I am confused because u is not a force but just acceleration.

In Fig 1, there are two forces acting on the ball. The component of mg that is parallel to the rod is cancelled by the normall force from the rod, so the net force at the ball is only mg sinθ.

You said:

acceleration of ball = acceleration of ball relative to hinge + acceleration of hinge

And that the acceleration of the ball is a = F/m = g sinθ. Therefore g sinθ must also equal Lα + u.

In Fig 5, the situation is similar except that u is now from an external force, F

_{u} = Mu. Is this situation the same? Why is/isn't F

_{u} part of the net force acting on the ball?

- Thanks