I'm thinking about this: what happens if a nonuniform circular motion goes on for a long time? centripetal acceleration=tangential speed^2/r. If there is an existent tangential acceleration: tangential speed increases. So, centripetal acceleration must also increase. Therefore, after a very long time, the limiting case would be that the tangential acceleration becomes very small and the acceleration points towards the centre of the circle (if the string that sustains the bob (just an example) can support the acceleration)! Does centripetal acceleration change overtime?