The Yo-Yo — Question 3

Question 3

Changing axle radius alters acceleration without changing body inertia

Two yo-yos have the same mass $M$ and inertia $I=4Mr^2$. Their axle radii are $r$ and $2r$. Each descends from rest on a fixed cord through the same height $h$.

1. Compare their downward accelerations.

2. Compare descent times and final speeds.

Question 3 - Solution

Changing axle radius alters acceleration without changing body inertia

(a) Derivation and calculation

For axle radius $r_a$, the fixed-cord acceleration is $$a=\frac{g}{1+I/(Mr_a^2)}.$$ Since the same inertia is $I=4Mr^2$, $$\boxed{a_1=\frac{g}{1+4}=g/5},\qquad \boxed{a_2=\frac{g}{1+4r^2/(2r)^2}=g/2}.$$

(b) Result and interpretation

For equal drops from rest, $$t_i=\sqrt{2h/a_i},\qquad v_i=\sqrt{2a_ih}.$$ Therefore $$\boxed{\frac{t_1}{t_2}=\sqrt{\frac{a_2}{a_1}}=\sqrt{\frac52}},\qquad \boxed{\frac{v_2}{v_1}=\sqrt{\frac52}}.$$ Increasing axle radius reduces the effective rotational mass $I/r_a^2$ and speeds the descent.

Physical check

The larger axle has one quarter the effective rotational mass, despite unchanged physical inertia.