Create a point on the tip of the object. Create Assembly where the curve (spiral) is a component and put it to the default position. Add the object with the point on thetip and give it only 1 constrain like a plane parallel to the plane of the curve (use the mechanism assembly).
Enter in mechanism. Use the slot follow tool. Create new slot follower (leave the name as the one of default).Pick the point on the tip as the point to join the path and the curve as the path to follow.
Now with mechanism if you drag the object it will follow the curve.
Create a servomotor in the center of the system and the application point is the axis of the object.