Then it just goes onto the acceleration
side of F = ma …
you add it to the other (rotational) acceleration.
I didn't understand this at all.
The net force on the ball is mg sinθ plus
the force from the rod.
Nothing is cancelled (what is it to be cancelled with?) …
the reason why you leave out the mg cosθ is not because its cancelled, but simply because you're only looking at tangential components.
isn't "part of the net force acting on the ball" because isn't acting on the ball
at all …
decide which body you're applying Netwon's second law to, and then only use the forces on
that body …
your book is using the ball as the body, so the only forces on
it are mg and the force from the rod …
any force on anything else is irrelevant.
This is much better … you've used the same method as your book, except that you've moved the mu cosθ from one side of the equation to the other, by using the accelerating frame.
(The tangential force is mg sinθ, and the tangential acceleration (times mass) is mαL + mu cosθ)
This isn't necessary
and it's risky, because it's so easy to get the ± sign wrong.
Your book just uses the stationary frame, and uses only tangential components, so that there's only one
component of force on the ball, and the acceleration is split into two parts.