I came up with a question in while studying pure rolling in mechanics. Consider a hollow cone on a flat surface with its curved side lying on the floor. On applying an impulse on the cone at the curved side, close to the centre of mass of the cone, it is observed that the cone undergoes pure rolling while it itself rotates about its apex. My question is that why does the centre of mass undergoes circular motion even after the impulse if apparently, there is no force acting on the cone. I know that there is no friction present after the impulse because frictional force acts only when there is net force on the mass and in this there is nothing like tension or something which provides centripetal force.EDIT: The cone seems to experiences this force only when the centre of mass has relative motion and that means the force might be velocity dependent. But we know neither gravitational force nor static friction are velocity dependent. So there might be some other force acting.
It’s experimentally observed that the cone moves in a circular path around its apex. It’s also known that any object moving in a circular path must have a force acting on it towards the center. This is enough to conclude that there is definitely a force acting on the center of mass of the cone pointing towards the center.
Now, what could this force be? Let’s see how this force actually arises. A cone has the natural tendency to move in a circular fasion as you have described, as it’s a property of it’s shape and also the fact that various rings at different lengths along the axis have different circumferences and hence will cover different distances when rolled. Now, this natural tendency of the cone causes static friction to come into the picture. It is static friction that acts inward to make sure that the cone executes it’s expected circular motion. Why is the force static friction? Because the lack of such a force would make the cone roll straight which would cause it to rub along the floor.