The Yo-Yo — Question 7

Question 7

A sleeping yo-yo uses a different contact model

A yo-yo spins with $I=8.0\times10^{-5}\,\mathrm{kg\,m^2}$ and angular speed $200\,\mathrm{rad/s}$ while its center hangs at rest from a frictionless axle bearing on a loop of string. Its mass is $0.10\,\mathrm{kg}$ and $g=9.8$.

1. Find the total upward support force and spin energy.

2. Explain why the fixed-cord unwinding constraint cannot be imposed here.

Question 7 - Solution

A sleeping yo-yo uses a different contact model

(a) Derivation and calculation

The center is in translational equilibrium: $$S-Mg=0\Rightarrow\boxed{S=.10(9.8)=.980\ \mathrm N}.$$ The spin energy is $$\boxed{K=\tfrac12I\omega^2=\tfrac12(8.0\times10^{-5})(200)^2=1.60\ \mathrm J}.$$

(b) Result and interpretation

The frictionless bearing transmits no rotor torque: $$\tau_C=0\Rightarrow\dot\omega=0,\qquad\dot K=0.$$ The rotor can rotate relative to the support, so the unwinding condition $v=r\omega$ does not apply. Imposing it with $v=0$ would incorrectly force $\omega=0$ and contradict the bearing model.

Physical check

An upward force can support the rotor weight without supplying a moment about its spin axis.