A smartphone

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 smartphone, modelled as a uniform rectangular block

0.146\,\mathrm{m}

long,

0.071\,\mathrm{m}

wide,

0.0078\,\mathrm{m}

thick, of mass

0.194\,\mathrm{kg}

, is tossed spinning (air resistance ignored, no torque once released) about each of its three principal axes in turn: the long axis, the width axis and the thickness (face-normal) axis, each time at

3.0

revolutions per second. ((a)) Find the three principal moments (using

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

for the axis along a side of length

r

, with

p,q

the other two lengths) and predict which single throw flips uncontrollably rather than merely wobbling. ((b)) For that throw, find the growth rate

\lambda

of a small misalignment and the time for a

1.0\%

deviation to grow to order unity.

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