A uniform thin rod of mass kg and length m lies along the axis with its centre of…

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

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

Problem

A uniform thin rod of mass M=0.60M = 0.60 kg and length L=0.80L = 0.80 m lies along the zz axis with its centre of mass at the origin. ((a)) Write down the inertia tensor about the centre of mass, using symmetry to state which entries are zero or equal. ((b)) Use the tensor parallel-axis theorem to find the inertia tensor about the point at one end of the rod, d=(0,0,L/2)\mathbf d = (0,0,L/2). ((c)) State the principal moments and axes at the end, and explain why the shift left the tensor diagonal. ((d)) The rod turns with Euler-angle rates ϕ˙=3.0\dot\phi = 3.0 rad/s, θ˙=1.0\dot\theta = 1.0 rad/s, ψ˙=0\dot\psi = 0 (3-1-3 convention) at θ=60.0∘\theta = 60.0^\circ, ψ=0∘\psi = 0^\circ. Find ω1,ω2,ω3\omega_1,\omega_2,\omega_3 in the principal-axis frame found in (c).

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