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## MUSICAL ACOUSTICS

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**MUSICAL ACOUSTICS**Chapter 2 VIBRATING SYSTEMS**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.**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.**Displacement of a damped vibrator whose amplitude decreases**with time**EVERY VIBRATING SYSTEM HAS**Inertia (mass) Elasticity (spring) Hooke’s Law F =Ky For a mass/spring In Chapter 1 we learned that KE= ½ mv2 Similarly, it can be shown that PE = ½ Ky2 If the vibrator has damping:**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.**SIMPLE VIBRATING SYSTEMS**A simple pendulum**A mass-spring system vibrates at a frequencyf**• If the mass is doubled: • The frequency will be 2f • The frequency will be √2f • The frequency will remain f • 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: • The frequency will be 2f • b) The frequency will be √2f • c)The frequency will remain f • The frequency will be f/√2 • e) The frequency will be f/2**SIMPLE VIBRATING SYSTEMS**A piston free to vibrate in a cylinder**SIMPLE VIBRATING SYSTEMS**A piston free to vibrate in a cylinder A Helmholtz resonator**SIMPLE VIBRATING SYSTEMS**A piston free to vibrate in a cylinder A Helmholtz resonator m=ρɑl K=ρɑ2l 2/V**Longitudinal vibrations of a three-mass vibrator Transverse**vibration of a three-mass vibratorTransverse vibrations for spring systems with multiple masses**SNARE**DRUM TIMPANI BASS DRUM**VIBRATING BARS**Both ends free One end clamped Arrows locate the nodes**CHLADNI PATTERNS OF A CIRCULAR PLATE**SALT COLLECTS AT THE NODES**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#modeshttp://www.phys.unsw.edu.au/jw/chladni.html#modes**VIBRATIONAL MODES OF A CYMBAL (top) AND A CIRCULAR PLATE**(bottom)**ANIMATIONS OF TUNING FORK VIBRATIONS AT DAN RUSSELL’S**WEBSITE http//www.acs.psu.edu/drussell/Demos/TuningFork/fork-modes.htmlhttp://www.acs.psu.edu/drussell/Demos/TuningFork/fork-mohttp://www.acs.psu.edu/drussell/Demos/TuningFork/fork-modes.htmldes.html HTTttp://www.acs.psu.edu/drussell/Demos/TuningFork/fork-modes.htmlhttp://www.acs.psu.edu/drussehttp://www.acs.psu.edu/drussell/Demos/TuningFork/fork-modes.htmlll/Demos/TuningFork/fork-modes.hthhttp://www.acs.psu.edu/drussell/Demos/TuningFork/fork-mohttp://www.acs.psu.edu/drussell/Demos/TuningFork/fork-modes.htmldes.html http://www.acs.psu.edu/drussell/Demos/TuningFork/fork-modes.html ttp://www.acs.psu.edu/drussell/Demos/TuningFork/fork-modes.html**ASSIGNMENT FOR MONDAY, Jan. 14**• READ CHAPTER 3 • EXERCISES IN CHAPTER 2: 1-7