The toy top of the worked example (a disc of mass and radius whose centre sits from a…

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

The toy top of the worked example (a disc of mass

0.15\,\mathrm{kg}

and radius

3.0\,\mathrm{cm}

whose centre sits

d=4.0\,\mathrm{cm}

from a fixed frictionless pivot, spun at 50 turns a second, with

g=9.81\,\mathrm{m\,s^{-2}}

) is again released at

\theta_0=30^\circ

from the upward vertical with

\dot\theta=0

. This time a sideways flick of the hand starts the axis with a precession rate

\dot\phi_0\ne0

. ((a)) For

\dot\phi_0=-2.0\,\mathrm{rad\,s^{-1}}

, opposite in sense to the slow steady precession at this tilt, find

p_\phi

and the second turning angle

\theta_1

. ((b)) Find the tilt

\theta^*

at which

\dot\phi

passes through zero, and decide whether the tip of the axis traces loops, cusps or smooth waves between the two turning angles. ((c)) Repeat parts (a) and (b) for

\dot\phi_0=+1.5\,\mathrm{rad\,s^{-1}}

. Then show that, for any release at rest in

\theta

at this tilt with

\dot\phi_0

below the slow steady rate, the sign of

\dot\phi_0

alone decides between loops and waves.

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