How to solve the physics problem in the picture?

Question

A uniform horizontal circular disc of mass M and radius R can freely rotate about a vertical axle passing through its centre. A particle of mass m is initially placed near the centre of the disc in a smooth groove made along the radius of the disc as shown. An initial angular velocity w₀ is imparted to the disc. Find the angular velocity of the disc, when the particle reaches the other end of the groove.

Answer 1

As external torque is zero just apply conservation of angular momentum.

I₁W₁ = I₂W₂.

So (MR²/2)w₀ = (MR²/2 + mR²)w.

So new w = Mw₀/(M+2m).