Movement Equation of a Particle in a Non Inertial System (polar coordinates)

Problem:
A particle of a mass [math]m[/math] is embedded in a circular rail, (radius: [math]R[/math]), without any friction. In a given moment, the particle finds itselfs without velocity at point C, and a force is applied on the rail, which starts moving with an [math] \vec A[/math] constant acceleration. Use a non-inertial system fixed to the rail to solve the problem.
Arrange the Newton's equations, and find the movement equation of the particle.

(It was originally in spanish that's why I only screenshoted the graph)
 

My attempt at a solution:
\( \vec F_v = bending force \)

So I manage to integrate \(\ddot \varphi \) so i get \( \dot \varphi (\varphi)\), but I guess i have to find \(\varphi (t)\) to get the movement equation, and I'm really stuck at this point.