I'm studying rotational dynamics and I've got a couple questions I can't answer. I want to describe de movement of the bodies in the cases below, the coefficient of friction in all the cases is [itex]\mu[/itex] . I will say what I think it would happen, and I would appreciate if you guys judge it right or wrong, as well as answering my other questions. a = translational acceleration (acceleration of the center of mass) γ = angular acceleration In (A) we have a sphere (I = 2/5 MR²) rotating without sliding in a horizontal plane. If we had friction to the right, a would increase and [itex]\gamma[/itex] would decrease, absurd because a = [itex]\gamma[/itex] R. If we had a frictions to the left, a would decrease and [itex]\gamma[/itex] increase. Absurd too. So will the sphere stay rotating forever? In (B) we have a sphere initially at rest at an inclined plane. If the sphere does not rotate, where is the friction? I would say the friction is upwards, as the sphere needs to increase both a and γ . Is it right? In (C) we have a sphere initially moving up an inclined plane, without sliding. Is it possible? I mean; if there was a friction upwards, a would be increasing and γ decreasing, but if there was a friction downwards, a would be decreasing and γ increasing, both things are imposible. So is there impossible to be a sphere rolling up an inclined plane without sliding? And a general question: How is the direction of the static friction determined generally? I've learned friction is opposite to the displacement of the body in relation to the plane of the friction but if we have a body rolling without sliding, the contact point have instantaneous velocity = 0, so there is no instantaneous displacement from the ground, where should the friction be?