Finding the movement equation (non intertial system)

[velvetmist]

Homework Statement

A particle of a mass $m$ is embedded in a circular rail, (radius: $R$), 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 $\vec A$ 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)

Homework Equations

$\vec F' = m\vec a - m \vec A$

The 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.

 

[kuruman]

What does your equation for $\ddot{\varphi}$ look like? Where is $\varphi$ measured from? I assume that the mass is instantaneously at rest relative to the circle's reference frame. It may help your thinking to consider that in a non-inertial frame accelerating with constant acceleration, there is an "effective" acceleration of gravity $g_{eff.}$ and find its magnitude and direction.