Say there's a cylinder laying on a flat surface, and the surface is rested on frictionless ice. Attached to the surface is an engine that is attached via a string to the center of the cylinder, and (starting from rest) creates a tension of F in the string. The static friction between the cylinder and surface is enough for rolling without slipping. Now after a certain time T, the engine stops working altogether. The cylinder is already rolling at a certain velocity and is not slipping across the surface. What I would like to know is if there is conservation of energy from now on? Is there a static friction force between the surface and the cylinder? If there is - does it do work? Is the velocity constant?
When an object is rolling without slipping, the point of contact with the surface always has zero velocity relative to the surface. Therefore the static friction does no work. (since the point of contact has no displacement caused by the static friction.) Since there are no non conservative forces now, energy can be conserved. and yes, the velocity of the rolling cylinder remains constant too. (Unless some external agent changes it.)