Classical Dynamics of Particles and Systems · Problem 11-18

Coupled Oscillations — Marion & Thornton

Physics — Mechanics · inertia-tensor-rotated-plate

Problem

If, in the previous problem, the coordinate axes are rotated through an angle θ\theta about the x3x_3-axis, show that the new inertia tensor is {I}=(A′−C′0−C′B′000A′+B′)\{\mathbf{I}\} = \begin{pmatrix} A' & -C' & 0 \\ -C' & B' & 0 \\ 0 & 0 & A' + B' \end{pmatrix} where A′=Acos⁡2θ−Csin⁡2θ+Bsin⁡2θA' = A\cos^2\theta - C\sin 2\theta + B\sin^2\theta B′=Asin⁡2θ+Csin⁡2θ+Bcos⁡2θB' = A\sin^2\theta + C\sin 2\theta + B\cos^2\theta C′=Ccos⁡2θ−12(B−A)sin⁡2θC' = C\cos 2\theta - \frac{1}{2}(B - A)\sin 2\theta and hence show that the x1x_1- and x2x_2-axes become principal axes if the angle of rotation is θ=12tan⁡−1(2CB−A)\theta = \frac{1}{2}\tan^{-1}\left(\frac{2C}{B - A}\right)

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