On the working of the centrifugal force I get that centrifugal force is what appears to force you out of a circular motion if you don't have the centripetal force required to stay in it. For example, a stone being whirled around on a string would go tangentially outward if let go due to its inertia and the fact that nothing's making it go in circular motion anymore.
But in the case of a frictionless object sliding radially outward on a frictionless rotating plate,
(1) Why is it radially outward and not tangential?
(2) Why does it move anyway; the surfaces in contact are frictionless; why doesn't it just stay there as the plate rotates, so that it doesn't move at all? Isn't that what inertia is?
(3) Generally, why does centrifugal force act outward on objects?
 A: We can analyse the path of the object in a non-rotating inertial reference frame or in a rotating non-inertial reference frame. Note that the physical path in space is the same in each case, but our point of view changes, so our description of the path changes.
As observed in a non-rotating inertial reference frame:

*

*When the centripetal force is present, this causes the object to move in a circle at constant speed. Its velocity vector is always tangential to the circle. When the centripetal force is removed there are no forces acting on the object so it continues to move tangentially in a straight line with constant velocity.

*The object continues to move because of inertia.

*There is no centrifugal force in an inertial reference frame.

As observed in a rotating non-inertial reference frame:

*

*When the centripetal force is present, this balances the centrifugal force. The net force on the object is zero so it remains at rest. When the centripetal force is removed there is an unbalanced centrifugal force acting on the object. This causes the object to initially move radially outwards. However, since the object is now no longer stationary its path is affected by Coriolis force, so it curves away from a radial path.

*The object starts to move because of the unbalanced centrifugal force. Its path is not a straight line because of Coriolis force.

*When the motion of the object in a non-rotating reference frame is converted to a rotating reference frame, a radially outward term is introduced because the basis vectors in the rotating reference frame are constantly changing their direction. This term is the centrifugal force.

