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:

.