A uniform wooden storage box

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 uniform wooden storage box, modelled as a uniform rectangular block, has sides

0.40\,\mathrm{m}

0.25\,\mathrm{m}

and

0.05\,\mathrm{m}

and mass

1.2\,\mathrm{kg}

. Free of any torque (thrown into the air over a short flight, air resistance ignored), it is spun about each of its three principal axes in turn, always at

3.0

revolutions per second, and released. ((a)) Using

I = M(p^2+q^2)/12

for the moment about the axis parallel to a side of length

r

, with

p

q

the other two side lengths, find the three principal moments and predict which of the three spins tumbles uncontrollably. Justify the choice from the moments alone, without assuming the answer. ((b)) For the unstable spin, find the growth rate

\lambda

of a small misalignment and estimate the time for a fractional deviation of

0.0010

in the transverse angular-velocity components to grow to order unity.

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