The Yo-Yo — Question 9

Question 9

A finite cord ends the validity of the smooth descent model

A yo-yo with $I=4Mr^2$ starts from rest and unwinds $1.0\,\mathrm m$ of usable cord on a fixed support. Use $g=10$ and $r=0.020\,\mathrm m$. No losses occur before the cord end is reached.

1. Find the time, speed and angular speed just before full extension.

2. Can the rebound speed be determined from these assumptions alone?

Question 9 - Solution

A finite cord ends the validity of the smooth descent model

(a) Derivation and calculation

Before the end event, fixed-cord unwinding gives $$a=\frac{g}{1+4}=2.00\ \mathrm{m/s^2}.$$ For a descent $h=1.0$ m from rest, $$\boxed{t=\sqrt{2h/a}=1.00\ \mathrm s},\qquad \boxed{v=\sqrt{2ah}=2.00\ \mathrm{m/s}},$$ $$\boxed{|\omega|=v/r=2/.020=100\ \mathrm{rad/s}}.$$

(b) Result and interpretation

These are the state variables just before full extension. Across the end event, impulses may change them: $$M(v_+-v_-)=J_y,\qquad I(\omega_+-\omega_-)=J_\tau.$$ Neither the impulse relation nor energy recovery is specified for that event. A restitution, cord-elasticity, or contact-transition model is required to determine the rebound speed.

Physical check

The lossless-descent assumption applies before full extension and does not prescribe the physics of the end transition.