Moments caused by centripetal force?

[TomW17]

Homework Statement

I was thinking about different scenarios in circular motion and came to this scenario. Suppose there's a car moving in a circle around some track. Obviously it's the frictional forces between the tyres and the road which provide the centripetal force. Now, suppose there's a rigid rod that is fastened to the floor of the car (fastened at one end in a way that the rod is upright initially), and let it be fastened in a way such that there are no reaction moments between the rod and the floor of the car (e.g. a ball and socket fastening). Here, it is the reaction forces between the fastening and the rod which provide the centripetal force, but here's my question. Does the centripetal force acting at the end (the bottom) of the rod cause it to rotate about its centre of mass if the fastening between the rod and the floor provides no reaction moments?

Homework Equations

$$F_c = m\frac{v^2}{\rho}$$, $$M = F \times r_{\perp}$$

The Attempt at a Solution

I think it would cause it to rotate, but I'm not sure here. My reasoning being that the centripetal force always acts in a direction traverse to the axis of the rod, which would end up causing it to rotate about its COM, but I'm not too sure.

 

[mfb]

If I understand your setup correctly, it would just fall down towards the outside.