No-circular motion on a turntable

Question

As the title says, I want to model the path of an object sliding on the surface of a turntable, as it is slowly flung off.

The final application of this is , modelling fine material moving along the grinding table of our vertical raw mill.

I'm thinking steady state for right now, so the particle just appears at some inner radius, with the same tangetial velocity as the table at that radius, but the max static friction in the radial direction is not enough to maintain a circular path so it begins to move.

Conditions :

• I would prefer this in a non-inertial reference frame. (i.e. I'm sitting to the side of the thing an watching this unfold) No drag from surrounding fluid atmosphere.

• Objects is microscopic compared to the radius of the table, and we are only focusing on a single particle.

• Object has mass and is a perfect sphere. I dont care about rolling vs sliding, so this shape can be changed for ease of calculation.

• There is friction between the turntable and the particle.

• The turntable is moving at a constant angular velocity and completely horizontal (after this, the next step is to include the peaks and valleys of the actual table topography).

Calculations :

We'll start with my crude diagrams :

Image (1) :

Image (2) :

Image (3) :

The first assumption for my approach is, at the moment $(t_1)$ the radial static friction shifts to kinetic friction, the object moves in the direction of the tangetial velocity of the table at $(r_1)$. Thus it experiences friction against the tangetial velocity, and moves to a new radii $(r_1 + dr =r_2)$ at a new angle $( \theta {_1} +d\theta = \theta_2)$.

At this new point, the table's tangential velocity is greater than the particle's so it is accelerated, and such and so forth.

My problem is trying to describe this mathematically describe the forces, it is keep confusing myself.

(a) Is the assumption that movement is in only the current tangential direction correct?

(b) Is the acceleration from the difference between the table and the object velocities, different from the angular friction of the movement in the first place?

(c) Should the sum of the accelerations at $r_1$ (or $t_1$ ) sum to the Centripedal acceleration at $r_2$ $(t_2)$?

So here are my sum of forces :

1. $F_z$ (Image (2), verical $z -$direction):

$m×a_z = F_n-F_g =0$ so , $F_n=m×g$

2. $F_o$ (Image (3), $\theta$ direction) :

$m×a_{\theta}×r=-u×F_n$ and $a_{\theta}×r =(v_2-v_1)/dt$

Or $m×a_ {\theta} =-u×F_n + m×(v_2-v_1)/dt$ ?

3. $F_r$ (Images (2) and (3) , radial direction) :

$m×a_r = -u×FN$ and $a_r = v^2/r$

Or something else ? I'm quite not sure how to account for the radial acceleration changing with poistion

4. Support equations :

   • $\rm a_{tangent} = a_o×r$
   • $\rm v_{tangent} = r× \omega$
   • $\rm a_{centripedal} = v_{tangent}^2/r = r× \omega^2$