Mathematical Methods for Physics and Engineering : A Comprehensive Guide

NORMAL MODES

under gravity. At timet,ABandBCmake anglesθ(t)andφ(t), respectively, with

the downward vertical. Find quadratic expressions for the kinetic and potential

energies of the system and hence show that the normal modes have angular

frequencies given by

ω^2 =

g

l

[

1+α±

√

α(1 +α)

]

.

Forα=1/3, show that in one of the normal modes the mid-point ofBCdoes
not move during the motion.
9.3 Continue the worked example, modelling a linear molecule, discussed at the end
ofsection9.1,forthecaseinwhichμ=2.

(a) Show that the eigenvectors derived there have the expected orthogonality

properties with respect to bothAandB.

(b) For the situation in which the atoms are released from rest with initial

displacementsx 1 =2,x 2 =−andx 3 = 0, determine their subsequent

motions and maximum displacements.

9.4 Consider the circuit consisting of three equal capacitors and two different in-
ductors shown in the figure. For chargesQion the capacitors and currentsIi

L 1 L 2

C C

C

I 1 I 2

Q 1 Q 2

Q 3

through the components, write down Kirchhoff’s law for the total voltage change

around each of two complete circuit loops. Note that, to within an unimportant

constant, the conservation of current implies thatQ 3 =Q 1 −Q 2. Express the loop

equations in the form given in (9.7), namely

AQ ̈+BQ= 0.

Use this to show that the normal frequencies of the circuit are given by

ω^2 =

1

CL 1 L 2

[

L 1 +L 2 ±(L^21 +L^22 −L 1 L 2 )^1 /^2

]

.

Obtain the same matrices and result by finding the total energy stored in the

various capacitors (typicallyQ^2 /(2C)) and in the inductors (typicallyLI^2 /2).

For the special caseL 1 =L 2 =Ldetermine the relevant eigenvectors and so

describe the patterns of current flow in the circuit.

9.5 It is shown in physics and engineering textbooks that circuits containing capaci-
tors and inductors can be analysed by replacing a capacitor of capacitanceCby a
‘complex impedance’ 1/(iωC) and an inductor of inductanceLby an impedance
iωL,whereωis the angular frequency of the currents flowing andi^2 =−1.
Use this approach and Kirchhoff’s circuit laws to analyse the circuit shown in