Part Two: Oscillations, Waves, & Fluids

1. Examples of oscillations & waves: Earthquake – Tsunami Electric guitar – Sound wave Watch – quartz crystal Radar speed-trap Radio telescope Part Two: Oscillations, Waves, & Fluids Examples of fluidmechanics: Flow speed vs river width Plane flight High-speed photo: spreading circular waves on water.

2. 13. Oscillatory Motion Describing Oscillatory Motion Simple Harmonic Motion Applications of Simple Harmonic Motion Circular & Harmonic Motion Energy in Simple Harmonic Motion Damped Harmonic Motion Driven Oscillations & Resonance

3. Dancers from the Bandaloop Project perform on vertical surfaces, executing graceful slow-motion jumps. What determines the duration of these jumps? Wilberforce Pendulum pendulum motion: rope length & g

4. Disturbing a system from equilibrium results in oscillatory motion. Absent friction, oscillation continues forever. Examples of oscillatory motion: Microwave oven: Heats food by oscillating H2O molecules in it. CO2 molecules in atmosphere absorb heat by vibrating  global warming. Watch keeps time thru oscillation ( pendulum, spring-wheel, quartz crystal, …) Earth quake induces vibrations  collapse of buildings & bridges . Oscillation

5. 13.1. Describing Oscillatory Motion Characteristics of oscillatory motion: • Amplitude A = max displacement from equilibrium. • PeriodT = time for the motion to repeat itself. • Frequencyf = # of oscillations per unit time. same period T same amplitude A [ f ] = hertz (Hz) = 1 cycle / s A, T, f do not specify an oscillation completely. Oscillation

6. Example 13.1. Oscillating Ruler An oscillating ruler completes 28 cycles in 10 s & moves a total distance of 8.0 cm. What are the amplitude, period, & frequency of this oscillatory motion? Amplitude = 8.0 cm / 2 = 4.0 cm.

7. 13.2. Simple Harmonic Motion Simple Harmonic Motion (SHM): 2nd order diff. eq  2 integration const. Ansatz: angular frequency  

8. A, B determined by initial conditions   ( t )  2 x  2A

9. Amplitude & Phase C = amplitude  = phase  Note: is independent of amplitude only for SHM. Curve moves to the right for < 0. Oscillation

10. Velocity & Acceleration in SHM |x| = max at v = 0 |v| = max at a = 0

11. GOT IT? 13.1. Two identical mass-springs are displaced different amounts from equilibrium & then released at different times. Of the amplitudes, frequencies, periods, & phases of the subsequent motions, which are the same for both systems & which are different? Same: frequencies, periods Different: amplitudes ( different displacement ) phases ( different release time )

12. Application: Swaying skyscraper Tuned mass damper : Damper highly damped , Overall oscillation overdamped. Taipei 101 TMD: 41 steel plates, 730 ton, d = 550 cm, 87th-92nd floor. Also used in: • Tall smokestacks • Airport control towers. • Power-plant cooling towers. • Bridges. • Ski lifts. Movie Tuned Mass Damper

13. Example 13.2. Tuned Mass Damper The tuned mass damper in NY’s Citicorp Tower consists of a 373-Mg (vs 101’s 3500 Mg) concrete block that completes one cycle of oscillation in 6.80 s. The oscillation amplitude in a high wind is 110 cm. Determine the spring constant & the maximum speed & acceleration of the block. 

14. 13.3. Applications of Simple Harmonic Motion • The Vertical Mass-Spring System • The Torsional Oscillator • The Pendulum • The Physical Pendulum

15. The Vertical Mass-Spring System Spring stretched by x1 when loaded. mass m oscillates about the new equil. pos. with freq

16. The Torsional Oscillator = torsional constant  Used in timepieces

17. The Pendulum Small angles oscillation: Simple pendulum (point mass m):

18. Example 13.3. Rescuing Tarzan Tarzan stands on a branch as a leopard threatens. Jane is on a nearby branch of the same height, holding a 25-m-long vine attached to a point midway between her & Tarzan. She grasps the vine & steps off with negligible velocity. How soon can she reach Tarzan? Time needed:

19. GOT IT? 13.2. • What happens to the period of a pendulum if • its mass is doubled, • it’s moved to a planet whose g is ¼ that of Earth, • its length is quadrupled? no change doubles doubles

20. Conceptual Example 13.1. Nonlinear Pendulum • A pendulum becomes nonlinear if its amplitude becomes too large. • As the amplitude increases, how will its period changes? • If you start the pendulum by striking it when it’s hanging vertically, • will it undergo oscillatory motion no matter how hard it’s hit? • If it’s hit hard enough, • motion becomes rotational. (a) sin increases slower than   smaller    longer period

21. The Physical Pendulum Physical Pendulum = any object that’s free to swing Small angular displacement  SHM

22. Example 13.4. Walking When walking, the leg not in contact of the ground swings forward, acting like a physical pendulum. Approximating the leg as a uniform rod, find the period for a leg 90 cm long. Table 10.2 Forward stride = T/2 = 0.8 s

23. 13.4. Circular & Harmonic Motion Circular motion: 2  SHO with same A &  but  = 90 x =  R x = R x = 0 Lissajous Curves

24. GOT IT? 13.3. The figure shows paths traced out by two pendulums swinging with different frequencies in the x- & y- directions. What are the ratios x : y ? 1 : 2 3: 2 Lissajous Curves

25. 13.5. Energy in Simple Harmonic Motion SHM: = constant Energy in SHM

26. Potential Energy Curves & SHM Linear force:  parabolic potential energy: Taylor expansion near local minimum:  Small disturbances near equilibrium points  SHM

27. GOT IT? 13.4. • Two different mass-springs oscillate with the same amplitude & frequency. • If one has twice as much energy as the other, how do • their masses & (b) their spring constants compare? • (c) What about their maximum speeds? • The more energetic oscillator has • twice the mass • twice the spring constant • (c) Their maximum speeds are equal.

28. 13.6. Damped Harmonic Motion sinusoidal oscillation Damping (frictional) force: Damped mass-spring: Amplitude exponential decay Ansatz:  

29.  At t = 2m / b, amplitude drops to 1/e of max value. (a) For  is real, motion is oscillatory ( underdamped ) (c) For  is imaginary, motion is exponential ( overdamped ) (b) For  = 0, motion is exponential ( critically damped ) Damped & Driven Harmonic Motion

30. Example 13.6. Bad Shocks A car’s suspension has m = 1200 kg & k = 58 kN / m. Its worn-out shock absorbers provide a damping constant b = 230 kg / s. After the car hit a pothole, how many oscillations will it make before the amplitude drops to half its initial value? Time  required is  # of oscillations: bad shock !

31. 13.7. Driven Oscillations & Resonance External force  Driven oscillator Let d= driving frequency ( long time ) Prob 75: = natural frequency Damped & Driven Harmonic Motion Resonance:

32. Buildings, bridges, etc have natural freq. If Earth quake, wind, etc sets up resonance, disasters result. Collapse of Tacoma bridge is due to self-excitation described by the van der Pol equation. Tacoma Bridge Resonance in microscopic system: • electrons in magnetron  microwave oven • Tokamak (toroidal magnetic field)  fusion • CO2 vibration: resonance at IR freq  Green house effect • Nuclear magnetic resonance (NMR)  NMI for medical use.