Mathematical Methods for Physics and Engineering - Matematica.NET

NORMAL MODES9.8 (It is recommended that the reader does not attempt this question until exercise 9.6has been studied.)Find a real linear transformation that simultaneously reduces the quadraticforms3x 2 +5y 2 +5z 2 +2yz +6zx − 2xy,5x 2 +12y 2 +8yz +4zxto diagonal form.9.9 Three particles of mass m are attached to a light horizontal string having fixedends, the string being thus divided into four equal portions each of length a andunder a tension T . Show that for small transverse vibrations the amplitudes x iof the normal modes satisfy =(maω 2 /T )x, whereB is the matrixBx⎛⎝ −1 2 −12 −1 0⎞⎠ .0 −1 2Estimate the lowest and highest eigenfrequencies using trial vectors (3 4 3)(Tand (3 − 4 3) T . Use also the exact vectors 1 √ ) T (2 1 and 1 − √ ) T2 1and compare the results.9.10 Use the Rayleigh–Ritz method to estimate the lowest oscillation frequency of aheavy chain of N links, each of length a (= L/N), which hangs freely from oneend. (Try simple calculable configurations such as all links but one vertical, orall links collinear, etc.)9.5 Hints and answers9.1 See figure 9.6.9.3 (b) x 1 = ɛ(cos ωt +cos √ 2ωt), x 2 = −ɛ cos √ 2ωt, x 3 = ɛ(− cos ωt +cos √ 2ωt).At various times the three displacements will reach 2ɛ, ɛ, 2ɛ respectively. For example,x 1 canbewrittenas2ɛ cos[( √ 2−1)ωt/2] cos[( √ 2+1)ωt/2], i.e. an oscillationof angular frequency ( √ 2+1)ω/2 and modulated amplitude 2ɛ cos[( √ 2−1)ωt/2];the amplitude will reach 2ɛ after a time ≈ 4π/[ω( √ 2 − 1)].9.5 As the circuit loops contain no voltage sources, the equations are homogeneous,and so for a non-trivial solution the determinant of coefficients must vanish.(a) I 1 =0,I 2 = −I 3 ; no current in PQ; equivalent to two separate circuits ofcapacitance C and inductance L.(b) I 1 = −2I 2 = −2I 3 ; no current in TU; capacitance 3C/2 and inductance 2L.9.7 ω =(2.634g/l) 1/2 or (0.3661g/l) 1/2 ; θ 1 = ξ + η, θ 2 =1.431ξ − 2.097η.9.9 Estimated, 10/17