Physicists tend to call a generic object a weight or a mass. Sometime early in my teaching career, I decided to stop doing that. Had I written the rotational motion pulley question, I would have referred to the object on the lower end of the rope as, "an object," "the lower object," "the falling object," or "the descending object;" or; better yet; I would have specified an object such as a ball or a rock and referred to it as such. Also, given that the pulley is functioning as a spool, I would have called it a spool. I like the suggestion that instead of rope, the writer of the problem statement use a band of uniform cross-section, negligible mass, and non-negligible thickness, made of limp inextensible material, wrapped around the spool in such a manner that each winding is directly on top of the preceding winding.