HRWP8p297

Problem 8 contains the relation, r << R, which means "r is much less than R" (see page A-9 for meanings of symbols). That means that you can approximate the smaller object as a point mass in relation to the larger object. In problem 8, this means that you can assume that the ball's center-of-mass rises 2R from the bottom of the loop to the top. In reality, the center-of-mass only rises 2R-2r because the CM is at r at the bottom of the loop and is r below the top of the loop when it is at the top.

HRWP15p298

The linear acceleration of the yo-yo is the same as the center-of-mass acceleration (because we assume the yo-yo is going straight down). You also have to assume that the string is rolling directly off the axle. That way, when you use the relationship which ties angular motion to linear motion, a = α r, you use r = radius of axle, not the radius of the yo-yo.