The direction of static friction on a rolling object depends on the point of force application and the object's moment of inertia. When a force is applied at the center, the object may slide without rotating, while a force at the top generates both translational and rotational motion, causing static friction to act in the same direction as the applied force. If the applied force creates more torque than necessary for the linear acceleration, the static friction will adjust to maintain rolling without slipping. The distinction between rolling without slipping and rolling with slipping is crucial, as the former requires the point of contact to be momentarily at rest. Understanding these dynamics is essential for analyzing motion in various physical scenarios.