plz answer rotational motion doubt

Question

A block of mass M with a semicircular track of radius R rests on a horizontal frictionless surface. A uniform cylinder of radius r and mass m is released from rest from the top point A. The cylinder slips on the semicircular frictionless track. The distance traveled by the block when the cylinder reaches the point B is

Answer 1

Given that initially, the system is at rest so v

_cm = 0. Now in slipping as no horizontal force is applied, the horizontal component of velocity of centre of mass of the system must remain constant. So, if v and V are the velocities of cylinder and block at B relative to the ground

               (mv + MV)/(m+M) = const. = 0

       i.e. mv + MV = 0                 as m + M = finite                ...... (1)

       or     m (∆r₁)/∆t+(∆r₂)/∆t M=0             [as v = (∆r₁)/∆t]

      or mΔr₁ + MΔr₂ = 0             as Δr₁ = d

       or md₁ - Md₂ = 0                as d₂ is opposite to d₁           ....... (2)

Now, when the cylinder reaches from A and B, it is displaced by a distance (R - r) horizontally to the right relative to the block. So, if the displacement of the block relative to ground is d₂ to the left, the displacement of cylinder relative to ground to the right will be

       d₁ = (R-r) - d₂                                                    ......(3)

Substituting d₁ from Equation (3) in(2), we get

M[(R-r)-d₂]-md₂ = 0

i.e. d₂ = (m(R-r))/(M+m) (to the left)

Therefore, the distance traveled by the block when the cylinder reaches the point B is

d₂ = (m(R-r))/(M+m) (to the left)