A heavy symmetric top has about the pivot

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 Euler's equations, free precession and the heavy symmetric top

Problem

A heavy symmetric top has I1=3.0×10−4 kg m2I_1=3.0\times10^{-4}\,\mathrm{kg\,m^2} about the pivot, I3=1.0×10−4 kg m2I_3=1.0\times10^{-4}\,\mathrm{kg\,m^2} about its symmetry axis, and Mgd=0.050 N mMgd=0.050\,\mathrm{N\,m}. It is held at a fixed tilt θ=40∘\theta=40^\circ from the upward vertical. ((a)) Find both steady-precession rates ϕ˙\dot\phi from I1cos⁡θ ϕ˙2−I3ω3ϕ˙+Mgd=0I_1\cos\theta\,\dot\phi^2 - I_3\omega_3\dot\phi + Mgd = 0 for spin rates ω3=100\omega_3 = 100, 300300 and 900 rad s−1900\,\mathrm{rad\,s^{-1}}. ((b)) State, with a reason from the two sets of roots, how the slow root and the fast root each vary as ω3\omega_3 increases, and check the slow root at ω3=900 rad s−1\omega_3=900\,\mathrm{rad\,s^{-1}} against the gyroscope formula Mgd/(I3ω3)Mgd/(I_3\omega_3).

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