Two point masses

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

Two point masses, each of mass m=0.050m = 0.050 kg, are fixed to a light rigid frame at r1=(0.12,0,0.09)\mathbf r_1 = (0.12, 0, 0.09) m and r2=(−0.12,0,−0.09)\mathbf r_2 = (-0.12, 0, -0.09) m, measured from the frame's centre of mass (which is at the origin by symmetry). ((a)) Compute the inertia tensor about the centre of mass, stating which entries vanish by symmetry. ((b)) Use the tensor parallel-axis theorem to find the inertia tensor about the point d=(0,0,0.05)\mathbf d = (0,0,0.05) m, a bearing 0.050.05 m along zz from the centre of mass. ((c)) Find the three principal moments of inertia about that bearing point. ((d)) In a separate orientation the frame turns with Euler-angle rates ϕ˙=2.0\dot\phi = 2.0 rad/s, θ˙=0.8\dot\theta = 0.8 rad/s, ψ˙=15.0\dot\psi = 15.0 rad/s at θ=55.0∘\theta = 55.0^\circ, ψ=35.0∘\psi = 35.0^\circ. Find the body components ω1,ω2,ω3\omega_1,\omega_2,\omega_3 of the angular velocity.

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