The four parts concern two unrelated bodies: a rotor in the first two

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

The four parts concern two unrelated bodies: a rotor in the first two, a symmetric body in the last two. A rigid rotor spins at constant angular speed ω\omega about the space zz axis, held by two bearings. At the instant considered, the body axes are aligned with the space axes, and about the centre of mass the inertia tensor has Izz=0.020 kg m2I_{zz} = 0.020\ \mathrm{kg\,m^2}, a single nonzero product of inertia Ixz=−3.0×10−3 kg m2I_{xz} = -3.0\times10^{-3}\ \mathrm{kg\,m^2}, and Ixy=Iyz=0I_{xy}=I_{yz}=0. The rotor turns at ω=150 rad s−1\omega = 150\ \mathrm{rad\,s^{-1}}. ((a)) Starting from Li=IijωjL_i = I_{ij}\omega_j with ω=(0,0,ω)\boldsymbol\omega = (0,0,\omega), write the general components of L\mathbf{L} in terms of IzzI_{zz}, IxzI_{xz} and ω\omega, and explain in one sentence why L\mathbf{L} is not parallel to ω\boldsymbol\omega. ((b)) Evaluate the components of L\mathbf{L}, its magnitude, and the angle between L\mathbf{L} and ω\boldsymbol\omega, for the stated numbers. ((c)) A symmetric body has principal moments I1=I2I_1=I_2 and I3I_3, and Euler angles ϕ,θ,ψ\phi,\theta,\psi (3-1-3 convention) describe its orientation. Starting from the body components ω1=ϕ˙sin⁡θsin⁡ψ+θ˙cos⁡ψ\omega_1=\dot\phi\sin\theta\sin\psi+\dot\theta\cos\psi, ω2=ϕ˙sin⁡θcos⁡ψ−θ˙sin⁡ψ\omega_2=\dot\phi\sin\theta\cos\psi-\dot\theta\sin\psi, ω3=ϕ˙cos⁡θ+ψ˙\omega_3=\dot\phi\cos\theta+\dot\psi, derive Trot=12I1(θ˙2+ϕ˙2sin⁡2θ)+12I3(ψ˙+ϕ˙cos⁡θ)2.T_{\rm rot}=\tfrac12 I_1(\dot\theta^2+\dot\phi^2\sin^2\theta)+\tfrac12 I_3(\dot\psi+\dot\phi\cos\theta)^2. ((d)) Evaluate TrotT_{\rm rot} for the symmetric body with I1=I2=3.4×10−4 kg m2I_1=I_2=3.4\times10^{-4}\ \mathrm{kg\,m^2}, I3=6.8×10−4 kg m2I_3=6.8\times10^{-4}\ \mathrm{kg\,m^2}, ϕ˙=4.0 rad s−1\dot\phi=4.0\ \mathrm{rad\,s^{-1}}, θ=40∘\theta=40^\circ, θ˙=1.5 rad s−1\dot\theta=1.5\ \mathrm{rad\,s^{-1}}, ψ˙=120 rad s−1\dot\psi=120\ \mathrm{rad\,s^{-1}}.

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