How Does Angular Velocity Influence Particle Motion in Rotational Dynamics?

[ensten]

A particle of mass m is inside a long, narrow tube which rotates
a constant angular velocity ω in the horizontal plane. (This means that you see the tube
from the top of the figure and not from the side.) At time t = 0 is the particle on the radial distance a from the rotational axis and the radial velocity is zero. Then begins the slide without friction. Determine the particle radial distance from the rotation axis and the horizontal normal force of the particle from the tube functions
of the time!Why is equation (10) found in this solution https://dl.dropboxusercontent.com/u/12645136/Losningar.pdf equal to 0?

If the particle moves from the center outwards isn't a force acting upon it in the radial direction? Hence giving it an acceleration in the same direction?

 

[Orodruin]

I will write this in English since it is public, PM me if you would perfer an explanation in Swedish:

No, assuming you solve this in a fixed coordinate system, there is no force acting in the radial direction (there simply is nothing to provide such a force). The reason you will have acceleration in the radial direction is that you use curvilinear coordinates, resulting in that a velocity in the tangential direction is translated into a radial component as the particle moves. This is described by the $r\dot\theta^2$ term in the radial equation.

If you instead chose to use a rotating coordinate system, the very same term will be provided by the centrifugal effect and you end up with the same equation. The tangential force will now be balanced by the Coriolis effect.