Mathematical Methods for Physics and Engineering - Matematica.NET

NORMAL MODESunder gravity. At time t, AB and BC make angles θ(t) andφ(t), respectively, withthe downward vertical. Find quadratic expressions for the kinetic and potentialenergies of the system and hence show that the normal modes have angularfrequencies given byω 2 = g [1+α ± √ ]α(1 + α) .lFor α =1/3, show that in one of the normal modes the mid-point of BC doesnot move during the motion.9.3 Continue the worked example, modelling a linear molecule, discussed at the endofsection9.1,forthecaseinwhichµ =2.(a) Show that the eigenvectors derived there have the expected orthogonalityproperties with respect to both A and B.(b) For the situation in which the atoms are released from rest with initialdisplacements x 1 =2ɛ, x 2 = −ɛ and x 3 = 0, determine their subsequentmotions and maximum displacements.9.4 Consider the circuit consisting of three equal capacitors and two different inductorsshown in the figure. For charges Q i on the capacitors and currents I iQ 1 Q 2CCQ 3L 1 C L 2I 1 I 2through the components, write down Kirchhoff’s law for the total voltage changearound each of two complete circuit loops. Note that, to within an unimportantconstant, the conservation of current implies that Q 3 = Q 1 − Q 2 . Express the loopequations in the form given in (9.7), namelyA¨Q + BQ = 0.Use this to show that the normal frequencies of the circuit are given byω 2 =1CL 1 L 2[L1 + L 2 ± (L 2 1 + L 2 2 − L 1 L 2 ) 1/2] .Obtain the same matrices and result by finding the total energy stored in thevarious capacitors (typically Q 2 /(2C)) and in the inductors (typically LI 2 /2).For the special case L 1 = L 2 = L determine the relevant eigenvectors and sodescribe the patterns of current flow in the circuit.9.5 It is shown in physics and engineering textbooks that circuits containing capacitorsand inductors can be analysed by replacing a capacitor of capacitance C by a‘complex impedance’ 1/(iωC) and an inductor of inductance L by an impedanceiωL, whereω is the angular frequency of the currents flowing and i 2 = −1.Use this approach and Kirchhoff’s circuit laws to analyse the circuit shown in330