A particle (p) moves along the curved surface (s). Show that p will remain in contact with s provided that at all times its speed v≥√rgcosteta.

Answer 1

Draw a figure. look at the forces :
centripetal force =mv^2/radius
gravity force along the radius: mg*CosTheta (angles measured from the top).

set them such that gravit>=cent force
v^2/r>=g*cosTheta
solve for v