QUESTION $2/3$

The wheel and axle below turn around a vertical axis with an angular velocity $$ as shown. The resulting spin of the wheel around the $x$ axis is defined as $$ as shown. The end of the axle $O$ is fixed and the point of contact of the wheel with the ground is no$-$slip. Therefore the relationship between $$ and $$ is:

$=-\frac{}{cos}$

$[$Please note that you can derive this relationship, but I am providing it to you.$]$

Treat the spokes and axle as massless, and treat the wheel as a thin ring with mass, $m,$ and a rectangular cross section $($area$).$

Find the angular momentum about the fixed point $0.$

ith permission from "Engineering

tics: Dynamics," McGill$/$King$,4^{th}$ Ed$,$

vec$(H{)}_O=m{r}^2(sin-\frac{1}{sin})hat(i)+m(b^2-\frac{r^2}{2})coshat(j)$

vec$(H{)}_O=m{r}^2(cos-\frac{1}{cos})hat(i)+m(b^2-\frac{r^2}{2})sinhat(j)$

vec$(H{)}_O=m\frac{r^2}{2}(cos-\frac{1}{cos})hat(i)+m(b^2-r^2)sinhat(j)$

None of the above