Mathematical Methods for Physics and Engineering : A Comprehensive Guide

8.19 EXERCISES 8.20 Demonstrate that the matrix A=   200 − 644 3 − 10   is defective, i.e. does not have three linearly inde ...

MATRICES AND VECTOR SPACES 8.26 Show that the quadratic surface 5 x^2 +11y^2 +5z^2 − 10 yz+2xz− 10 xy=4 is an ellipsoid with sem ...

8.19 EXERCISES is 2 and that an orthogonal base for the null space ofAis provided by any two column matrices of the form (2 +αi ...

MATRICES AND VECTOR SPACES 8.40 Find the equation satisfied by the squares of the singular values of the matrix associated with ...

8.20 HINTS AND ANSWERS 8.5 Use the property of the determinant of a matrix product. 8.7 (d)S= ( 0 −tan(θ/2) tan(θ/2) 0 ) . (e) N ...

9 Normal modes Any student of the physical sciences will encounter the subject of oscillations on many occasions and in a wide v ...

9.1 TYPICAL OSCILLATORY SYSTEMS corresponding to a kinetic energy, is positive definite; that is, whatever non-zero real values ...

NORMAL MODES P P P l l θ 1 θ 1 θ 1 θ 2 θ^2 θ 2 (a) (b) (c) Figure 9.1 A uniform rod of lengthlattached to the fixed pointPby a l ...

9.1 TYPICAL OSCILLATORY SYSTEMS coordinate is involved in this special motion. In general there will beNvalues ofωif the matrice ...

NORMAL MODES frequency corresponds to a solution where the string and rod are moving with opposite phase andx 1 :x 2 =9.359 :− 1 ...

9.1 TYPICAL OSCILLATORY SYSTEMS m μm m x 1 x 2 x 3 k k Figure 9.2 Three massesm,μmandmconnected by two equal light springs of fo ...

NORMAL MODES The final and most complicated of the three normal modes has angular frequency ω={(μ+2)/μ}^1 /^2 , and involves a m ...

9.2 SYMMETRY AND NORMAL MODES M M M M k k k k k k x 1 y 1 x 2 y 2 x 3 y 3 x 4 y 4 Figure 9.4 The arrangement of four equal masse ...

NORMAL MODES The potential matrix is thus constructed as B= k 4              3 − 1 − 2000 − 11 − 13000 − 21 − 1 − 2 ...

9.2 SYMMETRY AND NORMAL MODES (a)ω^2 =0 (b)ω^2 =0 (c)ω^2 =0 (d)ω^2 =2k/M (e)ω^2 =k/M (f)ω^2 =k/M (g)ω^2 =k/M (h)ω^2 =k/M Figure ...

NORMAL MODES neous equations forαandβ, but they are all equivalent to just two, namely α+β=0, 5 α+β= 4 Mω^2 k α; these have the ...

9.3 RAYLEIGH–RITZ METHOD and that this mode has the same frequency as three of the other modes. The general topic of the degener ...

NORMAL MODES be shown that they do possess the desirable properties (xj)TAxi=0 and (xj)TBxi=0 ifi=j. (9.16) This result is prov ...

9.4 EXERCISES
Estimate the eigenfrequencies of the oscillating rod of section 9.1. Firstly we recall that A= Ml^2 12 ( 63 32 ) ...

NORMAL MODES under gravity. At timet,ABandBCmake anglesθ(t)andφ(t), respectively, with the downward vertical. Find quadratic exp ...