Mechanics - Particle Sliding down a sphere

[Doogle]

Say I had a smooth sphere of radius r, and I placed a particle at the top and it started sliding down the sphere. At which point (lets say the angle to the horizontal with respect to the origin, taking the center of the sphere to be the origin) does the particle separate with (slide off of) the sphere? Assuming that the only force on the particle is the force due to gravity.

This is not a homework problem, but I'm just curious. Can anyone give me a complete solution? I'm more interested in seeing how it would be done than doing it myself!

Cheers in advance.

Oh I've just seen the same question here https://www.physicsforums.com/showthread.php?t=269635 - reply there instead. Okay don't, that post is a bit old. And they seem to think it's been posted before that even, but I can't find it. Anyone give me a link? Or just a solution? :D

 

[BigJDubs]

Assuming that the only force acting on the particle is gravity, the point at which it separates from the sphere is when the normal force of the surface of the sphere equals the gravitational force. The angle of the normal force depends on the radius and the angle of the surface, so we can calculate the angle by solving for the angle θ in the following equation: Fn = mg cosθWhere Fn is the normal force, m is the mass of the particle, g is the acceleration due to gravity, and θ is the angle of the surface. So, we can solve for the angle θ by rearranging the equation to get θ = cos-1(mg/Fn).Once we have the angle, we can then calculate the point at which the particle will separate from the sphere by finding the coordinates of the point using basic trigonometry. The x-coordinate of the point is simply rcosθ and the y-coordinate is rsinθ. Hope this helps!