A uniform heavy rope of length hangs from a hook and swings in small transverse…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Variational and asymptotic methods · Function spaces and operator theory

Conditions of Validity, Ch Eigenfunction expansions, convergence and the Rayleigh quotient

Problem

A uniform heavy rope of length

L=2.00\ \mathrm{m}

hangs from a hook and swings in small transverse oscillations. With

x

measured up from the free lower end, its mode shapes satisfy

-\big(xY'\big)'=\lambda Y,\qquad 0<x\le L,\qquad \lambda=\frac{\omega^2}{g},

with

Y(L)=0

at the hook and

Y

bounded at the free end, where

p=x

vanishes. Take

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

. Take as given that

R[Y]\ge\lambda_1

for every bounded trial function with

Y(L)=0

and

xY'\to0

x\to0

, where

R[Y]=\frac{\int_0^LxY'^2\,\mathrm dx}{\int_0^LY^2\,\mathrm dx}.

((a)) Evaluate

R

for the straight-line trial

Y=1-x/L

. ((b)) For the one-parameter family

Y_s=1-(x/L)^s

with

s>0

, show that

R[Y_s]=\dfrac{(s+1)(2s+1)}{4sL}

. Find the

s

that gives the best bound, and the bound itself. ((c)) Turn the best bound into an upper bound on the lowest angular frequency

\omega_1

. The exact eigenvalue, from Bessel functions, is

\lambda_1=1.4458/L

. By what percentage does your bound on

\omega_1

exceed the exact value, and why can it not fall below it?

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