Classical Dynamics of Particles and Systems · Problem 11-16

Coupled Oscillations — Marion & Thornton

Physics — Mechanics · inertia-tensor-rotation-diagonal

Problem

Consider the following inertia tensor: {I}=(12(A+B)12(A−B)012(A−B)12(A+B)000C)\{\mathbf{I}\} = \begin{pmatrix} \frac{1}{2}(A + B) & \frac{1}{2}(A - B) & 0 \\ \frac{1}{2}(A - B) & \frac{1}{2}(A + B) & 0 \\ 0 & 0 & C \end{pmatrix} Perform a rotation of the coordinate system by an angle θ\theta about the x3x_3-axis. Evaluate the transformed tensor elements, and show that the choice θ=π/4\theta = \pi/4 renders the inertia tensor diagonal with elements AA, BB, and CC.

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