A symmetric top has principal moments and

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 symmetric top has principal moments I1=I2=8.0×10−5 kg m2I_1=I_2=8.0\times10^{-5}\ \mathrm{kg\,m^2} and I3=1.6×10−4 kg m2I_3=1.6\times10^{-4}\ \mathrm{kg\,m^2}, and turns with Euler rates (3-1-3 convention) ϕ˙=5.0 rad s−1\dot\phi=5.0\ \mathrm{rad\,s^{-1}}, θ=50∘\theta=50^\circ, θ˙=0.80 rad s−1\dot\theta=0.80\ \mathrm{rad\,s^{-1}}, ψ˙=200 rad s−1\dot\psi=200\ \mathrm{rad\,s^{-1}}. Someone argues: "the body components of ω\boldsymbol\omega are just the three Euler-angle rates, so ω1=ϕ˙\omega_1=\dot\phi, ω2=θ˙\omega_2=\dot\theta, ω3=ψ˙\omega_3=\dot\psi," and computes Trot=12I1ϕ˙2+12I1θ˙2+12I3ψ˙2T_{\rm rot}=\tfrac12I_1\dot\phi^2+\tfrac12I_1\dot\theta^2+\tfrac12I_3\dot\psi^2. ((a)) Find this person's numerical value for TrotT_{\rm rot}. ((b)) Find the correct value of TrotT_{\rm rot} using the body components of ω\boldsymbol\omega from the Euler-angle rates. ((c)) State by what percentage the two values differ, and identify in one sentence where the argument breaks down.

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