How do I calculate this circular motion question?

Question

A particle moving with constant speed U inside a smooth spherical bowl of radius A describes a horizontal circle at a distance A/2 below its centre. Find the balue of U.

Answer 1

I assume the particle is maintaining its vertical position because of centrifugal force.  The vertical component of that force must balance the weight of the particle.  If the centrifugal force is Fc, the component of that force tangent to the sphere is Fc*cosθ, where θ is the half-angle subtended by the horizontal circle from the center of the sphere.  The vertical component of that force is Fc*cosθ*sinθ.  Fc is m*U²/r so the vertical force component is (m*U²/r)*cosθ*sinθ.  r is the radius of the horizontal circle which is r = A*sinθ.  the vertical force is (m*U²/A)*cosθ

From the geometry of the configuration, cosθ = ½ so the vertical component of centrifugal force is

½*m*U²/A

and the weight is m*g, so

m*g = ½*m*U²/A

U = √[2*g*A]