Mathematical Methods for Physics and Engineering - Matematica.NET

TENSORS(b) Find the principal axes and verify that they are orthogonal.26.17 A rigid body consists of eight particles, each of mass m, held together by lightrods. In a certain coordinate frame the particles are at positions±a(3, 1, −1), ±a(1, −1, 3), ±a(1, 3, −1), ±a(−1, 1, 3).Show that, when the body rotates about an axis through the origin, if the angularvelocity and angular momentum vectors are parallel then their ratio must be40ma 2 ,64ma 2 or 72ma 2 .26.18 The paramagnetic tensor χ ij of a body placed in a magnetic field, in which itsenergy density is − 1 µ 2 0M · H with M i = ∑ j χ ijH j ,is⎛⎝ 2k 0 0 3k 0k⎞⎠ .0 k 3kAssuming depolarizing effects are negligible, find how the body will orientateitself if the field is horizontal, in the following circumstances:(a) the body can rotate freely;(b) the body is suspended with the (1, 0, 0) axis vertical;(c) the body is suspended with the (0, 1, 0) axis vertical.26.19 A block of wood contains a number of thin soft-iron nails (of constant permeability).A unit magnetic field directed eastwards induces a magnetic moment inthe block having components (3, 1, −2), and similar fields directed northwardsand vertically upwards induce moments (1, 3, −2) and (−2, −2, 2) respectively.Show that all the nails lie in parallel planes.26.20 For tin, the conductivity tensor is diagonal, with entries a, a, and b when referredto its crystal axes. A single crystal is grown in the shape of a long wire of length Land radius r, the axis of the wire making polar angle θ with respect to the crystal’s3-axis. Show that the resistance of the wire is L(πr 2 ab) ( −1 a cos 2 θ + b sin 2 θ ) .26.21 By considering an isotropic body subjected to a uniform hydrostatic pressure(no shearing stress), show that the bulk modulus k, defined by the ratio of thepressure to the fractional decrease in volume, is given by k = E/[3(1−2σ)] whereE is Young’s modulus and σ is Poisson’s ratio.26.22 For an isotropic elastic medium under dynamic stress, at time t the displacementu i and the stress tensor p ij satisfy( ∂ukp ij = c ijkl + ∂u )l∂x l ∂x kand∂p ij= ρ ∂2 u i∂x j ∂t , 2where c ijkl is the isotropic tensor given in equation (26.47) and ρ is a constant.Show that both ∇ · u and ∇×u satisfy wave equations and find the correspondingwave speeds.980