The Yo-Yo — Question 8

Question 8

Cutting the cord changes acceleration but not instantaneous velocity

A yo-yo descends on a fixed cord with center speed $2\,\mathrm{m/s}$ and clockwise spin $100\,\mathrm{rad/s}$. The cord is cut without an impulse. Take $g=10$ and ignore air resistance.

1. Find center speed and angular speed $0.30\,\mathrm s$ later.

2. For axle radius $0.020\,\mathrm m$, test the old no-slip relation after the cut.

Question 8 - Solution

Cutting the cord changes acceleration but not instantaneous velocity

(a) Derivation and calculation

No cutting impulse means that center velocity and angular velocity remain continuous. After cutting, $$a=g,\qquad\tau_C=0\Rightarrow\alpha=0.$$ At $t=.30$ s, $$\boxed{v=2+10(.30)=5.00\ \mathrm{m/s}\quad\text{down}},\qquad \boxed{|\omega|=100\ \mathrm{rad/s}\quad\text{clockwise}}.$$

(b) Result and interpretation

The previous contact condition would require $$v=r|\omega|=.020(100)=2.00\ \mathrm{m/s}.$$ Instead, $$\boxed{v-r|\omega|=5-2=3.00\ \mathrm{m/s}}.$$ The relation no longer applies because the cord contact has been removed; gravity changes translation but supplies no torque about the center.

Physical check

The old constraint holds immediately before and at the cut, then ceases to govern subsequent motion.