(a) A frisbee spinning in flight is modelled as a thin flattened symmetric body with…

PHYSICS · Subjective · 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) A frisbee spinning in flight is modelled as a thin flattened symmetric body with principal moments I1=I2=1.2×10−3 kg m2I_1=I_2=1.2\times10^{-3}\,\mathrm{kg\,m^2} and I3=2.4×10−3 kg m2I_3=2.4\times10^{-3}\,\mathrm{kg\,m^2} about its centre of mass, spinning at ω3=30 rad s−1\omega_3=30\,\mathrm{rad\,s^{-1}} with a small wobble. Calculate the body precession rate Ωb\Omega_{\rm b}. (b) A desk top toy has mass M=0.10 kgM=0.10\,\mathrm{kg}, spin moment of inertia about its pivot I3=3.0×10−5 kg m2I_3=3.0\times10^{-5}\,\mathrm{kg\,m^2}, transverse moment about its pivot I1=1.5×10−4 kg m2I_1=1.5\times10^{-4}\,\mathrm{kg\,m^2}, and pivot-to-centre distance d=2.0 cmd=2.0\,\mathrm{cm}. Spun at ω3=400 rad s−1\omega_3=400\,\mathrm{rad\,s^{-1}} and released, spin only, at tilt θ=15∘\theta=15^\circ, write down pψp_\psi and pϕp_\phi. (c) Using the gyroscope approximation, estimate the slow steady-precession rate at θ=15∘\theta=15^\circ, taking g=9.8 m s−2g=9.8\,\mathrm{m\,s^{-2}}.

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