A uniform chain of length hangs freely from a rigid clamp at its top; its lower end…

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Coupled oscillators, normal modes and Fourier analysis · Superposition, boundaries and interference

Modes, Packets and Media, Ch The continuum limit, Fourier series and the spectrum of a measured signal

Problem

A uniform chain of length

L=2.00\,\mathrm{m}

hangs freely from a rigid clamp at its top; its lower end is free. Treat it as a limp, perfectly flexible string whose tension at height

z

above the free end is

\tau = \mu g z

, so its transverse modes obey

J_0\!\left(2\omega\sqrt{L/g}\right)=0

and

\nu_n = \dfrac{j_{0,n}}{4\pi}\sqrt{\dfrac{g}{L}}

, where

j_{0,n}

is the

n

th zero of

J_0

(

j_{0,1}=2.405

j_{0,2}=5.520

j_{0,3}=8.654

). Take

g=9.81\,\mathrm{m\,s^{-2}}

. ((a)) Compute the first three mode frequencies

\nu_1,\nu_2,\nu_3

. ((b)) Express

\nu_2,\nu_3

as ratios to

\nu_1

and explain in one sentence why they are not

2:1

and

3:1

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