(a) A rugby ball tumbling in flight, air resistance ignored, is modelled as a…

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 rugby ball tumbling in flight, air resistance ignored, is modelled as a symmetric rigid body with principal moments I1=I2=1.4×10−2 kg m2I_1=I_2=1.4\times10^{-2}\,\mathrm{kg\,m^2} and I3=6.0×10−3 kg m2I_3=6.0\times10^{-3}\,\mathrm{kg\,m^2} about its centre of mass, spinning about its long axis at ω3=15 rad s−1\omega_3=15\,\mathrm{rad\,s^{-1}} with a small wobble. Calculate the body precession rate Ωb\Omega_{\rm b} and, taking L≈I3ω3L\approx I_3\omega_3 for the small wobble, the space rate Ωs\Omega_{\rm s} at which a spectator in the stand sees the long axis circle L\mathbf L. Which of the two would a camera fixed to the ball record? (b) A museum demonstration top has mass M=0.30 kgM=0.30\,\mathrm{kg}, spin moment of inertia about its pivot I3=1.0×10−4 kg m2I_3=1.0\times10^{-4}\,\mathrm{kg\,m^2}, transverse moment about its pivot I1=4.0×10−4 kg m2I_1=4.0\times10^{-4}\,\mathrm{kg\,m^2}, and pivot-to-centre distance d=2.5 cmd=2.5\,\mathrm{cm}. Spun at ω3=300 rad s−1\omega_3=300\,\mathrm{rad\,s^{-1}} and released, spin only, at tilt θ=20∘\theta=20^\circ, write down pψp_\psi and pϕp_\phi. (c) Using the gyroscope approximation, estimate the slow steady-precession rate at θ=20∘\theta=20^\circ, taking g=9.8 m s−2g=9.8\,\mathrm{m\,s^{-2}}.

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