You can do this with vectors. There's 3 force vectors acting on the mass: N, the normal force, , the graviational force and , the centripetal force.

Notice that the gravitational force acts only down (no horizontal component) and that the centripetal force acts only in the horizontal position, but that the normal force has a both a horizontal and vertical component (since it's perpendicular to the surface). Moreover, those components have to cancel out the other two forces for the mass to be at equilibrium. So

vertical:

horizontal:

divide both:



For part b, we can see that r does not depend on the mass, so the problem wouldn't change at all if we increased the mass to 3m.