Question

a particle is made to move along a vertical circular path with radius r find the height reached by the bloock over the wedge if there is no friction

Answer 1

The particle has a velocity v before moving along a vertical circular path of radius r.

Let the center of the circular path be C.  Let the vertical radius through C be CB.  Let A be the point on the circle at which the particle stops finally.  The particle loses speed when it moves up, as gravity decelerates it.

Initial total energy of the particle = K E = 1/2 * m * v² 
final total energy of the particle = P E = m g h,  where h = r (1 - Cos θ)
   θ is the angle between CB and CA.

we apply the conservation of energy principle.

     m g h = 1/2 * m * v² 
       => h = r (1 - cos θ) = v² /(2 g)