How to use Centrifugal force in a rotating frame?

Question

Assume a stationary object (point object) on a smooth very large disc at a distance $r$ from its axis perpendicular and passing through the centre. (No external forces apart from gravity and Normal which are equally balanced). Now the disc is rotated with a constant angular velocity.

Now when the object is observed in an inertial frame O, The object remains stationary. But when observed in a non-inertial frame attached to the centre of the disc and rotating along with the disc, there exists a centrifugal force (pseudo force) pointing away from the centre and no force to counter it and hence must be moving away right? Yet this seems to be counter-intuitive to the idea that the object must be rotating in the opposite direction with the same magnitude. The above observation of the rotating frame is also against the observation of the object in the inertial frame where the distance from the centre remains constant.

Where am I wrong?

Answer 1

In the reference frame that rotates with the disc, centrifugal force is not the only pseudo-force acting on the object. There is also the Coriolis force which has magnitude $2m\omega v$ (where $v$ is the speed of the object in the rotating frame).

If the disc is rotating with constant angular speed $\omega$ and the object is stationary in the inertial reference frame then the object has speed $v=r \omega$ in the rotating reference frame, and its velocity vector is in the opposite direction to the rotation of the disc. In this case the Coriolis force has magnitude $2mr\omega^2$ and acts inwards, towards the centre of rotation.

So the horizontal pseudo-forces acting on the object in the rotating reference frame are centrifugal force $mr\omega^2$ radially outwards and Coriolis force $2m r \omega^2$ radially inwards. The net horizontal pseudo-force on the object is $m r \omega^2$ radially inwards, which makes the object travel in a circular path (relative to the rotating reference frame) with constant radius $r$ at constant angular speed $\omega$ in the opposite sense the the disc's direction of rotation. This is what we expect for an object that is stationary in the inertial reference frame.