4 : 3 9 : 7 4 : 3 *Superposition in 2D 3. Vertical Direction, Same Frequency Motion in an ellipse * Special case: rotational vector method 4. Vertical Direction, Different Frequency Lissajous Figures 30

  31. Damped harmonic motion Amplitude of real oscillator slowly decreases It is called damped harmonic motion Damping force depends on the speed F = F(v) (a simple case) Dynamic equation 31

  32. Solution of Damped motion where 1) Underdamped motion: ① frequency becomes lower ② amplitude keeps decreasing over time mean lifetime 32

  33. 2) Overdamped motion: No longer an oscillation! 3) Critical damped motion: No oscillation Car’s spring … 33

  34. Forced vibrations Damped External force to supply energy Natural frequency Driving force has its own frequency f and f isthe frequency of vibration! The amplitude becomes large when f is near f0 f = f0 → maximum vibration energy f0 : resonance frequency Resonance 34

  35. *Resonance Energy of driving force is efficiently transformed Resonance causes the collapse of bridge * Nuclear magnetic resonance (NMR) Selection of television channel 35

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