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Here is the question 'A table with smooth horizontal surface is turning at an angular speed ω about its axis. A groove is made on the surface along the radius and a particle is gently placed at a distance 'a' from the center. Find the speed of the particle as its distance from the center becomes L.'

enter image description here I have attached my version of approach and the solution. What I did was to write v/x (where v is velocity of particle at x distance from center) instead of ω in the integral. I'm not able to find out what was wrong in it. Treating ω as a constant was of course a great idea but I should've got same results if my method was correct. Where am I missing it? My version

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This is a bit tricky, but you are mixing radial velocity with tangential velocity.

Specifically, in your fourth equation, $V$ on the right hand side is radial-velocity while $V$ on the left hand side is tangential velocity.

The book is focusing on radial velocity by adopting the rotating coordinate system.

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