Question

Two beads(each of mass m) can move freely in a frictionless wire whose rotational inertia with respect to the vertical axis is I. The system is rotated with an angular velocity ${\omega }_0$ when the beads are at a distance $r/2$ from the axis. What is the angular velocity of the system when the beads are at a distance r from the axis?

Answer 1

Apply conservation of angular momentum :

The moment of inertia of the wire together with the beads,

$I_{wb}=I+2m{\left(\frac{r}{2}\right)}^2=I+m\frac{r^2}{2}...\left(1\right)$

the moment of inertia of the wire with beads at distance $r$.

$I_{wbf}=I+2m{r}^2...\left(2\right)$

hence,

By the conservation of angular momentum:

$I_{wb}{\omega }_0=I_{wbf}\omega$

Substitute the values:

$(I+m\frac{r^2}{2}){\omega }_0=(I+2m{r}^2)\omega \omega =\frac{(I+m\frac{r^2}{2})}{(I+2m{r}^2)}{\omega }_0$