A carpet of mass `M` is rolled along its length so as to from a cylinder of radius `R` and is kept on a rough floor. When a negligibly small push is given to the cylindrical carpet, it stars unrolling itself without sliding on the floor. Calculate horizontal velocity of cylindrical part of the carpet when its radius reduces to `R//2`.

Explanation

`Mass=densityxxvolume`
`M=rhopiR^(2)l`
`M'=rhopi((R)/(2))^(2)limpliesM'=(M)/(4)`

Applying the conservation of mechanical energy
`K_(A)+U_(A)=K_(B)+U_(B)`
`0+MgR=(1)/(2)(M)/(4)v^(2)(1+(k^(2))/((R//2)^(2)))+(M)/(4)(g)(R)/(2)`
`(7)/(8)MgR=(M)/(8)v^(2)(1+(1)/(2))`
`v=sqrt((14gR)/(3))`
Note: The factor `k^(2)//(radius)^(2)` depends on the shape. The factor of all soild cylinders is `1//2`, whatever be the mass or radius