What Additional Forces Might Influence the Motion of Particle P?

[eurekameh]

So I'm trying to write the equations of motion for the particle P. I've already figured the kinematics of the particle, tracking it with a position vector, deriving position to get velocity, and deriving velocity to get its inertial acceleration. For the forces acting on it, there is of course its weight mg acting downward, and also its two normal forces N1 and N2 both acting in the e,r direction, except they're both in opposite directions. Am I correct in thinking this, and are there more forces acting on the particle that I am not taking into account?

 

[Simon Bridge]

There's already an N1 labelled on the diagram.
The particle will have only one force from the sides of the tube unless it is a very tight fit.
Note: I'd be inclined to do this by Lagrangian mechanics.

 

[eurekameh]

Sorry for the confusion - the N1 labeled on the diagram is the name of one of the reference frames that I am using.
I'm not familiar with the method of Lagrangian mechanics, but I am using the Transport Theorem to relate one reference frame to another.
I wrote the inertial position vector as r,vector = r k,hat + r e,r,hat, where k,hat and e,r,hat are unit vectors.
I then derived these with the Transport Theorem to get inertial velocity and acceleration.
But I can see from the figure that the particle is very tightly fitted into the slot. In this case, would the two normal forces be equal and opposite in direction so that they both cancel?

 

[Simon Bridge]

In a tight fit - the normal forces would act like pressure ... one will be bigger than the other, which depends on position. It is probably easier to just deal with the unbalanced normal force.

Another one I'd try would be a rotating spherical coordinate system - which I suspect you are using. The object would experience centrifugal and coriolis forces as well as contact forces with the sides (and weight) - don't forget the tube the object sits in is 3D - there should be $\hat{e}_\phi$ forces too (if it's rotating at a constant speed, then they should cancel).

It is important to keep your notation clear.
It is common to use S to label reference frames - but try to pick something you don't use for anything else.