Past exam of the options II course of the University of Oxford / 2012 / H / H2 / b / Solution

For uniform density, the centroid is the volume average of position and coincides with the centre of mass:A point fixed in the body has ${\mathbf{r}}_{OQ}={\sum }_i{q}_i{\mathbf{e}}_i$ with time-independent body coordinates $q_i$. Since $O$ is fixed, part (a) givesThe centroid is fixed relative to the body as well, so ${\mathbf{v}}_G=\mathbf{\omega }\times {\mathbf{r}}_{OG}$. Using ${\mathbf{r}}_{OQ}={\mathbf{r}}_{OG}+{\mathbf{r}}_{GQ}$ therefore gives