A uniform rectangular plate of mass kg has sides m (along ) and m (along )

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Rotation, rigid bodies and accelerated frames

Action and Symmetry, Ch The inertia tensor, principal axes and Euler angles

Problem

A uniform rectangular plate of mass M=1.2M = 1.2 kg has sides a=0.40a = 0.40 m (along xx) and b=0.30b = 0.30 m (along yy), lies in the xyxy plane, and has its centre of mass at the origin. ((a)) Write down the inertia tensor about the centre of mass by symmetry. ((b)) Use the tensor parallel-axis theorem to find the inertia tensor about the corner at d=(a/2,b/2,0)\mathbf d = (a/2, b/2, 0). ((c)) Find the three principal moments of inertia at that corner. ((d)) The plate turns with Euler-angle rates ϕ˙=0.50\dot\phi = 0.50 rad/s, θ˙=2.0\dot\theta = 2.0 rad/s, ψ˙=10.0\dot\psi = 10.0 rad/s at θ=50.0∘\theta = 50.0^\circ, ψ=15.0∘\psi = 15.0^\circ. Find ω1,ω2,ω3\omega_1,\omega_2,\omega_3 in the principal-axis frame found in (c).

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