A ballerina begins a tour jeté with angular speed ${\mathrm{\omega }}_i$ and a rotational inertia consisting of two parts: ${\mathrm{I}}_{leg}=1.36 \mathrm{kg} {\mathrm{m}}^2$ for her leg extended outward at an angle of $\theta =90°$ to her body and ${\mathrm{I}}_{trunk}=0.620 \mathrm{kg} {\mathrm{m}}^2$ for the rest of her body (primarily her trunk). Near her maximum height, she holds both legs at an angle of $\theta =30°$ to her body and has an angular speed of ${\mathrm{\omega }}_f$. Assuming that ${\mathrm{I}}_{trunk }$has not changed, what is the ratio of $\frac{{\mathrm{\omega }}_f}{{\mathrm{\omega }}_i}$?

       

Two $2.00 \mathrm{kg}$ balls are attached to the ends of a thin rod of length $50.0 \mathrm{cm}$ and negligible mass. The rod is free to rotate in a vertical plane without friction about a horizontal axis through its center. With the rod initially horizontal (Figure), a $50.0 \mathrm{g}$ wad of wet putty drops onto one of the balls. Hitting it with a speed of $3.00 \mathrm{m} {\mathrm{s}}^{-1}$ and then sticking to it. (a) What is the angular speed of the system just after the putty wad hits? (b) What is the ratio of the kinetic energy of the system after the collision and that of the putty wad just before? (c) Through what angle will the system rotate before it momentarily stops?

In the figure shown below, a small $50 \mathrm{g}$ block slides down a frictionless surface through height $h=15 \mathrm{cm}$ and then sticks to a uniform rod of mass$100 \mathrm{g}$  and length $35 \mathrm{cm}$. The rod pivots about point $O$ through an angle $\mathrm{\theta }$ before momentarily stopping. Find $\mathrm{\theta }$.

The figure is an overhead view of a thin uniform rod of length $0.600 \mathrm{m}$ and mass $M$  rotating horizontally at $80.0 \mathrm{rad} {\mathrm{s}}^{-1}$ counter clockwise about an axis through its center. A particle of mass $M/3.00$ and traveling horizontally at speed $40.0 \mathrm{m} {\mathrm{s}}^{-1}$ hits the rod and sticks. The particle's path is perpendicular to the rod at the instant of the hit, at a distance $d$ from the rod's center. (a) At what value of $d$ are the rod and the particle stationary after the hit? (b) In which direction does the rod and the particle rotate if $d$ is greater than this value?