Problem $4(10$ pts$)$

The uniform bar $AB$ of mass $M$ and length $l$ has a small massless roller at each end and moves on a circular track as shown in figure $4.$ Using $$ as a generalized coordinate derive the equation of motion in the absence of friction. The length of $OG$ is $R.$ Using this equation of motion answer the following two questions concerning small motions about $=0.$

a$.$ Find the frequency of small oscillations about $=0.$

b$.$ Assume that this device is submerged in oil which exerts a significant drag force $($parallel to AB $)$ given by $-$bvec$(v{)}_G,$ where $b$ is a proportionality coefficient and vec$(v{)}_G$ is the velocity of point G$.$ Find the values of $b$ for which the bar will overshoot $=0$ if released from rest from $0.$

Figure $4$