In a frictionless scenario, a round-headed rod will slide down a slope without toppling if its center of mass aligns with the normal force at the point of contact. However, if the rod is not perfectly vertical or if the curvature of the rounded head causes the center of mass to shift, it may topple as it slides. The discussion highlights that the balance of torques around the center of mass is crucial; if the normal force does not pass through the center of mass, the rod is likely to rotate and fall. Real-world conditions, including friction, complicate this behavior, potentially leading to different outcomes. Ultimately, the rod's stability depends on its orientation and the slope's characteristics.