Explanation:

Introduction:
The scenario described in the question involves an observer sitting on a horizontal platform that is rotating with a constant angular velocity. The observer places an object on the smooth and frictionless floor of the platform, away from the axis of rotation, with zero initial velocity with respect to the observer. In this explanation, we will analyze the motion of the object from the perspective of the platform frame of reference.

Inertia:
The object placed on the smooth floor of the platform possesses inertia, which means it resists changes in its state of motion. In the absence of any external forces, the object will tend to maintain its state of rest or uniform motion in a straight line.

Centripetal Force:
Since the platform is rotating with a constant angular velocity, the object experiences a centripetal force that acts towards the axis of rotation. This force arises due to the rotational motion of the platform and is responsible for causing the object to move in a circular path.

Centripetal Acceleration:
The centripetal force acting on the object causes it to accelerate towards the axis of rotation. This acceleration is known as the centripetal acceleration and is directed towards the center of the circular path. The magnitude of the centripetal acceleration can be calculated using the formula:

a = (v^2) / r

Where "v" is the velocity of the object and "r" is the distance between the object and the axis of rotation.

Circular Path:
As the object accelerates towards the axis of rotation, it follows a circular path on the smooth floor of the platform. The radius of this circular path is determined by the distance between the object and the axis of rotation. The object will continue to move in this circular path as long as the centripetal force is acting on it.

Inertia and Velocity:
Since the object possesses inertia, it will continue to move in a straight line tangent to the circular path. However, due to the centripetal force acting on it, the object's motion will be curved towards the axis of rotation. As a result, the object's velocity vector will continuously change direction, but its magnitude will remain constant.

Conclusion:
In the frame of the platform, the object placed on the smooth floor will move in a circular path with a constant velocity. The centripetal force provided by the rotation of the platform causes the object to accelerate towards the axis of rotation, resulting in its circular motion. It is important to note that this explanation is based on the assumption of an idealized scenario with no external forces or friction.