If `rho` is cross sectional density of the cylinder carpet, then in the initial position, its mass, `M = pi R^(2) rho`, and
in the final position, when radius reduces to `R//2`, its mass, `m = pi (R//2)^(2)rho = (pi R^(2) rho)/(4) = (M)/(4)`
Decrease in `P.E.` of carpet `= MgR - mgr = MgR - (M)/(4)g. (R )/(2) = (7)/(8)MgR` ..(i)
Increase in total kinetic energy of carpet `= K_(t) + K_(r ) = (1)/(2)mv^(2) + (1)/(2)I omega^(2)`
`= (1)/(2)mv^(2) + (1)/(2) ((1)/(2)mr^(2))omega^(2) = (1)/(2)mv^(2) + (1)/(4)mv^(2) = (3)/(4)mv^(2) = (3)/(4) ((M)/(v))v^(2) = (3)/(16) M v^(2)` ..(ii)
Assuming that there is no stray loss of energy, we get from (i) and (ii),
`(3)/(16)M v^(2) = (7)/(8)MgR`
`v= sqrt((14)/(3)gR)`