(a) A cylindrical space-station module can be modelled as a uniform solid cylinder of…

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 cylindrical space-station module can be modelled as a uniform solid cylinder of radius

1.2\,\mathrm{m}

, length

3.0\,\mathrm{m}

and mass

2200\,\mathrm{kg}

. During a test it spins slowly about its long axis at

2.0

revolutions per minute. A small one-off nudge disturbs the spin axis and no further torque acts afterwards. Estimate the period with which the perpendicular part of

\boldsymbol\omega

circles the module's long axis, as seen by an astronaut riding inside the module (the body rate). (b) A small gyroscope toy top has a disc of mass

0.020\,\mathrm{kg}

and radius

2.0\,\mathrm{cm}

mounted with its centre

3.0\,\mathrm{cm}

from a fixed pivot. Treat the disc's own transverse moment as

\tfrac14MR^2

about its centre and use the parallel-axis rule to get the pivot moment

I_1

; use

I_3=\tfrac12MR^2

for the spin axis. Started spinning at

40

revolutions per second and standing upright, it slows under a constant bearing-friction torque of

2.0\times10^{-5}\,\mathrm{N\,m}

acting to reduce the spin. Taking

g=9.8\,\mathrm{m\,s^{-2}}

, estimate the spin rate at which it stops sleeping and the time after release at which this happens.

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