(a) A satellite's momentum wheel is a symmetric flywheel with principal moments and…

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 satellite's momentum wheel is a symmetric flywheel with principal moments I1=I2=4.0×10−3 kg m2I_1=I_2=4.0\times10^{-3}\,\mathrm{kg\,m^2} and I3=7.5×10−3 kg m2I_3=7.5\times10^{-3}\,\mathrm{kg\,m^2} about its centre of mass. Spinning torque-free at ω3=200 rad s−1\omega_3=200\,\mathrm{rad\,s^{-1}} with a small wobble, find the rate Ωb\Omega_{\rm b} at which the perpendicular part of ω\boldsymbol\omega turns about the symmetry axis, as seen in the body. (b) A separate toy top has mass M=0.12 kgM=0.12\,\mathrm{kg}, spin moment of inertia about its pivot I3=5.0×10−5 kg m2I_3=5.0\times10^{-5}\,\mathrm{kg\,m^2}, transverse moment about its pivot I1=2.0×10−4 kg m2I_1=2.0\times10^{-4}\,\mathrm{kg\,m^2}, and centre of mass a distance d=3.0 cmd=3.0\,\mathrm{cm} from the pivot. It is spun at ω3=200 rad s−1\omega_3=200\,\mathrm{rad\,s^{-1}}, held at tilt θ=40∘\theta=40^\circ from the vertical, and released with its spin only (ϕ˙=0\dot\phi=0 at release). Write down the conserved momenta pψp_\psi and pϕp_\phi. (c) Using the gyroscope approximation ϕ˙slow≈Mgd/(I3ω3)\dot\phi_{\rm slow}\approx Mgd/(I_3\omega_3), estimate the slow steady-precession rate of the toy top at θ=40∘\theta=40^\circ, taking g=9.8 m s−2g=9.8\,\mathrm{m\,s^{-2}}.

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