A 45 kg figure skater is spinning on the toes of her skates at 1.0 rev/s. Her arms are outstretched as far as they will go. In this orientation, the skater can be modeled as a cylindrical torso (40 kg, 20 cm average diameter, 160 cm tall) plus two rod-like arms (2.5 kg each, 66 cm long) attached to the outside of the torso. The skater then raises her arms straight above her head, where she appears to be a 45 kg, 20-cm-diameter, 200-cm-tall cylinder. What is her new angular velocity, in rev/s?

Mass of the skater 45 kg
Angular velocity of the skater= 1 rev/sec

Now details about the model

Mass of model =40 kg
Average diameter of model =20cm =0.2 m
so average radius of=0.1m
Length of the model =1.6 m
The specifications for the rod :
Mass of the rod =2.5 kg each
Length of arm =0.66 m

First consider when arms are stretched,

The torso acts like a solid cylindrical shape, and the arms stretched acts like a rod

Total moment of inertia = Moment of inertia of the solid cylinder + Moment of inertia of cylindrical rod

$$\begin{gathered} I_{\text{total}}=\frac{1}{2}M{r}^2+2M{r}^2 \\ {I}_{\text{total}}=\frac{(40\mathrm{kg})(0.10\mathrm{m}{)}^2}{2}+2(2.5\mathrm{kg})(0.33\mathrm{m}{)}^2=0.744kg{m}^2 \end{gathered}$$

angular momentum = Iω,

substitute the values we get

$L=(0.744kg.m^2)(1\text{revolution}/\sec )=0.744kg.m^2/s$

Now consider when her arms are above head:

Total mass is on center and total mass is 45 kg.

So the moment of inertia is

$I^{\text{'}}=\frac{M{r}^2}{2}$

Susbstitute the values

$$\begin{gathered} I^{\text{'}}=\frac{(45\mathrm{kg})(0.10m{)}^2}{2} \\ =0.225\mathrm{kg}{\mathrm{m}}^2 \end{gathered}$$

Calculate the momentum

$L^{\text{'}}=I^{\text{'}}\omega =0.225kg.m^2\omega ..................................\left(1\right)$

From the momentum conservation, equate equation(1) and equation(2)

$$\begin{gathered} L=L^{\text{'}}\Rightarrow I{\omega }_o=I^{\text{'}}\omega \\ \Rightarrow \omega =\frac{I{\omega }_o}{I^{\text{'}}} \end{gathered}$$

$\Rightarrow \omega =\frac{0.744kg.m^2/s}{0.225kg.m^2}=3.3\mathrm{rev}/\sec$