$(35$ points$).$ We have three objects all with mass $m$ and radius $r($but from different material$).$ One is a $($solid$)$ uniform sphere: S with $I_S=0\mathrm{.}4m{r}^2,$ one is a uniform $($solid$)$ cylinder C $,$ with $I_C=0\mathrm{.}5m{r}^2$ and the last one a thin pipe P with $I_P=m{r}^2.$ The surfaces are smooth so that there is no friction between the objects if they touch but there are placed on a relatively rough surface which will allow rolling but NO slipping.

a$.5$ pts Draw a set of free body diagrams for one of the three $($you pick$),$ if it were placed on the surface alone.

b$.15$ pts If placed on the incline one at a time, what would be the acceleration of each down the incline? Answers in terms of $g$ and $sin.$

c$.5$ pts Suppose we release them together in the arrangement$/$order shown below. Do they separate? Justify your answer briefly. Do you need to modify the free$-$body diagrams for any of them $($from part a$)?$ IF so$,$ show the modified ones.

d$.10$ pts Now suppose we release them with the positions of pipe and cylinder switched $($from the arrangement shown in the figure$).$ Do they separate? Do you need to modify the free$-$body diagrams of any of them $($from part a$)?$ IF so$,$ show the modified ones.

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