Hmm I don't think this is a very well defined problem. Could you be more specific or elaborate? Is there a diagram you could upload, perhaps?
A circular table with smooth horizontal surface is rotating at an angular speed ω about its axis. A groove is made on the surface along a radius and a small particle is gently placed inside the groove at a distance l from the centre. Find the speed of the particle with respect to the table as its distance from the centre becomes L.
The table is rotating.
In the co-rotating reference frame of the table top the particle is constrained to move on a fixed radius.
So taking for the co-rotating reference frame a coordinate system with origin at the centre of rotation and x axis oriented along the radius the particle moves under a psuedo-force:
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Unless im misinterpreting this, this is a relative velocity problem.
Let be the velocity of the table relative to Earth. Let be the velocity of the particle relative to Earth. And let be the velocity of the particle with repsect to the table.
The tangential speed of the table is given by: where is the radius of the table.
Since every point on a rotating object has the same angular speed, the tangential speed of the particle at radius L is given by: .
Hence the speed of the particle with repsect to the table is:
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I would check your notation, there appears to be some confusion about what denotes, is this the velocity of the table centre of mass with respect to the Earth, or the table surface at the point currently occupied by the particle, or some point on the rim of the table.
(and some confusion between vectors and scalars, or at least what kind of vector product is implied by juxtaposition)
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