Motion of a Particle on a Sphere

[schutte]

I have just recently been given an interesting problem to solve. It goes like this: Imagine a particle (point mass) on top of a sphere. If you perturb the particle from the very top, at what angle will it leave the sphere? Of course this is in a gravity field. Now, after writing the equations of motion, I came up with an angle of departure of theta = acos(2/3). Assuming the sphere is frictionless. However, when I assume some friction, mu, along the sphere things begin to become problematic. My equation of motion for the particle is listed in the attachment. It seems that this equation is only valid when the partilce is in motion. If you numerically integrate this equation for mu=0, the daparture angle matches exactly that of theta = acos(2/3). If you use a value of say mu=0.1, the motion plot doesn't look correct. The particle should leave at a greater angle with friction than without. I tried using an if-statement in my numerical integrator that says if the friction force is greater than the gravity force then theta_double_dot = 0, else it uses the equation listed in the attachment. This provided a greater angle than the frictionless case. In any sense, I was hoping to get someone's thoughts on this problem.

 

[Greg Bernhardt]

That is an interesting problem! It sounds like you have already done some good work on it, but are having some trouble with the friction component. I think it is a great idea to use an if-statement in your numerical integrator as you suggested. However, I would suggest also checking to make sure the equation of motion you are using is valid for both cases (friction and no-friction). Additionally, I think you should double check that the angles you are getting make sense and match with what you would expect from such a system (e.g. does the particle always leave at a greater angle when friction is present?). I hope this helps!