Answered: A few friends are goofing around with…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Question

A few friends are goofing around with some left-over helium party balloons, taking turns to witness how helium alters the pitch of one’s voice. A simplified model of how your voice is generated by your vocal cords, is to consider it as cords vibrating above a gas-filled cylindrical tube that is open only at one end. The quality of the voice depends on the harmonic frequencies generated by the tube; larger frequencies lead to higher-pitched voices. Consider two such tubes at 20°C. One is filled with air, in which the speed of sound is 343 m/s. The other is filled with helium, in which the speed of sound is 1.00 × 10³ m/s. To see the effect of helium on voice quality, calculate the ratio of the n^th natural frequency of the helium-filled tube to the n^th natural frequency of the air-filled tube.

Expert Solution

Introduction

$H a r m o n i c f r e q u e n c y i n a p i p e w i t h o n e e n d c l o s e d (s i m i l a i r t o o u r v o c a l c h o r d) i s g i v e n b y f_{n} = \frac{n \times v}{4 \times L} h e r e, n i s n t h h a r m o n i c (n = 1 r e p r e s e n t s n a t u r a l f r e q u e n c y, n = 2 r e p r e s e n t s 2^{n d} h a r m o n i c) v i s v e l o c i t y o f s o u n d i n t h e m e d i u m f i l l e d i n t h e p i p e L i s l e n g t h o f t h e p i p e G i v e n : v_{a i r} = 343 m / s v_{H e} = 10^{3} m / s I t i s a s k e d t o f i n d t h e r a t i o o f n^{t h} h a r m o n i c i n H e l i u m t o n^{t h} h a r m o n i c i n A i r$