1. MUSICAL ACOUSTICS Chapter 2 VIBRATING SYSTEMS

2. SIMPLE HARMONIC MOTION A simple vibrator consisting of a mass and a spring. At equilibrium (center), the upward force exerted by the spring and the force of gravity balance each other, and the net force F on the mass is zero.

3. Simple Harmonic Motion Graphs of simple harmonic motion: (a) Displacement versus time (b) Speed versus Time. Note that speed reaches its maximum when displacement is zero and vice versa.

4. Vibratory motion: y,v, and a all change with time.

5. Displacement of a damped vibrator whose amplitude decreases with time

6. EVERY VIBRATING SYSTEM HAS Inertia (mass) Elasticity (spring) For a mass/spring Hooke’s Law F = Ky In Chapter 1 we learned that KE= ½ mv 2 Similarly, it can be shown that PE = ½ Ky 2 If the vibrator has damping:

7. A mass hangs from a spring. You raise the mass 1 cm, hold it there for a short time and then let it drop Make a graph of its motion Make a graph of its total energy.

8. SIMPLE VIBRATING SYSTEMS A simple pendulum

9. A mass-spring system vibrates at a frequency f If the mass is doubled: a)The frequency will be 2 f b)The frequency will be √2 f c)The frequency will remain f d)The frequency will be f /√2 e) The frequency will be f /2 A mass swings on the end of a string at frequency f If the mass is doubled: a)The frequency will be 2 f b) The frequency will be √2 f c)The frequency will remain f d)The frequency will be f /√2 e) The frequency will be f/ 2

10. SIMPLE VIBRATING SYSTEMS A piston free to vibrate in a cylinder

11. SIMPLE VIBRATING SYSTEMS A piston free to vibrate in a cylinder A Helmholtz resonator

12. SIMPLE VIBRATING SYSTEMS A piston free to vibrate in a cylinder A Helmholtz resonator m= ρ ɑ l K=ρ ɑ 2 l 2 /V

13. SYSTEMS WITH TWO MASSES

14. Longitudinal vibrations of a three-mass vibrator Transverse vibration of a three-mass vibrator Transverse vibrations for spring systems with multiple masses

15. LINEAR ARRAY OF OSCILLATORS

16. MODES OF CIRCULAR MEMBRANES

17. BASS DRUM SNARE DRUM TIMPANI

18. VIBRATING BARS Both ends freeOne end clamped Arrows locate the nodes

19. CHLADNI PATTERNS OF A CIRCULAR PLATE SALT COLLECTS AT THE NODES

20. CHLADNI PATTERNS JOE WOLFE’S PHYSCLIPS ON MODES OF VIBRATION AND CHLADNI PATTERN CAN BE ACCESSED AT p://www.phys.unsw.edu.au/jw/chladni.html#modes http://www.phys.unsw.edu.au/jw/chladni.html#modes p://www.phys.unsw.edu.au/jw/chladni.html#modes

21. HOLOGRAPHIC INTERFEROMETRY

22. VIBRATIONAL MODES OF A CYMBAL (top) AND A CIRCULAR PLATE (bottom)

23. CYMBALS GONG TAM TAM

24. VIBRATIONS OF A TUNING FORK

25. ANIMATIONS OF TUNING FORK VIBRATIONS AT DAN RUSSELL’S WEBSITE http//www.acs.psu.edu/drussell/Demos/TuningFork/fork- modes.html http://www.acs.psu.edu/drussell/Demos/TuningFork/fork- mohttp://www.acs.psu.edu/drussell/Demos/TuningFork/fork-modes.html des.html http://www.acs.psu.edu/drussell/Demos/TuningFork/fork- mohttp://www.acs.psu.edu/drussell/Demos/TuningFork/fork-modes.html des.html HTT ttp://www.acs.psu.edu/drussell/De mos/TuningFork/fork-modes.html http://www.acs.psu.edu/drussehttp:// www.acs.psu.edu/drussell/Demos/Tu ningFork/fork-modes.html ll/Demos/TuningFork/fork- modes.hthhttp://www.acs.psu.edu/dru ssell/Demos/TuningFork/fork- mohttp://www.acs.psu.edu/drussell/D emos/TuningFork/fork-modes.html des.html http://www.acs.psu.edu/drussell/Dem os/TuningFork/fork-modes.html ttp://www.acs.psu.edu/drussell/Demo s/TuningFork/fork-modes.html

26. ASSIGNMENT FOR MONDAY, Jan. 12 READ CHAPTER 3 EXERCISES IN CHAPTER 2: 1-7