Starting from the highest point, a particle slides down the surface of a fixed smooth sphere. Determine the angle swept by the radius drawn to the particle at the instant it loses contact with the sphere.

Step-by-step solution

Let m be the mass of the particle and R be the radius of the sphere. Let be the angle made by the radius vector of the particle with the vertical at any instant. The vertical distance descended by the particle is h = R - R = R(1 - ) . Applying conservation of mechanical energy: 1 2 m v^2 = m g R (1 - ) v^2 = 2 g R (1 - ) The equation of motion along the radial direction is: m g - N = m v^2 R where N is the normal reaction. The particle loses contact with the sphere when N = 0 . m g = m v^2 R Substituting the value