Q:

Condition for a particle to move on a sphere

Is it possible for a particle of mass $m$ to complete a circular path on a sphere if it starts from the topmost point and no external force acts on it except gravity and frictional force between sphere and the particle?
If yes, what are the conditions for it?  

A:

Short Answer - No
Refer to the figure below:

If you draw free body diagram of the particle at a point beyond C you will notice that there is no force acting towards the surface of sphere. Hence it won't be able to complete the circle beyond point C whatsoever.

