A solid round object of radius R can roll down an incline that makes an angle \theta with the horizontal. Assume that the rotational inertia about an axis through the center of mass is given by I=\beta m R^{2}. The coefficient of kinetic and static friction between the object and the incline is \mu. The object moves from rest through a vertical distance h.
a. If the angle of the incline is sufficiently large, then the object will slip and roll; if the angle of the incline is sufficiently small, then the object with roll without slipping. Determine the angle \theta_{c} that separates the two types of motion.
b. Derive expressions for the linear acceleration of the object down the ramp in the case of
i. Rolling without slipping, and
ii. Rolling and slipping.